半导体物理与器件第四版课后习题答案4

文档鉴赏 Chapter 4 4 1 kT E NNn g ci exp 2 kT E T NN g OcO exp 300 3 where and are the values at 300 K cO N O N a Silicon K T eV kT cm i n 3 200 400 600 01727 0 03453 0 0518 0 4 1068 7 12 1038 2 14 1074 9 b Germanium c GaAs K T cm i n 3 cm i n 3 200 400 600 10 1016 2 14 1060 8 16 1082 3 38 1 9 1028 3 12 1072 5 4 2 Plot 4 3 a kT E NNn g ci exp 2 3 1919 2 11 300 1004 1 108 2105 T 3000259 0 12 1 exp T 3 3823 300 10912 2 105 2 T T0259 0 30012 1 exp By trial and error K 5 367 T b 25 2 122 105 2105 i n T T 0259 0 30012 1 exp 300 10912 2 3 38 By trial and error K 5 417 T 4 4 At K 200 T 300 200 0259 0 kT eV017267 0 At K 400 T 300 400 0259 0 kT eV034533 0 17 2 2 2 10 2 2 10025 3 1040 1 1070 7 200 400 i i n n 017267 0 exp 034533 0 exp 300 200 300 400 3 3 g g E E 034533 0 017267 0 exp8 gg EE 9578 289139 57exp810025 3 17 g E or 1714 38 8 10025 3 ln9561 28 17 g E or eV318 1 g E Now 3 2 10 300 400 1070 7 ocoN N 文档鉴赏 034533 0 318 1 exp 1721 10658 2 370 2 10929 5 ocoN N so cm 37 1041 9 ocoN N 6 4 5 kT kT kT An Bn i i 20 0 exp 90 0 exp 10 1 exp For K eV200 T017267 0 kT For K eV300 T0259 0 kT For K eV400 T034533 0 kT a For K 200 T 6 10325 9 017267 0 20 0 exp An Bn i i b For K 300 T 4 1043 4 0259 0 20 0 exp An Bn i i c For K 400 T 3 1005 3 034533 0 20 0 exp An Bn i i 4 6 a kT EE EEfg F cFc exp kT EE EE c c exp kT EE Fc exp Let xEE c Then kT x xfg Fc exp To find the maximum value kT x x dx fgd Fc exp 2 1 2 1 0exp 1 2 1 kT x x kT which yields 22 1 2 1 2 1 kT x kT x x The maximum value occurs at 2 kT EE c b kT EE EEfg F F exp1 kT EE EE exp kT EEF exp Let xEE Then kT x xfg F exp1 To find the maximum value 0exp 1 kT x x dx d dx fgd F Same as part a Maximum occurs at 2 kT x or 2 kT EE 4 7 文档鉴赏 kT EE EE kT EE EE En En c c c c 2 2 1 1 2 1 exp exp where and kTEE c 4 1 2 2 kT EE c Then kT EE kT kT En En 21 2 1 exp 2 4 5 3exp22 2 1 4exp22 or 0854 0 2 1 En En 4 8 Plot 4 9 Plot 4 10 ln 4 3 n p midgapFi m m kTEE Silicon op mm56 0 on mm08 1 eV0128 0 midgapFi EE Germanium op mm37 0 on mm55 0 eV0077 0 midgapFi EE Gallium Arsenide op mm48 0 on mm067 0 eV0382 0 midgapFi EE 4 11 c midgapFi N N kTEE ln 2 1 kTkT4952 0 108 2 1004 1 ln 2 1 19 19 K T eV kT eV midgapFi EE 200 400 600 01727 0 03453 0 0518 0 0086 0 0171 0 0257 0 4 12 a ln 4 3 n p midgapFi m m kTEE 21 1 70 0 ln0259 0 4 3 meV63 10 b 080 0 75 0 ln0259 0 4 3 midgapFi EE meV47 43 4 13 Let constant KEgc Then dEEfEgn F E co c dE kT EE K c E F exp1 1 dE kT EE K c E F exp Let so that kT EE c dkTdE We can write cFcF EEEEEE so that 文档鉴赏 expexpexp kT EE kT EE FcF The integral can then be written as d kT EE kTKn Fc o 0 expexp which becomes kT EE kTKn Fc o exp 4 14 Let for cc EECEg 1c EE Then dEEfEgn F E co c dE kT EE EE C c E F c exp1 1 dE kT EE EEC F E C c exp 1 Let so that kT EE c dkTdE We can write FccF EEEEEE Then kT EE Cn Fc o exp 1 dE kT EE EE c E c c exp or kT EE Cn Fc o exp 1 dkTkT exp 0 We find that 11expexp 0 0 d So kT EE kTCn Fc o exp 2 1 4 15 We have 1 m m a r o r o For germanium 16 r o mm55 0 Then 53 0 29 55 0 1 16 1 o ar or o Ar 4 15 1 The ionization energy can be written as eV 6 13 2 s o o m m E eV 029 0 6 13 16 55 0 2 E 4 16 We have 1 m m a r o r o For gallium arsenide 1 13 r o mm067 0 Then o Ar10453 0 067 0 1 1 13 1 The ionization energy is 6 13 1 13 067 0 6 13 2 2 s o o m m E or eV0053 0 E 4 17 a o c Fc n N kTEEln 文档鉴赏 15 19 107 108 2 ln0259 0 eV2148 0 b FcgF EEEEE eV90518 0 2148 0 12 1 c kT EE Np F o exp 0259 0 90518 0 exp1004 1 19 cm 3 1090 6 3 d Holes e i o FiF n n kTEEln 10 15 105 1 107 ln0259 0 eV338 0 4 18 a o F p N kTEE ln 16 19 102 1004 1 ln0259 0 eV162 0 b EEEEE FgFc eV958 0 162 0 12 1 c 0259 0 958 0 exp108 2 19 o n cm 3 1041 2 3 d i o FFi n p kTEEln 10 16 105 1 102 ln0259 0 eV365 0 4 19 a o c Fc n N kTEEln 5 19 102 108 2 ln0259 0 eV8436 0 FcgF EEEEE 8436 0 12 1 eV2764 0 EEF b 0259 0 27637 0 exp1004 1 19 o p cm 14 10414 2 3 c p type 4 20 a eV 032375 0 300 375 0259 0 kT 032375 0 28 0 exp 300 375 107 4 2 3 17 o n cm 14 1015 1 3 28 0 42 1 FcgF EEEEE eV14 1 032375 0 14 1 exp 300 375 107 2 3 18 o p cm 3 1099 4 3 b 14 17 1015 1 107 4 ln0259 0 Fc EE eV2154 0 2154 0 42 1 FcgF EEEEE eV2046 1 0259 0 2046 1 exp107 18 o p cm 2 1042 4 3 文档鉴赏 4 21 a eV 032375 0 300 375 0259 0 kT 032375 0 28 0 exp 300 375 108 2 2 3 19 o n cm 15 1086 6 3 28 0 12 1 FcgF EEEEE eV840 0 032375 0 840 0 exp 300 375 1004 1 2 3 19 o p cm 7 1084 7 3 b o c Fc n N kTEEln 15 19 10862 6 108 2 ln0259 0 eV2153 0 eV9047 0 2153 0 12 1 EEF 0259 0 904668 0 exp1004 1 19 o p cm 3 1004 7 3 4 22 a p type b eV28 0 4 12 1 4 g F E EE kT EE Np F o exp 0259 0 28 0 exp1004 1 19 cm 14 1010 2 3 EEEEE FgFc eV84 0 28 0 12 1 kT EE Nn Fc co exp 0259 0 84 0 exp108 2 19 cm 5 1030 2 3 4 23 a kT EE nn FiF io exp 0259 0 22 0 exp105 1 10 cm 13 1033 7 3 kT EE np FFi io exp 0259 0 22 0 exp105 1 10 cm 6 1007 3 3 b kT EE nn FiF io exp 0259 0 22 0 exp108 1 6 cm 9 1080 8 3 kT EE np FFi io exp 0259 0 22 0 exp108 1 6 cm 2 1068 3 3 4 24 a o F p N kTEE ln 15 19 105 1004 1 ln0259 0 eV1979 0 b EEEEE FgFc eV92212 0 19788 0 12 1 c 0259 0 92212 0 exp108 2 19 o n 文档鉴赏 cm 3 1066 9 3 d Holes e i o FFi n p kTEEln 10 15 105 1 105 ln0259 0 eV3294 0 4 25 eV 034533 0 300 400 0259 0 kT 2 3 19 300 400 1004 1 N cm 19 10601 1 3 2 3 19 300 400 108 2 c N cm 19 103109 4 3 19192 10601 1103109 4 i n 034533 0 12 1 exp 24 106702 5 cm 12 10381 2 i n 3 a o F p N kTEE ln 15 19 105 10601 1 ln034533 0 eV2787 0 b eV84127 0 27873 0 12 1 Fc EE c 034533 0 84127 0 exp103109 4 19 o n cm 9 10134 1 3 d Holes e i o FFi n p kTEEln 12 15 10381 2 105 ln034533 0 eV2642 0 4 26 a 0259 0 25 0 exp107 18 o p cm 14 1050 4 3 eV17 1 25 0 42 1 Fc EE 0259 0 17 1 exp107 4 17 o n cm 2 1013 1 3 b eV034533 0 kT 2 3 18 300 400 107 N cm 19 10078 1 3 2 3 17 300 400 107 4 c N cm 17 10236 7 3 o F p N kTEE ln 14 19 1050 4 10078 1 ln034533 0 eV3482 0 eV072 1 3482 0 42 1 Fc EE 034533 0 07177 1 exp10236 7 17 o n cm 4 1040 2 3 4 27 a 0259 0 25 0 exp1004 1 19 o p cm 14 1068 6 3 eV870 0 25 0 12 1 Fc EE 0259 0 870 0 exp108 2 19 o n 文档鉴赏 cm 4 1023 7 o n 3 b eV034533 0 kT 2 3 19 300 400 1004 1 N cm 19 10601 1 3 2 3 19 300 400 108 2 c N cm 19 10311 4 3 o F p N kTEE ln 14 19 1068 6 10601 1 ln034533 0 eV3482 0 eV7718 0 3482 0 12 1 Fc EE 034533 0 77175 0 exp10311 4 19 o n cm 9 1049 8 3 4 28 a Fco FNn 2 1 2 For 2kTEE cF 5 0 2 kT kT kT EE cF F Then 0 1 2 1 F F 0 1108 2 2 19 o n cm 19 1016 3 3 b Fco FNn 2 1 2 0 1107 4 2 17 cm 17 1030 5 3 4 29 Fo FNp 2 1 2 F F 2 1 1919 1004 1 2 105 So 26 4 2 1 F F We find kT EE F F 0 3 eV 0777 0 0259 0 0 3 F EE 4 30 a 4 4 kT kT kT EE cF F Then 0 6 2 1 F F Fco FNn 2 1 2 0 6108 2 2 19 cm 20 1090 1 3 b 0 6107 4 2 17 o n cm 18 1018 3 3 4 31 For the electron concentration EfEgEn Fc The Boltzmann approximation applies so c n EE h m En 3 2 3 24 kT EE F exp or kT EE h m En Fcn exp 24 3 2 3 kT EE kT EE kT cc exp Define 文档鉴赏 kT EE x c Then xxKxnEn exp To find maximum set xnEn xxK dx xdn exp 2 1 0 2 1 xxexp1 2 1 or xxKx 2 1 exp0 2 1 which yields kTEE kT EE x c c 2 1 2 1 For the hole concentration EfEgEp F 1 Using the Boltzmann approximation EE h m Ep p 3 2 3 24 kT EEF exp or kT EE h m Ep F p exp 24 3 2 3 kT EE kT EE kT exp Define kT EE x Then xxKxp exp To find maximum value of xpEp set Using the results from 0 xd xdp above we find the maximum at kTEE 2 1 4 32 a Silicon We have kT EE Nn Fc co exp We can write FddcFc EEEEEE For eV and 045 0 dc EE eVkTEE Fd 3 we can write 3 0259 0 045 0 exp108 2 19 o n 737 4 exp108 2 19 or cm 17 1045 2 o n 3 We also have kT EE Np F o exp Again we can write EEEEEE aaFF For and kTEE aF 3 eV045 0 EEa Then 0259 0 045 0 3exp1004 1 19 o p 737 4exp1004 1 19 or cm 16 1012 9 o p 3 b GaAs assume eV0058 0 dc EE Then 3 0259 0 0058 0 exp107 4 17 o n 224 3 exp107 4 17 or cm 16 1087 1 o n 3 Assume eV0345 0 EEa Then 3 0259 0 0345 0 exp107 18 o p 332 4exp107 18 or 文档鉴赏 cm 16 1020 9 o p 3 4 33 Plot 4 34 a cm 1515 10310154 o p 3 cm 4 15 2 10 105 7 103 105 1 o n 3 b cm 16 103 do Nn 3 cm 3 16 2 10 105 7 103 105 1 o p 3 c cm 10 105 1 ioo npn 3 d 3 19192 300 375 1004 1 108 2 i n 3750259 0 30012 1 exp cm 11 10334 7 i n 3 cm 15 104 ao Np 3 cm 8 15 2 11 1034 1 104 10334 7 o n 3 e 3 19192 300 450 1004 1 108 2 i n 4500259 0 30012 1 exp cm 13 10722 1 i n 3 2 13 2 1414 10722 1 2 10 2 10 o n cm 14 10029 1 3 cm 12 14 2 13 1088 2 10029 1 10722 1 o p 3 4 35 a 1515 10104 dao NNp cm 15 103 3 cm 3 15 2 62 1008 1 103 108 1 o i o p n n 3 b cm 16 103 do Nn 3 cm 4 16 2 6 1008 1 103 108 1 o p 3 c cm 6 108 1 ioo npn 3 d 3 18172 300 375 100 7107 4 i n 3750259 0 30042 1 exp cm 8 10580 7 i n 3 cm 15 104 ao Np 3 cm 2 15 2 8 1044 1 104 10580 7 o n 3 e 3 18172 300 450 100 7107 4 i n 4500259 0 30042 1 exp cm 10 10853 3 i n 3 cm 14 10 do Nn 3 cm 7 14 2 10 1048 1 10 10853 3 o p 3 4 36 a Ge cm 13 104 2 i n 3 i 2 2 22 i dd o n NN n 2 13 2 1515 104 2 2 102 2 102 or cm 15 102 do Nn 3 文档鉴赏 15 2 132 102 104 2 o i o n n p cm 11 1088 2 3 ii 1516 10710 dao NNp cm 15 103 3 15 2 132 103 104 2 o i o p n n cm 11 1092 1 3 b GaAs cm 6 108 1 i n 3 i cm 15 102 do Nn cm 3 15 2 6 1062 1 102 108 1 o p 3 ii cm 15 103 dao NNp 3 cm 3 15 2 6 1008 1 103 108 1 o n 3 c The result implies that there is only one minority carrier in a volume of cm 3 10 3 4 37 a For the donor level kT EEN n Fdd d exp 2 1 1 1 0259 0 20 0 exp 2 1 1 1 or 4 1085 8 d d N n b We have kT EE Ef F F exp1 1 Now FccF EEEEEE or 245 0 kTEE F Then 0259 0 245 0 1exp1 1 EfF or 5 1087 2 EfF 4 38 a p type da NN b Silicon 1313 101105 2 dao NNp or cm 13 105 1 o p 3 Then cm 7 13 2 102 105 1 105 1 105 1 o i o p n n 3 Germanium 2 2 22 i dada o n NNNN p 2 13 2 1313 104 2 2 105 1 2 105 1 or cm 13 1026 3 o p 3 Then 13 13 2 132 1076 1 10264 3 104 2 o i o p n n cm 3 Gallium Arsenide cm 13 105 1 dao NNp 3 and cm 216 0 105 1 108 1 13 2 62 o i o p n n 3 4 39 a n type ad NN b 1515 102 1102 ado NNn cm 14 108 3 cm 5 14 2 102 1081 2 108 105 1 o i o n n p 3 文档鉴赏 c daao NNNp 151515 102102 1104 a N cm 15 108 4 a N 3 cm 4 15 2 10 10625 5 104 105 1 o n 3 4 40 cm 15 5 2 102 10125 1 102 105 1 o i o p n n 3 n type oo pn 4 41 3 18192 300 250 100 61004 1 i n 3002500259 0 66 0 exp 24 108936 1 cm 12 10376 1 i n 3 22 22 4 1 4 io o i o i o nn n n p n n io nn 2 1 So cm 11 1088 6 o n 3 Then cm 12 1075 2 o p 3 2 2 22 i aa o n NN p 2 12 2 10752 2 a N 24 2 108936 1 2 a N 2 1224 2 10752 2105735 7 a a N N 24 2 108936 1 2 a N so that cm 12 10064 2 a N 3 4 42 Plot 4 43 Plot 4 44 Plot 4 45 2 2 22 i adad o n NNNN n 2 102 1102 101 1 1414 14 2 2 1414 2 102 1102 i n 2 2 13 2 1314 104104101 1 i n 22727 106 1109 4 i n so cm 13 1074 5 i n 3 cm 13 14 272 103 101 1 103 3 o i o n n p 3 4 46 a p type da NN Majority carriers are holes 1616 105 1103 dao NNp cm 16 105 1 3 Minority carriers are electrons cm 4 16 2 102 105 1 105 1 105 1 o i o p n n 3 b Boron atoms must be added 文档鉴赏 daao NNNp 161616 105 1103105 a N So cm 16 105 3 a N 3 cm 3 16 2 10 105 4 105 105 1 o n 3 4 47 a n type io np b o i o o i o p n n n n p 22 cm o n 16 4 2 10 10125 1 102 105 1 3 electrons are majority carriers cm 4 102 o p 3 holes are minority carriers c ado NNn 1516 10710125 1 d N so cm 16 10825 1 d N 3 4 48 i o FFi n p kTEEln For Germanium K T eV kT cm i n 3 200 400 600 01727 0 03453 0 0518 0 10 1016 2 14 1060 8 16 1082 3 and 2 2 22 i aa o n NN p cm 15 10 a N 3 K T cm o p 3 eV FFi EE 200 400 600 15 100 1 15 1049 1 16 1087 3 1855 0 01898 0 000674 0 4 49 a d c Fc N N kTEEln d N 19 108 2 ln0259 0 For cm eV 14 10 3 3249 0 Fc EE cm eV 15 10 3 2652 0 Fc EE cm eV 16 10 3 2056 0 Fc EE cm eV 17 10 3 1459 0 Fc EE b i d FiF n N kTEEln 10 105 1 ln0259 0 d N For cm eV 14 10 3 2280 0 FiF EE cm eV 15 10 3 2877 0 FiF EE cm eV 16 10 3 3473 0 FiF EE cm eV 17 10 3 4070 0 FiF EE 4 50 a 2 2 22 i dd o n NN n cm 15 1005 1 05 1 do Nn 3 2 1515 105 01005 1 2 2 15 105 0 i n so 282 1025 5 i n Now 3 19192 300 1004 1 108 2 T ni 文档鉴赏 3000259 0 12 1 exp T 3 3828 300 10912 2 1025 5 T T 973 12972 exp By trial and error K 5 536 T b At K 300 T o c Fc n N kTEEln 15 19 10 108 2 ln0259 0 Fc EE eV2652 0 At K 5 536 T eV 046318 0 300 5 536 0259 0 kT 2 3 19 300 5 536 108 2 c N cm 19 10696 6 3 o c Fc n N kTEEln 15 19 1005 1 10696 6 ln046318 0 Fc EE eV5124 0 then eV 2472 0 Fc EE c Closer to the intrinsic energy level 4 51 i o FFi n p kTEEln At K eV200 T017267 0 kT K eV400 T034533 0 kT K eV600 T0518 0 kT At K 200 T 3 19192 300 200 1004 1 108 2 i n 017267 0 12 1 exp cm 4 10638 7 i n 3 At K 400 T 3 19192 300 400 1004 1 108 2 i n 034533 0 12 1 exp cm 12 10381 2 i n 3 At K 600 T 3 19192 300 600 1004 1 108 2 i n 0518 0 12 1 exp cm 14 10740 9 i n 3 At K and K 200 T400 T cm 15 103 ao Np 3 At K 600 T 2 2 22 i aa o n NN p 2 14 2 1515 10740 9 2 103 2 103 cm 15 10288 3 3 Then K eV200 T4212 0 FFi EE K 400 T eV2465 0 FFi EE K 600 T eV0630 0 FFi EE 4 52 a 6 108 1 ln0259 0 ln a i a FFi N n N kTEE 文档鉴赏 For cm 14 10 a N 3 eV4619 0 FFi EE cm 15 10 a N 3 eV5215 0 FFi EE cm 16 10 a N 3 eV5811 0 FFi EE cm 17 10 a N 3 eV6408 0 FFi EE b aa F NN N kTEE 18 100 7 ln0259 0 ln For cm 14 10 a N 3 eV2889 0 EEF cm 15 10 a N 3 eV2293 0 EEF cm 16 10 a N 3 eV1697 0 EEF cm 17 10 a N 3 eV1100 0 EEF 4 53 a ln 4 3 n p midgapFi m m kTEE 10ln0259 0 4 3 or eV0447 0 midgapFi EE b Impurity atoms to be added so eV45 0 Fmidgap EE i p type so add acceptor atoms ii eV4947 0 45 0 0447 0 FFi EE Then kT EE np FFi io exp 0259 0 4947 0 exp105 or cm 13 1097 1 ao Np 3 4 54 kT EE NNNn Fc cado exp so 0259 0 215 0 exp108 2105 1915 d N 1515 1095 6 105 or cm 16 102 1 d N 3 4 55 a Silicon i d c Fc N N kTEEln eV 2188 0 106 108 2 ln0259 0 15 19 ii eV1929 0 0259 0 2188 0 Fc EE kT EE NN Fc cd exp 0259 0 1929 0 exp108 2 19 cm 16 10631 1 d N 3 15 106 d N cm 16 10031 1 d N 3 Additional donor atoms b GaAs i 15 17 10 107 4 ln0259 0 Fc EE eV15936 0 ii eV13346 0 0259 0 15936 0 Fc EE 0259 0 13346 0 exp107 4 17 d N cm 15 10718 2 3 15 10 d N cm 15 10718 1 d N 3 Additional 文档鉴赏 donor atoms 4 56 a a FFi N N kTEE ln 16 19 102 1004 1 ln0259 0 eV1620 0 b d c FiF N N kTEEln eV 1876 0 102 108 2 ln0259 0 16 19 c For part a cm 16 102 o p 3 16 2 10