Introduction to the Quantum Theory of Solids2.pdf

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1、Chapter3IntroductiontotheQuantumTheoryofSolidsWeknowjxecosxjsinxjxecosxjsinx1jxjx1jxjxcos()xee,sin()xee2j21xx1xxcoshxee,sinhxee22coshjxcosxsinhjxjsinxcosjxcoshxsinjxjsinhx22()sinasinbcosacosbcoskab.3(19)2.3(19)

2、22()sinasinjbcosacosjbcoskab2jj22()sinasinhbcosacoshbcoskab.3(21)21Assumebathusaba,andVo2m2VoEbremainsfiniteb很小22mE2因為22mVoE2222m因此2Vo其中(3.21)這個部分，(22)sinasinhb2分子分母同乘以ab(22)absinasinhb

3、2abmVoba12Assumebathusaba,andVo(3.19)(3.19)進一步簡化mVobasinacosacoska(3.22)2amVobaDefineP'2跟位能阱的寬度有關的常數sinaP'cosacoska(3.24)aCase.1freeelectronV(x)potentialVo,0'P0cosacoskax稍早前有對作定義2mEkpf2wherepfmomentumoffree

4、particlemomentumoffreeelectronandp2k2Efff2m2m電子能量對應電子之機率波的空間頻率作圖，呈現連續能量。TheEnergyEiscontinuousversuskforfreeelectron.Case.2PeriodicalpotentialV(x)potentialxa+bGraphicalmethodsina1coska1'Pcosacoska.3(24)aPlotlookingforandktostatisfyplotsina'Pv.s.aacosav.s.acoska

5、v.s.asina'Pv.s.aacosav.s.a圖(a)+圖(b)1coska1PlotE-kdiagramka2a2coska12mE2ifisgivencoska1a.182a.0733isEalsogivenka222Ev.s.k2mka2a0.82coska1coska1coska0aa.1540ka有解ka2有解0ka/有解a/k2a/有解.....22PlotEv.s.k2m220.730.73k0時，,

6、因此Ea2ma220.820.82k時,,因此Ea2a2maE22k時,,因此Eaa2mak對k而言每隔2/a會有相同的物理意義。coskacoska2mcoska2mm0,1,2,3....Figure11showsthecaseinwhichtheentireEversuskplotiscontainedwithin-/a

7、igure3.11DirectBandgapandIndirectBandgapElectroninfreespaceomomentumpmkoooV(x)potentialxElectroninGaAsMaterialpkV(x)potentialGaAswherepxa+bcrystalmomentummomentumofelectronGaAswherekissolvedbySchrodingesr'equationocrystalmomentum