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Chapter 4
4.1
??Eg ni2?NcN?exp??kT??? ?? ?N?T?3cON?O??300??exp???Eg??? ?kT??
where NcO and N?O are the values at 300 K.
(a) Silicon T(K) kT(eV) n?3i(cm) 200 0.01727 7.68?104 400 0.03453 2.38?1012 600 0.0518 9.74?1014 (b) Germanium (c) GaAs T(K) n33i(cm?) ni(cm?) 200 2.16?1010 1.38 400 8.60?1014 3.28?109 600 3.82?1016 5.72?1012 _______________________________________ 4.2
Plot
_______________________________________ 4.3
(a) n2N??Eg?i?cN?exp??kT?? ?? ?5?1011?2??2.8?1019??1.04?1019??3?T??300??
?exp???1.12???0.0259??T300???
2.5?1023??2.912?1038??3?T??300??
?exp????1.12??300????0.0259??T???
By trial and error, T?367.5K
(b)
n2i??5?1012?2?2.5?1025
?3?2.912?1038???T????1.?300?12??300???exp???0.0259??T???
By trial and error, T?417.5K
_______________________________________ 4.4
At T?200K, kT??0.0259???200??300??
?0.017267eV
At T?400K, kT??0.0259???400??300??
?0.034533eV
n2i?400??7.70?1010?217n2i?200???1.40?102?2?3.025?10
??400?3exp??Eg???300????0.034533???3?
?200??300??exp???Eg??0.017267??
?8exp??EgEg??0.017267?0.034533?
?
3.025?1017?8exp?Eg?57.9139?28.9578??
or
?3.025?1017E?g?28.9561??ln????8???38.1714 or Eg?1.318eV
Now ?7.70?1010?23?N?400?coN?o??300??
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??x??exp???1.318??0.034533?? 5.929?1021?N?17coN?o?2.370??2.658?10? so NcoN?o?9.41?1037cm?6
_______________________________________ 4.5
???1.10nexp???i?Bn??kT??exp???0.20?? i?A?exp???0.90??kT??kT?? For T?200K, kT?0.017267eV For T?300K, kT?0.0259eV For T?400K, kT?0.034533eV (a) For T?200K, ? niB?n?exp???0.20????9.325?10?6 i?A??0.017267(b) For T?300K, ? ni?B??0.20??nA??exp??0.0259???4.43?104 i?(c) For T?400K, ? niB?n??exp???0.20??0.034533???3.05?10?3 i?A_______________________________________ 4.6
(a) gf?E???E?EF??cF?Ecexp??kT??
?E?E???E?Ec??cexp??kT??
?exp????Ec?EF???kT?? Let E?Ec?x
Then gcfF?xexp??kT??
To find the maximum value: d?gcfF?dx?12x?1/2exp???x??kT??
?1kT?x1/2exp???x??kT???0 which yields
1x1/2 kT2x1/2?kT?x?2 The maximum value occurs at
E?EkTc?2
(b)
g?E???EF?E????1?fF???Eexp??kT??
?E???E??E????Eexp??kT?? ?exp?? ??EF?E????kT?? Let E??E?x
Then g??x???1?fF??xexp??kT??
To find the maximum value ?
dg??1?fF??dx?d?dx??xexp???x??kT??????0 Same as part (a). Maximum occurs at
x?kT2
or
E?EkT??2
_______________________________________ 4.7
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E???E1?Ec??1?Ecexp?
n?E1??kT??n?E?2?EE???EE
2?c??2?cexp??kT?? where
EEkT1?c?4kT and E2?Ec?2 Then
n?E1?4kTkTexp?n?E?2????E1?E2???kT?? 2
?22exp??????1???4?2?????22exp??3.5?
or
n?E1?n?E?0.0854
2?_______________________________________ 4.8
Plot
_______________________________________ 4.9
Plot
_______________________________________ 4.10
E3?Fi?Emidgap?4kTln?m*p???m*? n?? Silicon: m**p?0.56mo, mn?1.08mo
EFi?Emidgap??0.0128eV Germanium: m*p?0.37mo,
m*n?0.55mo
EFi?Emidgap??0.0077eV Gallium Arsenide: m*p?0.48mo,
m*n?0.067mo EFi?Emidgap??0.0382eV _______________________________________ 4.11 E1Fi?Emidgap??kT?ln??N??2???Nc??
?12?kT?ln???1.04?1019???2.8?1019????0.4952?kT? T(K) kT(eV) (EFi?Emidgap)(eV) 200 0.01727 ?0.0086 400 0.03453 ?0.0171 600 0.0518 ?0.0257 _______________________________________ 4.12
(a) E3?Fi?Emidgap?4kTln?m*p???*?m n?? ?3?0.70?4?0.0259?ln??1.21??
??10.63meV
(b) E3?0.75?Fi?Emidgap?4?0.0259?ln??0.080??
??43.47meV
_______________________________________ 4.13
Let gc?E??K?constant Then
? no?gc?E?fF?E?dE
E?c? ?K?1exp???dE
Ec1??E?EF?kT???? ?K?exp????E?EF??Ec?kT??dE Let ??E?EckT so that dE?kT?d?
We can write E?EF??Ec?EF???E?Ec?
so that
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exp????E?EF???kT???exp????Ec?EF???kT???exp???? The integral can then be written as
nkT?exp????E?c?EF??o?K??kT???exp????d? 0 which becomes
n???Ec?EF??o?K?kT?exp??kT?? _______________________________________ 4.14
Let gc?E??C1?E?Ec? for E?Ec Then
? no?E?gc?E?fF?E?dE c? ?C??E?Ec?1Ec1?exp???E?EF?dE
?kT???
??C1??E?E????E?EF??Cexp?Ec?kT??dE Let
??E?EckT so that dE?kT?d? We can write
E?EF??E?Ec???Ec?EF?
Then
n???Ec?EF??o?C1exp??kT??
????E?E?E?Ec??c?exp???Ec?kT??dE or
n???Ec?EF??o?C1exp??kT?? ????kT?????exp??????kT?d?
0 We find that
???exp????d??exp????????1????1
00 So
n?C2???Ec?EF??o1?kT?exp??kT?? _______________________________________ 4.15
We have r1a????mo?r?m*??? o? For germanium, ?r?16, m*?0.55mo Then
r?1?1??16???0.55??ao??29??0.53?
or
ro1?15.4A
The ionization energy can be written as 2 E???m*??m??????o?????13.6???eV o?s? ?0.55?16?2?13.6??E?0.029eV
_______________________________________ 4.16
We have r1a???mo?r?o??m*??? For gallium arsenide, ?r?13.1,
m*?0.067mo Then
r1???1?o1??13.?0.067???0.53??104A
The ionization energy is
2 E???m*??????o???m?13.6??0.0672?13.6? o?????s???13.1? or
E?0.0053eV
_______________________________________ 4.17
(a) EE?Nc?c?F?kTln????no?? 文档鉴赏
??0.0259?ln???2.8?1019???7?1015??
?0.2148eV
(b) EF?E??Eg??Ec?EF?
?1.12?0.2148?0.90518eV
(c) p???EF?E???o?N?exp??kT??
??1.04?1019?exp???0.90518??0.0259?? ?6.90?103cm?3 (d) Holes
(e) EE?no?F?Fi?kTln????ni??
??0.0259?ln???7?1015???1.5?1010??
?0.338eV
_______________________________________ 4.18
(a) E?kTln??N???F?E???po??
??0.0259?ln???1.04?1019???2?1016??
?0.162eV
(b)
Ec?EF?Eg??EF?E?? ?1.12?0.162?0.958eV
(c) n??1019?exp???0.958?o?2.8?0.0259??
?2.41?103cm?3
(d) E?po?Fi?EF?kTln????ni??
??0.0259?ln???2?1016??1.5?1010???
?0.365eV
_______________________________________ 4.19
(a) E?Nc?c?EF?kTln????no??
??0.0259?ln???2.8?1019???2?105??
?0.8436eV EF?E??Eg??Ec?EF? ?1.12?0.8436 EF?E??0.2764eV (b)
p??0.27637?o??1.04?1019?exp??0.0259??
?2.414?1014cm?3
(c) p-type
_______________________________________ 4.20
(a) kT??0.0259???375??300???0.032375eV
no??4.7?1017??3/2?375??300??exp???0.28??0.032375?? ?1.15?1014cm?3
EF?E??Eg??Ec?EF??1.42?0.28 ?1.14eV
/2p??1018??375?3o?7??300??exp???1.14??0.032375?? ?4.99?103cm?3
(b) E??4.7?1017?c?EF??0.0259?ln???1.15?1014??
?0.2154eV
EF?E??Eg??Ec?EF??1.42?0.2154 ?1.2046eV
p?18??1.2046?o?7?10?exp??0.0259??
?4.42?10?2cm?3
半导体物理与器件第四版课后习题答案4
