半导体物理与器件第四版课后习题答案2

1、Chapter22.1Sketch_2.2Sketch_2.3Sketch_2.42xFromProblem2.2,phaset=constantThen2dxdx0,p2dtdtFromProblem2.3,phase2xt=constantThen2dx0,dxdtp2dt_2.5EhhchcEGold:E4.90eV4.901.61019JSo,34106.625103102.54105cm4.901.61019or0.254mCesium:E1.90eV1.901.61019JSo,6.6251034310106.54105cm1.901.61019or0.654m_2.6(a)ph6

2、.62510345501091027kg-m/s1.205p1.204510271.32103m/sm9.111031105cm/sor1.32(b)ph6.62510344401091.5061027kg-m/sp1.505710271.65103m9.111031m/s105cm/sor1.65Yes_2.7(a)(i)p2mE29.1110311.21.610195.9151025kg-m/sh6.62510341.129p5.915102510moor11.2Ap29.111031121.610191.871024kg-m/s6.62510343.541010m1.87041024oo

3、r3.54A3119(iii)p29.11101201.61010345.9151024kg-m/s6.6251.121010m5.9151024oor1.12A(b)p21.671027191.21.6102310342.53210kg-m/s6.6252.6210112.5321023moor0.262A_2.8Eavg3kT30.02590.03885eV22Nowpavg2mEavg29.1110310.038851.61019orpavg1.0641025kg-m/sNow34h6.625106.2259p1.064102510moro62.25A_2.9EphhcppNowEepe

4、2and2mh1h2Eepe2meeSetEpEeandp10eThen2hc1h2110hp2me2mpwhichyieldsp100h2mc2mc2EpEhchc2mcp100h10029.113138210101001.641015J10.25keV_2.10(a)ph6.62510348510107.7941026kg-m/sp7.79410268.56104m/sm9.111031or8.56106cm/sE1m219.1110318.561042223.331021JorE3.33410212.08102eV11.61019(b)E9.1110318103222.9151023Jo

5、rE2.91510231.82104eV1.61019pm9.11103181037.2881027kg-m/sh6.62510359.09108mp7.2881027oor909A_2.11(a)Ehhc6.62510343108110101.9915JNow101015EeVE1.99V1.61019eV1.24104V12.4kV(b)p2mE29.1110311.9910156.021023kg-m/sThenh6.62510341.1011mp6.02102310oro11A_2.12p1.0541034x1061.0541028kg-m/s_2.13px(a)(i)1.054103

6、48.7831026kg-m/sp121010p2(ii)EdEdppdp2mdp2pppp2mmNowp2mE291031161.610192.1471024kg-m/ssoE2.146610248.7831026910312.0951019JorE2.09510191.31eV1.6101926(b)(i)p8.78310kg-m/s(ii)p251028161.610195.061023kg-m/sE5.0610238.7831026510288.8881021JorE8.88810215.551021.61019eV_2.14p1.05410341.0543210210kg-xm/s1

7、032pmp1.054m15001036m/s_2.15Et(a)t1.05410348.231016s0.81.61019(b)p1.0541034x1.5101025kg-m/s7.0310_2.16(a)If1x,tand2x,taresolutionstoSchrodingerswaveequation,then22x,t1x,t1Vx1x,tj2mx2tand222x,t2x,tx2x,tj2mx2VtAddingthetwoequations,weobtain222mx21x,t2x,tVx1x,t2x,tjt1x,t2x,twhichisSchrodingerswaveequat

8、ion.So1x,t2x,tisalsoasolution.(b)If1x,t2x,twereasolutiontoSchrodingerswaveequation,thenwecouldwrite222mx212Vx12jt12whichcanbewrittenas222211222m1x222xxxVxj211212ttDividingby12,wefind21212221122m2x21x212xxVx1211jtt21Since1isasolution,then21211Vx12m2jt1x1Subtractingtheselasttwoequations,wehave21222122

9、m2x212xx12jtSince2isalsoasolution,wehave2122122mx2Vxjt22Subtractingtheselasttwoequations,weobtain22122m12xVx0 xThisequationisnotnecessarilyvalid,whichmeansthat12is,ingeneral,notasolutiontoSchrodingerswaveequation._2.173xA2cos2dx112A2xsinx31221A231122soA212orA12_2.181/2A2cos2nxdx11/2A2xsin2nx1/2124n1

10、/2A2111A21442orA2_2.19Notethat*dx10Functionhasbeennormalized.(a)Nowao24P2expxdx0aoaoao242xexpdxao0ao2ao2xao4aoexpao20orP1exp2ao11exp124aowhichyieldsP0.393(b)ao222xPdxexpaoaoao4o222xexpdxaoaoao42ao2xao2expao2aoao4orP1exp1exp12whichyieldsP0.239(c)2ao2xPexpdx0aoao2ao2xexpdxao0ao2aoexp2xaoao2ao01exp21wh

11、ichyields0.865_2.202Pxdxa/4x(a)2cos2dxa202xsina/42xaa240aasin242a24a2a1aa84orP0.409a/2(b)P2cos2xdxaaa/4sin2xa/22xaa24a/4a2asinasin2a4484aa21011484orP0.0908a/22x(c)Pcos2dxa/2aa2x2xsinaa/2a24a/2a2asinasina4444aaorP1_2.21a/42sin22xdx(a)P0aasin4xa/42xaa2240a2asina88aorP0.25a/2xdx(b)P2sin22aaa/44xsina/22

12、xaa224a/4a2asin2asina4888aaorP0.25a/2(c)P2sin22xdxa/2aa4xa/2sin2xaa242a/2a2asin2asin2a4848aaorP1_2.221012(a)(i)pk8104m/s8108orp106cm/s2827.854109mk108oor78.54A(ii)pm9.1110311049.111027kg-m/sE1m219.1110311042224.5551023JorE4.55510232.85104eV1.610191013(b)(i)p1.5104m/s1.5109korp106cm/s224.19109mk1.510

13、9oor41.9Ap9.111027kg-m/s2.85104eV_2.23Aejkxt(a)x,t(b)E0.0251.610191m2219.11103122so9.37104m/s9.37106cm/sForelectrontravelinginxdirection,9.37106cm/spm9.1110319.371048.5371026kg-m/s34h6.625107.769p8.537102610mk221098.097108m17.76k8.0971089.37104or7.5861013rad/s_2.24pm9.11103151044.5551026kg-m/sh6.625

14、10348(b)hcEp4.55510261.45410mE3.3851.5041.610193.0110J19k221084.32108m11.454108104k4.3252.161013rad/s(b)p9.1110311069.111025kg-m/s6.62510347.2710109.111025mk210108.64109m17.2728.641091068.641015rad/s_2.252n22n21.054103422En2ma229.1110317510102Enn21.06981021Jorn21.06981021En1.61019n23orEn6.68610eVThe

15、nE16.69103eVE22.67102eVE36.02102eV_2.2622n22n21.05410342(a)En2ma229.1110311010102n26.0181020Jorn21020En6.018n20.3761eV1.61019ThenE10.376eVE21.504eVE33.385eV6.625103431083.0110196.604107mor660.4nm_2.27(a)En2n222ma22n21.05410342151032151031.2102215103n22.5381062orn7.6881029(b)En115mJNo_2.28Foraneutron

16、andn1:2221.05410342E12ma221.66102710142or3.30251013J1013E13.30252.06106eV1.61019Foranelectroninthesamepotentialwell:1.054103422E129.111031101426.01771010Jor1010E16.01773.76109eV1.61019_2.29Schrodingerstime-independentwaveequation2x2x2m2EVxx0Weknowthatx0forxaandxa22WehaveaaVx0forx22sointhisregion2x2m

17、Ex0 x22ThesolutionisoftheformxAcoskxBsinkxwherek2mE2Boundaryconditions:x0atxa,xa22Firstmodesolution:1xA1cosk1xwhere22k1aE12ma2Secondmodesolution:2xB2sink2xwhere22k22E24a22maThirdmodesolution:3xA3cosk3xwhere22k33E39a2ma2Fourthmodesolution:4xB4sink4xwhere22k44E416a2ma2_2.30The3-Dtime-independentwaveeq

18、uationincartesiancoordinatesforVx,y,z0is:222x,y,zx,y,zx,y,zx2y2z22mE2x,y,z0Useseparationofvariables,soletx,y,zXxYyZzSubstitutingintothewaveequation,weobtainYZ2XXZ2YXY2Zx2y2z22mE02XYZDividingbyXYZandlettingk22mE2,wefind(1)12X12Y12Zk20Xx2Yy2Zz2Wemayset221Xkx2Xkx2X0Xx2x2SolutionisoftheformXxAsinkxxBcos

19、kxxBoundaryconditions:X00B0andXxa0nxkxnx1,2,3.awhereSimilarly,let2Z12Y212Yy2kyandz2kzZApplyingtheboundaryconditions,wefindkyny,ny1,2,3.akznz,nz1,2,3.aFromEquation(1)above,wehavekx2ky2kz2k20or222k22mEkxkykz2sothat2222EEnxnynz22nxnynz2ma_2.31(a)222mEx,yx,yx,y0 x2y22Solutionisoftheform:x,yAsinkxxsinkyy

20、Wefindx,yxAkxcoskxxsinkyy2x,yAkx2sinkxxsinkyyx2x,yAkysinkxxcoskyyy2x,yAky2sinkxxsinkyyy2Substitutingintotheoriginalequation,wefind:(1)222mE0kxky2Fromtheboundaryconditions,oAsinkxa0,wherea40ASonx,nx1,2,3,.kxaoAlsoAsinkyb0,whereb20ASony,ny1,2,3,.kybSubstitutingintoEq.(1)above22222Enxnynxny2ma2b2(b)Ene

21、rgyisquantized-similarto1-Dresult.Therecanbemorethanonequantumstatepergivenenergy-differentthan1-Dresult._2.32(a)Derivationofenergylevelsexactlythesameasinthetext222E2ma2n2n1Forn22,n11ThenE3222ma2o(i)Fora4A31.054103422E21.6710274101026.1551022JorE6.15510223.85103eV1.61019(ii)Fora0.5cmE31.05410342221

22、.6710270.510223.9391036JorE3.93910362.461017eV1.61019_2.33x0(a)ForregionII,22x2m2EVO2x0 x2Generalformofthesolutionis2xA2expjk2xB2expjk2xwherek22mEVO2TermwithB2representsincidentwaveandA2termwithrepresentsreflectedwave.RegionI,x021x2mEx0 x221Generalformofthesolutionis1xA1expjk1xB1expjk1xwherek12mE2Te

23、rminvolvingB1representsthetransmittedwaveandtheterminvolvingA1representsreflectedwave:butifaparticleistransmittedintoregionI,itwillnotbereflectedsothatA10.Then1xB1expjk1x2xA2expjk2xB2expjk2x(b)Boundaryconditions:(1)1x02x0(2)12xx0 xx0Applyingtheboundaryconditionstothesolutions,wefindB1A2B2k2A2k2B2k1B

24、1Combiningthesetwoequations,wefindk2k1B2A2k1k22k2B2B1k1k2ThereflectioncoefficientisA2A2*k2k12Rk2k1B2B2*ThetransmissioncoefficientisT1RT4k1k2k12k2_2.342xA2expk2x2Pxexp2k2xA2A2*wherek22mVoE229.1110313.52.81.610191.05410341k24.286109mo1010m(a)Forx5A5Pexp2k2xexp24.285910951010o0.0138(b)For10mx15A1510Pex

25、p24.28591091510102.61106o1010m(c)Forx40A40Pexp24.28591094010101.291015_2.35T16E1Eexp2k2aVoVowherek22mVoE229.1110311.00.11.610191.0541034ork24.860109m1(a)Fora41010mT160.110.1exp24.85976109410101.01.00.02951010m(b)Fora12T160.110.1exp24.859761091210101.01.01.24105(c)JNte,whereNtisthedensityoftransmitte

26、delectrons.E0.1eV1.61020J1m219.1110312221.874105m/s1.874107cm/s1.2103Nt1.610191.874107Nt4.002108electrons/cm3Densityofincidentelectrons,Ni4.0021081.3571010cm30.0295_2.36T16E1Eexp2k2aVOVO(a)Form0.067mok22mVOE21/220.0679.1110310.80.21.610191.05410342or109m1k21.027ThenT160.210.20.80.8exp21.027109151010

27、orT0.138(b)Form1.08mok2=10311/221.089.110.80.21.610191.05410342orm1k24.124109ThenT160.210.20.80.8exp24.124109151010or1.27105_T16E1Eexp2k2aVoVowherek22mVoE221.6710271211061.610191.0541034(a)7.2741014m1T16111exp27.2741014101412121.222exp14.5485.875107(b)T105.8751071.222exp27.2741014a27.2741014aln1.222

28、65.87510ora0.8421014m_2.38RegionIx0,V0;RegionII0 xa,VVORegionIIIxa,V0(a)RegionI:1xA1expjk1xB1expjk1x(incident)(reflected)wherek12mE2RegionII:2xA2expk2xB2expk2xwhere2mVOEk22RegionIII:2.373xA3expjk1xB3expjk1x(b)InRegionIII,theB3termrepresentsareflectedwave.However,onceaparticleistransmittedintoRegionI

29、II,therewillnotbeareflectedwavesothatB30.(c)Boundaryconditions:Atx0:12A1B1A2B2d1d2dxdxjk1A1jk1B1k2A2k2B2Atxa:23A2expk2aB2expk2aA3expjk1ad2d3dxdxk2A2expk2ak2B2expk2ajk1A3expjk1aThetransmissioncoefficientisdefinedasA3A3*TA1A1*sofromtheboundaryconditions,wewanttosolveforA3intermsofA1.SolvingforA1interm

30、sofA3,wefindA1jA3k22k12expk2aexpk2a4k1k22jk1k2expk2aexpk2aexpjk1aWethenfind*A3A3*22expk2aA1A14k1k22k2k12expk2a4k12k22expk2aexpk2a2Wehavek22mVOE2IfweassumethatVOE,thenk2awillbelargesothatexpk2aexpk2aWecanthenwrite*A3A3*222A1A14k1k22k2k1expk2a4k12k22expk2a2whichbecomesA1A1*A3A3*2k22k12exp2k2a4k1k2Subs

31、titutingtheexpressionsfork1andk2,wefind222mVOk1k22and222mVOE2mEk1k2222m2VOEE22m2EVO12EVOThen2mVO2*exp2k2a*A3A32A1A12E162mVO12EVOA3A3*E1Eexp2k2a16VOVOFinally,A3A3*E1ET16exp2k2aA1A1*VOVO_2.39RegionI:V021x2mE1x0 x221xA1expjk1xB1expjk1xreflectedincidentwherek12mE2RegionII:VV122x2mEV1x0 x2222xA2expjk2xB2

32、expjk2xtransmittedreflectedwherek22mEV12RegionIII:VV223x2mEV2x0 x2233xA3expjk3xtransmittedwherek32mEV22ThereisnoreflectedwaveinRegionIII.Thetransmissioncoefficientisdefinedas:3A3A3*k3A3A3*TA1A1*k1A1A1*1Fromtheboundaryconditions,solveforA3intermsofA1.Theboundaryconditionsare:Atx0:12A1B1A2B212xxk1A1k1

33、B1k2A2k2B2Atxa:23A2expjk2aB2expjk2aA3expjk3a23xxk2A2expjk2ak2B2expjk2ak3A3expjk3aButk2a2nexpjk2aexpjk2a1Then,eliminatingB1,A2,B2fromtheboundaryconditionequations,wefindk34k124k1k3Tk1k322k1k1k3_2.40VOE,wecanwrite(a)RegionI:Since2x2mVOE11x0 x22RegionII:V0,so22x2mEx0 x222RegionIII:V30Thegeneralsolutionscanbewritten,keepinginmindthat1mustremainfiniteforx0,as1xB1expk1x2xA2sink2xB2cosk2x3x0wherek12mVOEand22mEk22(b)BoundaryconditionsAtx0:12B1B212k1B1k2A2xxAtxa:23A2sink2aB2cosk2a0orB2A2tank2a(c)k1B1k2A2A2k1B1k2andsinceB1B2,then