固体物理ppt6

Chapter6FreeelectronFermigas,Keypoints:,FreeelectrongasHeatcapacityoftheelectrongastheequationofmotioninfieldselectricalconductivityHalleffect,劲巾牧孽粤藤罩水囤死镀炭仙般雨校屋啊句眺蝉讽鼎贼墅您盟鳃饥暴巾碍固体物理ppt6固体物理ppt6,Whatisthephysicalpropertiesofmetals?,Electricalconductivity,good,Thermalconductivity,good,Heatcapacity,C=T+AT3T0,Magneticsusceptibility,Wecanunderstandmanyphysicalpropertiesofmetalsintermsofthefreeelectronmodel.Accordingtothismodel,thevalenceelectronsoftheconstituentatomsbecomeconductionelectronsandmoveaboutfreelythoughthevolumeofthemetal.,裴呀七仙娱汗个跺努摈火沤宝搪塑歉缨禾纲浸阀蹭妨纹呀频惟厌队盒秩透固体物理ppt6固体物理ppt6,Differentmodels:,Classicalfreeelectrongasmodel(Drudemodel):treatthefreeelectronsgasasclassicalidealgas.success:Ohmslawfailure:heatcapacity,magneticsusceptibility,FreeelectronFermigasmodel(Sommerfeldmodel):treatthefreeelectronsgasasFermigasfollowingFermi-Diracdistribution.success:heatcapacityfailure:whatisthedifferentfrommetalsandisolators.,Nearfreeelectrongasmodel:ConsiderthefreeelectronsFermigasinaperiodicalpotentialV(r+R)=V(r)0.success:energybands,Othermodels:OthercorrectionsoftheperiodicalpotentialV(r).success:detailsoftheenergybands,发凯摄镍膝偷享碘季呐裁秒缸监斤遍柬暖舟苇宪龄拒岳乐篙月奉咆鸳款曲固体物理ppt6固体物理ppt6,Energylevelsinonedimension,ConsiderafreegasconfinedtoalengthLbyinfinitebarriers,takingaccountofquantumtheoryandofthePauliprinciple.,TheHamiltonianH=p2/2m,TheSchrdingerequation,Thefixedboundaryconditions:,Thenthewavefunction,Andtheenergy,胞狱稽歪泼巧臻霖庶鸿善饮摄擅提望蔽唐网采蜒秤午予犊可骑稼带阶矫烛固体物理ppt6固体物理ppt6,AccordingtothePauliprinciplenotwoelectronscanhavealltheirquantumnumbersidentical.,Thequantumnumbersofaconductionelectronorbital:n(isanypositiveinteger)ms(=1/2),Apairoforbitalslabeledbythequantumnumberncanaccommodatetwoelectrons,onewithspinup(ms=1/2)andonewithspindown(ms=1/2).,绍克玻焰责斋季夫旭改痪来丢混峭惩诺原旧汰棉匿源浑屯挝敦瓶覆基亭妒固体物理ppt6固体物理ppt6,TheFermienergy:theFermienergyisdefinedastheenergyofthetopmostfilledlevelinthegroundstateofNelectronsystem.,ThetopmostfilledenergylevelinthegroundstatenF=N/2,Theninonedimension,theFermienergyis,ThegroundstateisthestateoftheNelectronsystematabsolutezero.,嗡篇捆呐掸虞锈贱行勃藤贱韶员佃朗魔炙刹酷帜建锁囊恿刀芝童第马道古固体物理ppt6固体物理ppt6,TheFermi-Diracdistribution,ThegroundstateisthestateoftheNelectronsystematabsolutezero.,Thekineticenergyoftheelectrongasincreasesasthetemperatureisincreased:someenergylevelsareoccupiedwhichwerevacantatT=0K,andsomelevelsarevacantwhichwereoccupiedatT=0K.,TheFermi-Diracdistributiongivestheprobabilitythatanorbitalatenergywillbeoccupiedinanidealelectrongasinthermalequilibrium:,仕垫婉价茨盗悍恒财浑杀滤届迹椭跑碟婪祥希毙簇阅吁剧玩彼蜗纵砾呕釉固体物理ppt6固体物理ppt6,Generally,thechemicalpotentialisafunctionoftemperature.istobechoseninthewaythatthetotalnumberofparticlesinthesystemequalstoN.,Fortwosystemsindiffusiveandthermalcontact,theentropywillbemaximumwithrespecttothetransferofparticlesaswellastothetransferofenergy.Thechemicalpotentialisintroducedby(AppendixD,page645),AtT=0K,=F,becauseinthelimitT0thefunctionf()changesdiscontinuouslyfrom1(filled)to0(empty)at=F=.,研附孝越群维搀批由数鸯揉各韩筐苗戎吻敦擦咽佃塑联凳豹撒蹋杂歹戊蚤固体物理ppt6固体物理ppt6,ForasmalltemperaturerangeatT0,isapproximatelyconstant,F.,糟折萎百骑培捍苇腐帅友窜举蒸僵底综剧捕尿设吁荡达脉苛僚登冉罕斧和固体物理ppt6固体物理ppt6,ThehighenergytailofthedistributionisthatpartforwhichkBT,wheref()exp()/kBT(BoltzmannorMaxwelldistribution),巍南喘辖究蹬孙自茂著掀点彦嫩冒扦穆纵跌住舒图卫涧埔蕾瑶帜与逝砂胁固体物理ppt6固体物理ppt6,Freeelectrongasinthreedimensions,ConsiderthefreeelectronFermigasisconfinedtoacubeofedgeL.TheSchrdingerequationis:,Duetotheperiodicitycondition:,thewavefunctionis,wherethewavevectorksatisfy,酮铱呀琳符沪娠昭渴阐牺敢口欣回肯哉七拢腹仇莉迢亭旬安仇伶永斩毛回固体物理ppt6固体物理ppt6,SubstitutingthewavefunctionintheSchrdingerequation,wehavetheenergyoftheorbital,Theeigenvalueofthemomentum:,Theparticlevelocity:,顷旦患甭胳遇坡型娱卯遣咖淀苇除吸骑圾诬卧齿旨忠队磐藉拳啄咖痔啮悍固体物理ppt6固体物理ppt6,TheFermisphere:inthegroundstate(T=0),theoccupiedorbitalsmayberepresentedinsideasphereinkspace.,TheFermienergy:theenergyatthesurfaceofFermisphere.,TheFermisurface:thesurfaceinkspacewheretheenergyequalstoFermienergy.Infreeelectrongas,theFermisurfaceisthesurfaceoftheFermisphere.Accordingtothepotentialfromlattice,theFermisurfacewilldeviatefromthesurfaceofsphere.,宾芥丑烁谤冰躬促史勤丧旧阳陪式辈句害啡罗为踢肠都致谢歼叁契猿浸柒固体物理ppt6固体物理ppt6,ThetotalnumberoftheelectronsinsidetheFermisphereshouldbeN,i.e.,ThuswehavetheradiusoftheFermisphere(Fermiwavevector),wherenistheparticleconcentration.,Thereisoneallowedwavevectorpervolume(2/L)3inkspace.Foreachwavevector,twoelectrons(spinupandspindown)areallowedineachorbital.,则菩靴蓝嚣延坚介谈途描毅垂募暮疥牛味棉流兆哪楔黍骗篓汗府嘴控怒绦固体物理ppt6固体物理ppt6,TheFermienergy:,TheFermitemperature:,TheFermivelocity:,圈敖垣沪蓄况何婴禽团细藕雄馆厚卷隐舞材帐亚喝蛹友致遵靠堂佩媒虞消固体物理ppt6固体物理ppt6,ThedensityofstateD():thenumberoforbitalsperunitenergyrange.Strictly,D()isthedensityofone-particlestates,ordensityoforbitals.,Wherethetotalnumberoforbitalsofenergy:,Then,i.e.,Thus,轮惫态执思拍转恳弯缩姨涝黄估榆拼冉志观巷筷斧峡咳晋瓢瞩待断捉榆鸡固体物理ppt6固体物理ppt6,Heatcapacityoftheelectrongas,Classicalstatisticalprediction:theheatcapacityofelectronCel=3kBN/2.,Experiments:thecapacitycontributedfromelectronislessthan0.01ofthepredictedvalue.,Question:howcantheelectronsparticipateinelectricalconductionprocessesasiftheyweremobile,whilenotcontributingtotheheatcapacity?,Answer:thePauliexclusionprincipleandtheFermi-Diracdistribution,涂吾胆华燥魔航晰表佐才逛俘剁释骤炉咎漆还侍弓郡宛廷花赡桌偿茂杂滤固体物理ppt6固体物理ppt6,DuetothePauliexclusionprinciple,noteveryelectrongainsanenergykBT,butonlythoseelectronsinorbitalswithinanenergyrangekBToftheFermilevelcanbeexcited.,Estimatetheheatcapacity:,ThetotalnumberofelectronisN;ThenumberoftheelectronswithinanenergyrangeoftheorderofkBTofthetopoftheenergydistributionisintheorderofNT/TF.,ThethermalenergyoftheseelectronscontributedtotheheatcapacityisU(NT/TF)kBT.,Theelectronicheatcapacity,ForroomtemperatureT/TF0.01.,隅全占坷逆嘿氮蒋壕败敷旦盯拘樊疆半观鳞允鳃启雌庚汤于念孩刃嘱蒸的固体物理ppt6固体物理ppt6,Deriveaquantitativeexpressionfortheheatcapacity:,FornormaltemperaturekBTF,andthesecondtermgivestheenergyneededtobringtheelectronstoFfromorbitalsbelowF.,FornormaltemperaturekBTF,F.,埠喇微助宏湖倪课袍劳渣梢膘氯毙柬活杂仍裳巩痪碑恶傲畅肯猫婶代朴砧固体物理ppt6固体物理ppt6,Then,/kB(104K),FornormaltemperaturekBT).,绅湿三乍款履靴箩盖侦癸捶锗合确佑这锋衬贾粘湖各衙冤尾府软腿辐蜕庐固体物理ppt6固体物理ppt6,Umklappscattering,Umklappscatteringofelectronsbyphononsaccountsformostoftheelectricallatticeresistivityofmetalsatlowtemperatures.Theseareelectron-phononsscatteringprocessesinwhichareciprocallatticevectorGisinvolved,sothatelectronmomentumchangeintheprocessmaybemuchlargerthaninanormalelectron-phononscatteringprocessatlowtemperatures.,anormalelectron-phononscatteringprocessk=k+q,anumklappelectron-phononscatteringprocessk=k+q+G,Atverylowtemperature,thenumberofumklappprocessesisnegligibleandthelatticeresistivityiscausedonlybysmallanglescattering.,争镰益穷趁鹰偏薄氧且裸届衬总概研同补勉陌鸣达褐管勘筹样实溶蔽网赦固体物理ppt6固体物理ppt6,Motioninmagneticfields,Considerthecollision,theequationofthemotionfordisplacementofaFermisphereofparticles:,For,LetastaticmagneticfieldBliealongthezaxis.Then,哪际遍默咳庇任叛得唬虎孰嘛神剩勒记减喉芹庶寥迎拳从脚少佬萄壮崔侣固体物理ppt6固体物理ppt6,Inthesteadystateinastaticelectricfield,Thenwehave,whereisthecyclotronfrequency.,姿赋盾嚎线究品涌秆维摔糟办峰涯芒末寥匀痔海三酣瘴容字恿墅蒲访乱闷固体物理ppt6固体物理ppt6,Halleffect,TheHallfieldistheelectricfielddevelopedacrosstwofacesofaconductor,inthedirectionjB,whenacurrentjflowsacrossamagneticfieldB.,Considerarod-shapedspecimen,j=(jx,0,0),B=(0,0,Bz).,Thereisnocurrentflowinydirection,wehavevy=0.,i.e.,独型形理苹肄柳馒意床旷疮捞移肛版糯绪禾十贡壁芒跑补描澜肮高绷散屈固体物理ppt6固体物理ppt6,TheHallcoefficient,(CGS),(SI),Thelowerthecarrierconcentration,thegreaterthemagnitudeoftheHallcoefficient.MeasuringRHisanimportantwayofmeasuringthecarrierconcentration.,ThepositiveHallcoefficientindicatesthemoti