Semicondutores - UFPRfisica.ufpr.br/kleber/semicondutores.pdf · Ao aplicar um campo elétrico sobre os buracos, ... A massa do elétron é substituída pela massa ... Há um núcleo

SemicondutoresCF086 - Introduccedilatildeo a Fiacutesica do Estado Soacutelido 1

Introduccedilatildeo

Vimos que um dado soacutelido pode apresentar bandas de energia que podem estar totalmente vazias totalmente preenchidas ou parcialmente preenchidas

2


Isolante em 119879 = 0 bandas vazias ou totalmente preenchidas A uacuteltima banda preenchida num isolante estaacute separada da banda vazia imediatamente superior por um gap

Num isolante o gap eacute alto (simgt 3 eV) e a resistividade eleacutetrica tambeacutem (120602 sim 1020 Ω sdot m)

3


Para gaps da ordem de alguns deacutecimos de eV ateacute sim 3 eV temos semicondutores As resistividades satildeo menores (120602 sim 10minus5 minus 107 Ω sdotm) (para metais 120602 sim 10minus8Ω sdot m)

4


Por causa do gap menor quando 119879 ne 0 haacute uma probabilidade de ocupaccedilatildeo dos estados maior de modo que os eleacutetrons que ldquoatravessamrdquo o gap eacute um pouco maior rArr material conduz para 119879 ne 0

5


Outra grandeza importante eacute a densidade de portadores

Vamos comparar os comportamentos para isolantes e condutores

6


Densidade de estados para eleacutetrons livres (3D)

119967(120598) fornece o nuacutemero de estados mas natildeo diz se ele estaacute ocupado

Distribuiccedilatildeo de Fermi-Dirac

7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583


Nuacutemero meacutedio de eleacutetrons 119967 120598 119891(120598)

8

Em metais 119879 = 0 banda

semipreenchida

Os niacuteveis satildeo preenchidos ateacute uma

energia maacutexima energia de Fermi

119864119865


9

Quando 119879 ne 0 119967(120598) eacute o mesmo mas

119891(120598) muda em torno de 119864119865 = 120583(119879 =0) na faixa plusmn2119896119861119879 em torno de 119864119865

Os eleacutetrons excitados termicamente

datildeo origem agrave corrente eleacutetrica

Para metais ocorre excitaccedilatildeo em

qualquer 119879 Estimativa do nuacutemero

relativo de eleacutetrons Δ119873119879119873 sim 1(119879 = 300 K quadro)

Semicondutores

A ideia eacute similar para semicondutores e isolantes As bandas satildeo separadas por um gap

Uacuteltima banda preenchida banda de valecircncia (BV)

Banda vazia superior agrave BV banda de conduccedilatildeo (BC)

Diferenccedila entre o topo da BV e a base da BC gap 119864119892

10

Semicondutores

Potencial quiacutemico 120583(119879) ou niacutevel de Fermi proacuteximo ao meio do gap varia pouco com 119879 (veremos depois)

11

120583(119879) regula o fluxo de

partiacuteculas (Termodinacircmica)

119967 120598 = 0 no gap Natildeo eacute dada

apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892

Natildeo haacute eleacutetrons de conduccedilatildeo

Para 119879 gt 0 119891(120598) permite alguns

eleacutetrons na BC

Estimativa (119879 = 300 K quadro)

Δ119873119873 sim 10minus52 119864119892 = 6 eV Δ119873119873 sim

10minus9 119864119892 = 1 eV

Semicondutores

Note que quando 119879 cresce o nuacutemero de eleacutetrons excitados termicamente cresce e a condutividade aumenta

Esse comportamento eacute oposto ao exibido por metais onde 120602 cresce com 119879 (e a condutividade decresce) Em metais

e 120591119890119897 diminui com 119879 pois haacute mais colisotildees

Semicondutores tecircm coeficientes de temperatura para a resistividade negativos rArr propriedade que os fez se destacarem no seacutec XIX

13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores

A excitaccedilatildeo teacutermica pode fazer com que eleacutetrons passem da BV para a BC Essa eacute a origem da condutividade intriacutenseca do semicondutor

Semicondutor intriacutenseco condutividade eacute dominada por efeitos teacutermicos

14

Semicondutores

Ao excitar um eleacutetron da BV para a BC a BV fica com a ausecircncia de um eleacutetron

A BV estava totalmente preenchida Assim o momento total dela era nulo Natildeo haacute movimento ordenado de cargas ao aplicar um campo eleacutetrico fraco (119864campo ≪ 119896119861119879)

Com um eleacutetron a menos o momento total deixa de ser nulo e torna-se possiacutevel alterar estados eletrocircnicos na BV

15

Semicondutores

A ausecircncia de um eleacutetron se comporta como uma carga positiva chamada ldquoburacordquo

O buraco tem carga oposta agrave do eleacutetron

Eleacutetrons no topo da BV tecircm massas efetivas negativas pois a BV eacute cocircncava para baixo

Ausecircncia dos eleacutetrons (buracos) tecircm massas efetivas positivas

16

Semicondutores

Ao aplicar um campo eleacutetrico sobre os buracos eles se comportam de maneira ldquonormalrdquo rArr movem-se no mesmo sentido que o campo

Haacute outros modos de alterar a condutividade aleacutem do teacutermico

Incidindo radiaccedilatildeo com energia da ordem do gap ou maior os eleacutetrons da BV podem ser fotoexcitados rArr fotocondutividade

Semicondutores podem ser usados como fotodetectores

17

Semicondutores

Haacute outro modo de alterar a condutividade rArr impurezas

Suponha que uma pequena quantidade de arsecircnio (As) seja introduzida na rede de germacircnio (Ge)

18

Semicondutores

Ge tetravalente As pentavalente

Ao trocar Ge rArrAs o As faz 4 ligaccedilotildees com o Ge e um eleacutetron fica ldquoflutuandordquo em torno do As

O As se comporta como um nuacutecleo com uma carga positiva +119890 envolto por um eleacutetron ldquoorbitandordquo esse nuacutecleo rArr aacutetomo de hidrogecircnio com niacuteveis discretos dentro do gap

Haacute algumas diferenccedilas

19

Semicondutores

O meio blinda a forccedila eleacutetrica rArr permissividade eleacutetrica 120598 e natildeo 1205980 Permissividade eleacutetrica relativa 120598119903

A massa do eleacutetron eacute substituiacuteda pela massa efetiva 119898lowast

Estimativa 1198641119889 sim 001 eV para 119898lowast = 02119898 120598119903 = 16

20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores

O pequeno valor da energia dos niacuteveis doadores faz com que seja faacutecil excitar termicamente os portadores para a BC

Os niacuteveis doadores ficam logo abaixo da BC no niacutevel 119864119889 abaixo de 119864119888 As eacute uma impureza doadora

21

Semicondutores

Considere a substituiccedilatildeo de um Ge por um gaacutelio (Ga)

Ga trivalente

A ideia eacute similar mas agora temos falta de um eleacutetron e natildeo excesso

Haacute um nuacutecleo de carga minus119890 e um buraco orbitando esse nuacutecleo

A situaccedilatildeo eacute simeacutetrica mas os niacuteveis satildeo proacuteximos agrave BV

22

Semicondutores

Os niacuteveis satildeo niacuteveis de buracos Assim eleacutetrons podem passar da BV para estes niacuteveis aceitadores (de eleacutetrons) por excitaccedilatildeo teacutermica

Surge um niacutevel aceitador 119864119886 logo acima de 119864119907 da BV

23

Semicondutores

Quando eleacutetrons passam da BV para um niacutevel aceitador 119864119886 geram buracos na BV que podem conduzir

Ga impureza aceitadora

Semicondutor tipo 119899 (negativo) niacuteveis doadores

Semicondutor tipo 119901 (positivo) niacuteveis aceitadores

24

Semicondutores

Representaccedilatildeo graacutefica

Semicondutor extriacutenseco semicondutor dopado

25

Semicondutores

Num semicondutor intriacutenseco (natildeo dopado) o nuacutemero de estados vazios na BV eacute igual ao nuacutemero de eleacutetrons na BC

120583 (niacutevel de Fermi) fica proacuteximo ao centro do gap

Bandas simeacutetricas

26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores

Semicondutor extriacutenseco situaccedilatildeo mais complicada

Semicondutor tipo 119899 em 119879 = 0 120583 fica entre 119864119889 e 119864119888 Quando 119879cresce eleacutetrons passam de 119864119889 para 119864119888 e 120583 cai

Quando metade dos eleacutetrons passa de 119864119889 para 119864119888 120583 = 119864119889

Aumentando 119879 eleacutetrons da BV passam para a BC e 120583 cai ainda mais indo em direccedilatildeo a 119864119907 + 1198641198922

27

Semicondutores

Semicondutor tipo 119901 em 119879 = 0 120583 fica entre 119864119907 e 119864119886 Quando 119879cresce buracos passam de 119864119889 para 119864119907 e 120583 cresce indo em direccedilatildeo a 119864119907 + 1198641198922 de forma similar ao que ocorre no tipo 119899

28

Semicondutores

O proacuteprio gap varia com 119879

As dimensotildees da rede se alteram e com isso as bandas e os intervalos entre elas tambeacutem mudam

A distribuiccedilatildeo de focircnons varia com 119879 e isso influi nas bandas

29

Semicondutores

Haacute dois tipos de gaps

Gap direto miacutenimo da BC e maacuteximo da BV ocorrem para o mesmo valor de

119896 Assim eles estatildeo verticalmente alinhados num graacutefico ℰ times 119896

30

Note que o momento transferido

pelo foacuteton eacute muito pequeno quando

comparado com o do eleacutetron

Os vetores 119896 para o eleacutetron antes e

depois da transiccedilatildeo satildeo iguais

Semicondutores

Gap indireto miacutenimo da BC e maacuteximo da BV ocorrem para valores diferentes

de 119896

Para haver conservaccedilatildeo de momento um focircnon deve participar da transiccedilatildeo

O focircnon pode ser absorvido + ou criado minus

31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores

Os niacuteveis de energia importantes em transiccedilotildees eletrocircnicas satildeo os que ficam proacuteximos ao topo da BV e na base da BC

De forma geneacuterica estas bandas pode ser escritas como

32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 eleacutetrons

ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores

ℰ119888 base da BC ℰ119907 topo da BV

෩119872 tensor de massa efetiva Na forma diagonal temos

As bandas satildeo elipsoides de energia constante (em 119896)

33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 eleacutetrons

ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores

Si 6 bandas de conduccedilatildeo em lang1 0 0rang

Bandas de valecircncia degeneradas em 119896 = 0

34

Semicondutores

Ge

35

Propriedades dos Buracos

Buracos carga oposta agrave do eleacutetron

Vetor de onda do eleacutetron 119896119890 Vetor de onda do buraco (hole)

Niacutevel de energia zero no topo da BV Eleacutetrons nela tecircm energias negativas Assim a energia do buraco eacute

36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890


Velocidade do buraco

Massa efetiva

Equaccedilatildeo de movimento sob campo eletromagneacutetico

37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ


Densidade de corrente

eleacutetrons na BC Ԧ119869119890 buracos na BV Ԧ119869ℎ as duas orientam-se no mesmo sentido

Notar que

Eacute importante poder determinar 119899 rArr efeito Hall

38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall

Na presenccedila de um campo magneacutetico temos

ou reescrevendo

ി119903 tensor resistividade eleacutetrica

39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall

Para determinar 119903119894119895 usa-se a configuraccedilatildeo abaixo chamada

configuraccedilatildeo padratildeo

40

Efeito Hall

Na situaccedilatildeo estacionaacuteria 119869119910 = 0 Assim

Desenvolvendo

Magnetoresistividade longitudinal

41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall

Magnetoresistividade transversal (resistividade Hall)

Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1

um tipo de portador (eleacutetrons) banda isotroacutepica densidade numeacuterica 119899 e massa efetiva 119898lowast

meio dissipativo modelado por um tempo de relaxaccedilatildeo 120591

Equaccedilatildeo de movimento

43

119898lowast119889 Ԧ119907

119889119905= minus119890119864 minus 119890 Ԧ119907 times ℬ minus

119898lowast

120591Ԧ119907

Efeito Hall

Situaccedilatildeo estacionaacuteria 119889119907

119889119905= 0

Frequecircncia ciacuteclotron

44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591

119898lowast Ԧ119907 times ℬ

Efeito Hall

Desenvolvendo chega-se a (quadro)

Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall


Magnetoresistividade transversal

Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall

Modelo 2 dois portadores

eleacutetrons massa efetiva 1198981 = 1198981lowast densidade numeacuterica 119899 tempo de relaxaccedilatildeo

1205911 frequecircncia ciacuteclotron 1205961

buracos massa efetiva 1198982 = 1198982lowast densidade numeacuterica 119901 tempo de relaxaccedilatildeo


47

Efeito Hall

Magnetocondutividade soma das duas contribuiccedilotildees

onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e

Note que 119903119909119909 ℬ gt 119903119909119909(0) para qualquer ℬ ne 0

49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos

Nesse caso 119903119909119909 rarr infin se ℬ rarr infin Se 119899 ne 119901 119903119909119909 satura quando ℬ rarr infin

50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall

Para a magnetoresistividade transversal temos

Quando ℬ rarr infin temos (verificar)

e o coeficiente Hall fica

51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890

Efeito Hall Quacircntico

Em condutores bidimensionais em baixas temperaturas e campos magneacuteticos intensos ocorre o efeito Hall quacircntico

Curva da resistividade Hall (119903119910119909) em funccedilatildeo de ℬ exibe platocircs que satildeo

muacuteltiplos de

Correspondentemente 119903119909119909 = 0 nesses platocircs

52

ℎ

1198902= 25812806 Ω


Os valores de 119903119910119909 satildeo dados por

ℓ isin ℤ efeito Hall quacircntico usual

ℓ =119875

119876 onde 119875 119876 isin ℤ 119876 iacutempar

efeito Hall quacircntico fracionaacuterio

53

119903119910119909 =ℎ

11989021

ℓ

Portadores em funccedilatildeo de 119931

Queremos o nuacutemero de portadores por volume em funccedilatildeo de 119879 Impurezas influenciam nos valores mas eacute possiacutevel obter resultados gerais que natildeo dependem disso

Na BC temos 119899119888 eleacutetrons por volume e densidade de estados por

volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598


Na BV temos 119901119907 buracos por volume e densidade de estados por

volume 119892119907 120598 =119967119907

119881 Entatildeo

120583 eacute influenciado pelas impurezas Para conhecer 120583 eacute preciso ter informaccedilotildees sobre elas

55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer

eacute possiacutevel obter resultados importantes sem conhecer 120583 precisamente

Se essa condiccedilatildeo eacute vaacutelida temos um semicondutor natildeo degenerado Se natildeo eacute valida entatildeo o semicondutor eacute degenerado e natildeo eacute possiacutevel usar os resultados abaixo

56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879


Considerando a condiccedilatildeo de natildeo degenerescecircncia temos (quadro)

Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo

De forma similar temos

e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879


As energias envolvidas para os eleacutetrons e buracos satildeo da ordem de 119896119861119879 Com isso eacute possiacutevel escrever

e considerando as bandas com forma paraboacutelica em torno da base da BC e do topo da BV temos para a BC

59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888


Com isso achamos (quadro)

e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)


A partir disso obtemos

que eacute a lei de accedilatildeo de massas

Para semicondutor intriacutenseco

62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo

Aleacutem disso achamos tambeacutem (quadro)

63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64


Semicondutor extriacutenseco

Lei de accedilatildeo de massas eacute vaacutelida e escrevemos

Desenvolvendo chegamos a (quadro)

65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem

que indica qual a importacircncia das impurezas para o nuacutemero de portadores

Note que

Δ119899119899119894 soacute eacute apreciaacutevel quando 120583 natildeo eacute comparaacutevel a 120583119894

66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879


Semicondutor natildeo degenerado 120583 sim 119874(120583119894) e Δ119899

119899119894≪ 1 rArr niacuteveis de

impurezas natildeo satildeo importantes

Nesse caso

A concentraccedilatildeo do portador majoritaacuterio eacute Δ119899

119899119894

2vezes maior que a do

outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2


Se Δ119899 gt 0 119899119888 ≫ 119901119907 e temos um semicondutor tipo 119899 (excesso de eleacutetrons)

Se Δ119899 lt 0 119901119907 ≫ 119899119888 e o semicondutor eacute tipo 119901 (excesso de buracos)

68


Vamos agora estimar a influecircncia de 119879 nos niacuteveis das impurezas

Os niacuteveis das impurezas doadoras ficam em 120598119889 logo abaixo de 120598119888 Os niacuteveis aceitadores ficam em 120598119886 logo acima de 120598119907

Haacute 119873119889 impurezas doadoras por volume e 119873119886 impurezas aceitadoras por volume

Portadores em niacuteveis de impurezas natildeo interagem (hipoacutetese)

69


Cada niacutevel doador pode estar vazio ter um eleacutetron ou dois eleacutetrons Essa uacuteltima configuraccedilatildeo tem energia muito alta e eacute pouco provaacutevel

Nuacutemero meacutedio de ocupaccedilatildeo de um niacutevel qualquer (Termodinacircmica grande-canocircnico)

70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)


Para um dado niacutevel doador temos 119873119895 = 0 ou 119873119895 = 1 (uarr ou darr)

Assim o nuacutemero meacutedio de eleacutetrons nos niacuteveis doadores eacute

71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)


Para os niacuteveis aceitadores a ideia eacute similar trocando-se eleacutetrons por buracos Assim

Queremos generalizar a condiccedilatildeo 119899119888 = 119901119907 vaacutelida para equiliacutebrio teacutermico em semicondutores intriacutensecos

72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)


Conservaccedilatildeo de carga

Condiccedilatildeo para semicondutor natildeo degenerado

73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879


Se esta condiccedilatildeo eacute verificada ocorre

Assim praticamente todos os niacuteveis de impurezas estatildeo ionizados (vazios ndash doadores com eleacutetrons ndash aceitadores)

Conservaccedilatildeo de carga fica

74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)


Regime intriacutenseco 119899119894 ≫ |119873119889 minus 119873119886|

Regime extriacutenseco 119899119894 ≪ |119873119889 minus 119873119886|

76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942

119873119889 minus 1198731198862119873119889 minus119873119886 plusmn

1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889


Em baixas temperaturas ou para altas concentraccedilotildees de impurezas

uma das fraccedilotildees 119899119889

119873119889ou

119901119886

119873119886(natildeo ambas) pode deixar de ser despreziacutevel

rArr niacutevel natildeo estaacute totalmente ionizado

A densidade de portadores dominante diminui com 119879

78


Outro efeito em 119879 baixa eacute a possibilidade de ocorrer tunelamento entre os niacuteveis de impurezas por causa da superposiccedilatildeo das funccedilotildees de onda rArr hopping

79


80

Ge dopado com Sb

Junccedilatildeo 119953119951

Vamos agora investigar um dispositivo formado por semicondutores junccedilatildeo pn

81


Hipoacuteteses

1 Semicondutor tipo n apenas niacuteveis doadores tipo p apenas niacuteveis aceitadores

2 1D (direccedilatildeo x)

3 Junccedilatildeo ocorre em 119909 = 0 Regiatildeo de depleccedilatildeo minus119889119901 lt 119909 lt 119889119899

4 Impurezas tipo n densidade 119873119886(119909) Tipo p densidade 119873119889(119909)

5 Densidades dadas por

6 Impurezas ionizadas rArr saturaccedilatildeo longe da junccedilatildeo

82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0


Considere os semicondutores separados

Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886

Diferenccedila 119890Δ120601

83


Colocando os semicondutores em contato

84


A diferenccedila no potencial quiacutemico gera fluxo de eleacutetrons do lado 119899para o 119901

85

O lado 119901 fica negativo

na regiatildeo da junccedilatildeo O

lado 119899 fica positivo

Surge campo eleacutetrico na

junccedilatildeo

BV e BC na regiatildeo 119901 satildeo

mais altos que na regiatildeo

119899 Diferenccedila 119890Δ120601


O campo eleacutetrico orienta-se de 119899 rarr 119901 O potencial eleacutetrico correspondente aumenta de 119901 rarr 119899

Eles aparecem numa regiatildeo chamada de regiatildeo de depleccedilatildeo Fora dela o potencial eacute constante e o campo eacute nulo

Em geral haacute circulaccedilatildeo de cargas nos dois sentidos

86


Num dado instante algum eleacutetron da BV na regiatildeo 119901 eacute excitado termicamente agrave BC da regiatildeo 119901

Posteriormente ele pode seguir para a BC da regiatildeo 119899 Isso daacute origem agrave corrente teacutermica

Noutro instante algum eleacutetron num niacutevel da BC do lado 119899 abaixo da BC do lado 119901 pode sofrer flutuaccedilatildeo em energia e atingir energia compatiacutevel com a BC-119901 passando para ela Essa eacute a corrente de recombinaccedilatildeo

87


No equiliacutebrio teacutermico sem potencial externo a corrente total eacute nula

A aplicaccedilatildeo de uma tensatildeo externa modifica o comportamento da corrente de recombinaccedilatildeo sem alterar a corrente teacutermica

88


O potencial eleacutetrico altera o hamiltoniano do sistema da seguinte forma

Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909

119899119888 119909 = 119873119888 119879 119890minus120573(120598119888minus119890120601 119909 minus120583)

119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909


Saturaccedilatildeo (longe da junccedilatildeo)

Graficamente

90

119873119889 = 119899119888 infin = 119873119888 119879 119890minus120573(120598119888minus119890120601 infin minus120583)

119873119886 = 119901119907 minusinfin = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 minusinfin


Definindo

temos a condiccedilatildeo (quadro)

que eacute uma condiccedilatildeo de contorno para o problema

91

Δ120601 = 120601 infin minus 120601(minusinfin)

119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907


Para achar 120601 119909 eacute preciso trabalhar com a equaccedilatildeo de Poisson (em 1D)

Na saturaccedilatildeo

Em princiacutepio combinar estas equaccedilotildees resulta numa equaccedilatildeo diferencial soluacutevel numericamente

92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598

120602 119909 = minus119890[119899119888 119909 minus 119901119907 119909 + 119873119886 119909 minus 119873119889(119909)


Estimativa mais uacutetil (quadro) potencial soacute varia na regiatildeo de depleccedilatildeo minus119889119901 le 119909 le 119889119899 A densidade fica

Para 119909 lt minus119889119901 e 119909 gt 119889119899 o campo eacute nulo e o potencial eacute constante

93

120602 119909 = ൞

0 119909ltminus119889119901minus119890119873119886 minus119889119901lt119909lt0

119890119873119889 0lt119909lt1198891198990 119909gt119889119899


Apoacutes resolver a equaccedilatildeo de Poisson o potencial fica

94

120601 119909 = 120601 minusinfin 119909 lt minus119889119901

120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95


A continuidade do campo e do potencial em 119909 = 0 fornece

e

Com isso eacute possiacutevel determinar a largura da regiatildeo de depleccedilatildeo

96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos

e para 119889119899 ficamos com

97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total

Ex para Δ120601 sim 1 V 120598 = 10minus10 Fm e 119873119886 119873119889 na faixa 1014 a 1018

portadorescm3 temos 119908 sim 102 minus 104 Å e campos da ordem de 105 minus 107 Vm

98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12


Na junccedilatildeo pn usual 119873119886 sim 119873119889 e ambos natildeo satildeo muito grandes

Eacute interessante considerar o caso onde 119873119886 119873119889 ou ambos satildeo grandes Temos as junccedilotildees

p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886

p+n+ ambos satildeo grandes

99

Junccedilatildeo homopolar

Podemos combinar portadores de mesmo tipo formando junccedilotildees homopolares onde apenas um tipo de portador eacute relevante

p+p acuacutemulo de buracos no lado p

n+n acuacutemulo de eleacutetrons no lado n

Elas facilitam a conduccedilatildeo ao contraacuterio da junccedilatildeo pn

100

Junccedilatildeo sob Tensatildeo Externa

Vamos agora considerar o efeito da aplicaccedilatildeo de uma tensatildeo externa agrave junccedilatildeo

Ocorre deslocamento do equiliacutebrio sob tensatildeo externa

Recordando na junccedilatildeo eleacutetrons passam naturalmente do lado 119899 para o 119901

Considere uma fonte de tensatildeo contiacutenua com terminais (+) e (-)

101


Ao conectar o terminal (+) no lado 119899 e o (-) no 119901 o campo eleacutetrico na regiatildeo da junccedilatildeo fica mais intenso

A altura da barreira de potencial aumenta

O efeito eacute dificultar a passagem de eleacutetrons no sentido 119899 rarr 119901

Com isso a corrente de recombinaccedilatildeo diminui

102


A corrente teacutermica natildeo eacute alterada pois depende apenas de 119879

Assim eleacutetrons vatildeo de 119901 rarr 119899 e corrente flui de 119899 rarr 119901

Essa eacute a corrente de polarizaccedilatildeo reversa e eacute pequena

103


Conectando o terminal (+) ao lado p e o (-) ao n a situaccedilatildeo se inverte

A altura de barreira diminui o campo eleacutetrico fica menos intenso e a corrente de recombinaccedilatildeo cresce muito

Haacute corrente eleacutetrica no sentido 119901 rarr 119899 Esta eacute a corrente de polarizaccedilatildeo direta que eacute 4-5 ordens de grandeza maior que a reversa (tipicamente)

104


Vamos considerar que 119881 gt 0 corresponde agrave polarizaccedilatildeo direta e 119881 lt0 agrave reversa

105


Diferenccedila de potencial na junccedilatildeo sob tensatildeo externa

Δ1206010 ddp para 119881 = 0 (situaccedilatildeo de equiliacutebrio)

A aplicaccedilatildeo da tensatildeo externa altera os limites da regiatildeo de depleccedilatildeo

106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total

Interessa-nos investigar as correntes na regiatildeo da junccedilatildeo pn

Convenccedilatildeo

119895 densidade de corrente eleacutetrica

119973 densidade de corrente numeacuterica

107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12


Relaccedilatildeo entre as densidades

Quando 119881 = 0 119973119890 = 119973ℎ = 0 rArr compensaccedilatildeo entre corrente teacutermica e de recombinaccedilatildeo

Para eleacutetrons

119973119890term densidade numeacuterica de corrente teacutermica em 119881 = 0

108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos


Densidade resultante (vetorial)

Como determinar os coeficientes Outra abordagem

109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1


Correntes podem ser geradas por campos eleacutetricos ou por gradientes de concentraccedilatildeo de portadores (correntes de difusatildeo) Em 1D

Estas equaccedilotildees combinam as relaccedilotildees

110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ


120583119890 e 120583ℎ mobilidade de eleacutetrons e buracos (120583119894 gt 0)

A mobilidade eacute dada por

De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)


119863119901 e 119863119899 coeficientes de difusatildeo para buracos e eleacutetrons

Satildeo grandezas positivas e relacionadas com 120583119894 pelas relaccedilotildees de Einstein

112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879


A condutividade pode ser modelada por

Com isso a mobilidade pode ser escrita como

120591119894col tempo meacutedio de colisotildees para o portador i

119898119894 massa efetiva do portador i

113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ


Equaccedilatildeo de continuidade

Escrevendo em termos das densidades numeacutericas temos

Entretanto estas equaccedilotildees natildeo consideram a transferecircncia de cargas entre as bandas

114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909


Levando em conta as transferecircncias de cargas obtemos

Iacutendice g-r geraccedilatildeo ndash recombinaccedilatildeo Modelo para essas taxas

1198991198940 valores de equiliacutebrio

120591119894 tempo meacutedio de recombinaccedilatildeo

115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901


Note que em geral 120591119894 ≫ 120591119894col pois as colisotildees satildeo intrabandas e as

recombinaccedilotildees satildeo interbandas

Tipicamente 120591119894col sim 10minus12 - 10minus13 s e 120591119894 sim 10minus3 - 10minus8 s

Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909


Situaccedilatildeo estacionaacuteria para 119881 ne 0

Caso particular Ԧℰ pequeno e concentraccedilatildeo de portadores majoritaacuterios constante

117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0


Comprimentos de difusatildeo distacircncias caracteriacutesticas em que as concentraccedilotildees voltam aos valores de equiliacutebrio

Soluccedilatildeo considerando que estamos do lado 119899 da junccedilatildeo

118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901

119901119907 = 119901119907 infin + 119901119907 1199090 minus 119901119907 infin 119890minus(119909minus1199090)119871119901


Estimativa para as densidades numeacutericas de corrente

A densidade de corrente fica

Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892


Corrente de saturaccedilatildeo 119881 rarr minusinfin

120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901


Se 119881 gt 0 rarr polarizaccedilatildeo direta corrente apreciaacutevel

Se 119881 lt 0 rarr polarizaccedilatildeo reversa corrente baixa

121

retificador

Transistor

119901+ altamente dopada emissor

119899 fracamente dopada e fina base

119901 moderadamente dopada coletor

122

119881119864 gt 0

polarizaccedilatildeo

direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor

Praticamente toda a corrente que entra no emissor segue para o coletor 119868119862 sim 119868119864

Como 119881119862 ≫ 119881119864 temos um amplificador de potecircncia

123


Vimos que um dado soacutelido pode apresentar bandas de energia que podem estar totalmente vazias totalmente preenchidas ou parcialmente preenchidas

2




3



4



5




6





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




3



4



5




6





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



4



5




6





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



5




6





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




6





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





7

119967 120598 =2119898

ℏ2

32 119881

4120587212059812

119891 120598 =1

119890120573 120598minus120583 + 1119891 120598 119879 = 0 = ቊ

1 120598 lt 1205830 120598 gt 120583



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



8


semipreenchida



119864119865


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


9








Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





10

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores


11




apenas por 119967 120598 prop 1205981

2

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores

12

Para 119879 = 0 119891 120598 = 0 se 120598 gt 120583(0)

Logo 119891 120598 = 0 para 120598 gt 119864119892



eleacutetrons na BC



10minus9 119864119892 = 1 eV

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





13

120590met =1198902

119898

119873

119881120591119890119897

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



14

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




15

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





16

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





17

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



18

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





19

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




20

119864119899119889 = minus

1

4120587120598

2119898lowast1198904

2ℏ21198992=

119898lowast

1198981205981199032 119864119899

119867

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



21

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores


Ga trivalente




22

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



23

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





24

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



25

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




26

120583 119879 = 0 = 119864119907 +119864119892

2

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores





27

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores


28

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




29

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




30






Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores


de 119896



31

119896119888 = 119896119894 + Ԧ119902119864119892 = ℏ120596 plusmn ℏΩ

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



32

ℰ 119896 = ℰ119888 +ℏ2

2

120583120584


ℰ 119896 = ℰ119907 minusℏ2

2

120583120584

119896120583 ෩119872minus1 119896120584 buracos

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores




33

ℰ 119896 = ℰ119888 + ℏ211989612

21198981+

11989622

21198982+

11989632


ℰ 119896 = ℰ119888 minus ℏ211989612

21198981+

11989622

21198982+

11989632

21198983 buracos

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores



34

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Semicondutores

Ge

35





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





36

119896ℎ = minus119896119890

120598ℎ 119896ℎ = minus120598119890 119896119890



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



Massa efetiva


37

Ԧ119907ℎ = Ԧ119907119890 rArr 120571ℎ120598ℎ 119896ℎ = 120571119890120598119890 119896119890

119898ℎlowast = minus119898119890

lowast

ℏ119889119896ℎ119889119905

= 119890 119864 + Ԧ119907ℎ times ℬ




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




Notar que


38

Ԧ119869 = 120590119864 Ԧ119869 = 120602 Ԧ119907 120602 =119873

119881119876 = 119899119876 119876 = plusmn119890

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


ou reescrevendo


39

Ԧ119869 = ി120590 ℬ sdot 119864

119864 = ി119903 ℬ sdot Ԧ119869

119903119894119895 = ി120590minus1 119894119895

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall



40

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


Desenvolvendo


41

119903119909119909 ℬ =119864119909119869119909

119864119909119864119910

=119903119909119909 119903119909119910119903119910119909 119903119910119910

1198691199090

119864119909 = 119903119909119909 ℬ 119869119909 119864119910 = 119903119910119909 ℬ 119869119909

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


Coeficiente Hall

Tensatildeo Hall

42

119903119910119909 ℬ =119864119910

119869119909

119881119867 = 119884119864119910

119877119867 =119903119910119909

ℬ=

119864119910

119869119909ℬ

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall

Modelo 1




43

119898lowast119889 Ԧ119907


119898lowast

120591Ԧ119907

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


119889119905= 0


44

120596119888 =119890ℬ

119898lowast

Ԧ119907 = minus119890120591

119898lowast 119864 minus119890120591


Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


Entatildeo

45

ി119903 ℬ =119898lowast

1198991198902120591

1 120596119888120591minus120596119888120591 1

119869119909 =1198991198902120591

119898lowast

119864119909 minus120596119888120591119864119910

1 + 12059611988821205912

119869119910 =1198991198902120591

119898lowast

120596119888120591119864119909 + 119864119910

1 + 12059611988821205912

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall



Coeficiente Hall

46

119903119909119909 =119898lowast

1198991198902120591

119903119910119909 = minusℬ

119899119890

119877119867 = minus1

119899119890

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall






47

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


onde

48

ി120590 ℬ =1198601 minus11986211198621 1198601

+1198602 minus11986221198622 1198602

=1198601 + 1198602 minus1198621 minus 11986221198621 + 1198622 1198601 + 1198602

119860119894 =120590119894

1 + 1205961198942120591119894

2

120590119894 =119899119894119890

2120591119894119898119894

119862119894 =120590119894120596119894120591119894

1 + 1205961198942120591119894

2

120596119894 =119890ℬ

119898119894

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall


Se ℬ = 0 1205961 = 1205962 = 0 e


49

119903119909119909 ℬ =1205901 + 1205902 + 12059011205962

212059122 + 12059021205961

212059112

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119909119909 ℬ = 0 =1

1205901 + 1205902

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall

Se ocorrer

temos


50

120590112059621205912 = 120590212059611205911

119899 = 119901

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Efeito Hall




51

119903119910119909 =120590212059621205912 1 + 1205961

212059112 minus 120590112059611205911 1 + 1205962

212059122

1205901 + 12059022 + 120590112059621205912 minus 120590212059611205911

2

119903119910119909 =ℬ

119901 minus 119899 119890

119877119867 =1

119901 minus 119899 119890




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




muacuteltiplos de


52

ℎ

1198902= 25812806 Ω




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




ℓ =119875



53

119903119910119909 =ℎ

11989021

ℓ




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




volume 119892119888 120598 =119967119888

119881 Entatildeo

54

119899119888 119879 = න120598119888

infin

119892119888 1205981

119890120573(120598minus120583) + 1119889120598



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



volume 119892119907 120598 =119967119907

119881 Entatildeo


55

119901119907 119879 = නminusinfin

120598119907

119892119907 1205981

119890120573(120583minus120598) + 1119889120598


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Se ocorrer



56

120598119888 minus 120583 ≫ 119896119861119879

120583 minus 120598119907 ≫ 119896119861119879



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



Definimos

57

119899119888 119879 = 119890minus120573(120598119888minus120583)න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598

119873119888 119879 = න120598119888

infin

119892119888 120598 119890minus120573(120598minus120598119888) 119889120598


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Entatildeo


e

58

119899119888 119879 = 119890minus120573 120598119888minus120583 119873119888 119879

119875119907 119879 = නminusinfin

120598119907

119892119907 120598 119890minus120573(120598119907minus120598) 119889120598

119901119907 119879 = 119890minus120573 120583minus120598119907 119875119907 119879




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




59

119892119894 120598 =1

212058722119898119894

ℏ2

32

120598 minus 120598119894

120598119896 = 120598119888 +ℏ21198962

2119898119888



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



e

60

119873119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119875119907 119879 =1

4

2119898119907

120587120573ℏ2

32


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Entatildeo obtemos

e

61

119899119888 119879 =1

4

2119898119888

120587120573ℏ2

32

119890minus120573(120598119888minus120583)

119901119907 119879 =1

4

2119898119907

120587120573ℏ2

32

119890minus120573(120583minus120598119907)





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





62

119899119888 119879 119901119907 119879 = 119873119888 119879 119875119907 119879 119890minus120573119864119892

119899119888 119879 = 119901119907 119879 = 119899119894 119879


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Entatildeo


63

119899119894 119879 =1

4

2

120587120573ℏ2

32

11989811988811989811990734119890minus1205731198641198922

120583 119879 = 120598119907 +119864119892

2+3

4119896119861119879 ln

119898119907

119898119888


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Ex

64





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





65

119899119888 minus 119901119907 = Δ119899 (ne 0)

119899119888 119879 119901119907 119879 = 1198991198942

119899119888119901119907

=Δ119899 2 + 4119899119894

2

2plusmnΔ119899

2


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Obtemos tambeacutem


Note que


66

Δ119899

119899119894= 2 sinh 120573 120583 minus 120583119894

120598119888 minus 120583119894 ≫ 119896119861119879

120583119894 minus 120598119907 ≫ 119896119861119879





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





Nesse caso


119899119894


outro portador

67

119899119888119901119907

=Δ119899

2+

1198991198942

Δ119899 2 plusmnΔ119899

2




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




68






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






69




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




70

119899 =σ119873119895119890

minus120573(119864119895minus120583119873119895)

σ119890minus120573(119864119895minus120583119873119895)




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




71

119899 =1

1 +12 119890

120573(120598119889minus120583)

119899119889 =119873119889

1 +12 119890

120573(120598119889minus120583)




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




72

119901119886 =119873119886

1 +12 119890

120573(120583minus120598119886)




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




73

119899119888 + 119899119889 minus 119901119907 + 119901119886 = 119873119889 minus 119873119886

120598119889 minus 120583 ≫ 119896119861119879

120583 minus 120598119886 ≫ 119896119861119879





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





74

119899119888 + 119899119889 = 119873119889 minus 119873119886 = Δ119899

119899119889 ≪ 119873119889 119901119886 ≪ 119873119886


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Entatildeo

e

75

119873119889 minus 119873119886119899119894

= 2 sinh 120573 120583 minus 120583119894

119899119888119901119907

=1

2119873119889 minus 119873119886

2 + 41198991198942 12

plusmn1

2(119873119889 minus 119873119886)




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




76

119899119888119901119907

= 119899119894 plusmn1

2(119873119889 minus119873119886)

119899119888119901119907

=1

2119873119889 minus 119873119886 +

1198991198942


1

2119873119889 minus 119873119886


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Se 119873119889 gt 119873119886

Se 119873119889 lt 119873119886

77

119899119888 = 119873119889 minus119873119886 119901119907 =1198991198942

119873119889 minus 119873119886

119899119888 =1198991198942

119873119886 minus 119873119889 119901119907 = 119873119886 minus 119873119889




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




119873119889ou

119901119886




78



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



79


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


80

Ge dopado com Sb



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



81


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Hipoacuteteses







82

119873119889 119909 = ቊ119873119889 119909 gt 00 119909 lt 0

119873119886 119909 = ቊ0 119909 gt 0119873119886 119909 lt 0



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



Tipo 119899 120583 entre 119864119888 e 119864119889

Tipo 119901 120583 entre 119864119907 e 119864119886


83



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



84



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



85





junccedilatildeo








86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





86





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





87




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




88



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



Com isso temos

e

89

ℋ119899 = ℰ119899 119896 minus 119890120601 119909


119901119907 119909 = 119875119907 119879 119890minus120573 120583minus120598119907+119890120601 119909



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



Graficamente

90




Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Definindo



91


119890Δ120601 = 119864119892 + 119896119861119879 ln119873119886119873119889119873119888119875119907





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





92

1205712120601 =1198892120601

1198891199092= minus

120602 119909

120598





93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




93

120602 119909 = ൞


119890119873119889 0lt119909lt1198891198990 119909gt119889119899



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



94


120601 119909 = 120601 infin 119909 gt 119889119899

120601 119909 = 120601 minusinfin +1198901198731198862120598

119909 + 1198891199012 minus119889119901lt 119909 lt 0

120601 119909 = 120601 infin minus1198901198731198892120598

119909 minus 1198891198992 0 lt 119909 lt 119889119899


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Graficamente

95



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



e


96

Δ120601 =119890

2120598(119873119886119889119901

2 + 1198731198891198891198992)

119873119886119889119901 = 119873119889119889119899


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Para 119889119901 temos


97

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598Δ120601

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598Δ120601

119890

12


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Largura total



98

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598Δ120601

119890

12




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




p+n119873119886 ≫ 119873119889

pn+ 119873119889 ≫ 119873119886


99






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






100






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






101






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






102





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





103





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





104



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



105





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





106

Δ120601 = Δ1206010 minus 119881

119889119901 =119873119889119873119886

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12

119889119899 =119873119886119873119889

1

119873119889 + 119873119886

2120598 Δ1206010 minus 119881

119890

12


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Largura total


Convenccedilatildeo



107

119908 = 119889119899 + 119889119901 =119873119886 + 119873119889119873119886119873119889

2120598 Δ1206010 minus 119881

119890

12




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




Para eleacutetrons


108

119895119890 = minus119890119973119890 119895ℎ = 119890119973ℎ

119973119890 = 119973119890term 119890120573119890119881 minus 1


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123


Para buracos




109

119973ℎ = 119973ℎterm 119890120573119890119881 minus 1

119895 = 119890 119973ℎterm + 119973119890

term 119890120573119890119881 minus 1




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




110

119973ℎ = 120583ℎ119901119907119864 minus 119863119901119889119901119907119889119909

Ԧ119895 = 120590119864

119973119890 = minus120583119890119899119888119864 minus 119863119899119889119899119888119889119909

Ԧ119895 = minus119863120571ϱ




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




De

sai

111

120583 =Ԧ119907

119864

Ԧ119895 = 120602 Ԧ119907

120590119890 = 119890119899120583119890 120590ℎ = 119890119901120583ℎ 120590 = 120590ℎ + 120590119890 = 119890(120583119890119899 + 120583ℎ119901)




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




112

120583119890 =119890119863119899119896119861119879

120583ℎ =119890119863119901119896119861119879






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






113

120590 =1198991198902120591

119898

120583119890 =119890120591119890

col

119898119890120583ℎ =

119890120591ℎcol

119898ℎ





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





114

120571 sdot Ԧ119895 +120597120602

120597119905= 0

120597119899119890120597119905

= minus120597119973119890120597119909

120597119901119907120597119905

= minus120597119973ℎ120597119909






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123






115

120597119899119890120597119905

=119889119899119888119889119905

119892minus119903

minus120597119973119890120597119909

119889119899119888119889119905

119892minus119903

= minus119899119888 minus 119899119888

0

120591119899

120597119901119907120597119905

=119889119901119907119889119905

119892minus119903

minus120597119973ℎ120597119909

119889119901119907119889119905

119892minus119903

= minus119901119907 minus 119901119907

0

120591119901





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123





Com isso temos

116

120597119899119890120597119905

= minus119899119888 minus 119899119888

0

120591119899minus120597119973119890120597119909

120597119901119907120597119905

= minus119901119907 minus 119901119907

0

120591119901minus120597119973ℎ120597119909




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




117

120597119973119890120597119909

+119899119888 minus 119899119888

0

120591119899= 0

120597119973ℎ120597119909

+119901119907 minus 119901119907

0

120591119901= 0

11986311989911988921198991198881198891199092

minus119899119888 minus 119899119888

0

120591119899= 0 119863119901

11988921199011199071198891199092

minus119901119907 minus 119901119907

0

120591119901= 0




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




118

119871119899 = 119863119899120591119899 119871119901 = 119863119901120591119901





Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




Lembrar que

119

119973ℎ119905119890119903119898 =

1198991198942

119873119889

119871119901120591119901

119973119890119905119890119903119898 =

1198991198942

119873119886

119871119899120591119899

119895 = 1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901

119890120573119890119881 minus 1

1198991198942 =

1

4

2119898119888

120587120573ℏ2

321

4

2119898119907

120587120573ℏ2

32

119890minus120573119864119892



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123



120

119895 = minus1198901198991198942 119863119899

119873119886119871119899+

119863119901119873119889119871119901




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123




121

retificador

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Transistor




122

119881119864 gt 0


direta emissor-

base

119881119862 ≪ 0


reversa base-

coletor

Transistor



123

Transistor



123

Documents

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