固體物理英文版課件：Chapter 7 Energy Bands

Chapter7 EnergyBandsKeypoints:

NearlyFreeelectronmodelEnergybandsBlochtheoremandBlochfunctionsCentralequationEnergybandsnearBrillouinzoneboundaryMetalsandinsulatorsTheproblemsofthefreeelectronFermigasmodel:Thedistinctionbetweenmetals,semiconductors,andinsulatorsPositivevalueofHallcoefficientTherelationofconductionelectronsinmetalstothevalenceelectronsoffreeatomsManytransportpropertiesTounderstandthedifferencebetweeninsulatorsandconductors,wemustextendthefreeelectronmodeltotakeaccountoftheperiodiclatticeofthesolid.Thepossibilityofabandgapisthemostimportantnewpropertythatemerges.MetalSemimetalSemiconductorInsulatorEnergyEnergybandValencebandConductionbandEnergygap(bandgapsorforbiddenband)NearlyfreeelectronmodelOnthefreeelectronFermigasmodel,wehaveThewavefunctionsarewhere,fortheperiodicboundaryconditionsTheenergyThemomentumConsidertheperiodicpotentialsinthelatticeasaperturbation,wehavethe

nearlyfreeelectronmodel.where .絕熱近似：假設(shè)晶體中原子實(shí)是固定不動(dòng)，則晶體中的周期勢(shì)場(chǎng)是不隨時(shí)間變化的。單電子近似：忽略價(jià)電子間的相互作用，則價(jià)電子的運(yùn)動(dòng)是相互獨(dú)立的，其波函數(shù)可以用單電子在周期勢(shì)場(chǎng)中的波函數(shù)來(lái)描述。ForanelectronInsidetheBrillouinzone,bytheperturbationtheorywehavewithThenwehaveAttheBrillouinzoneboundary,wehaveThenwehavetheeigenfunctionwithTheSchr?dingerequationisMultiplyandfromleftsiderespectively,withSet

thenWehaveThehomogenouslinearequationshaveasolutiononlyifthedeterminantofthecoefficientsoftheunknownA,Bvanishes.ThereforeTheperiodicpotentialofthelatticecausestheenergygapattheBrillouinzoneboundary.Thelowenergyportionsofthebandstructurein(b)aresimilartofreeelectronsin(a),butwithanenergygapattheBrillouinzoneboundary.DuetotheBraggreflection,attheBrillouinzoneboundary,thewavefunctionsoftheelectronsarenottravelingwaves,butarestandingwavesmadeupofequalpartsofwavestravelingtotherightandtotheleft.Inalinearlatticewiththelatticeconstanta,thestandingwavesare:Braggreflectionofelectronwavesincrystalsisthecauseofenergygaps.TheBraggreflectionisacharacteristicfeatureofwavepropagationincrystals.Thetwostandingwave(+)and()pileupelectronsatdifferentregion.Thereforethesetwowavehavedifferentvaluesofthepotentialenergy.Thisistheoriginoftheenergygap.Forwhichconcentratestheelectronsonthepositiveioncenterswherethepotentialenergyislowest.Forwhichconcentratestheelectronsawayfromthepositiveioncoreswherethepotentialenergyishighest.OriginoftheenergygapMagnitudeoftheenergygapWhenwecalculatetheaveragevalueofthepotentialenergyoverthesethreechargedistributions,wefindthatthepotentialenergyof(+)islowerthanthatofthetravelingwavewhereasthepotentialenergyof()ishigherthanthatofthetravelingwave.ThewidthoftheenergygapEgisthepotentialdifferenceof(+)and().SupposethepotentialenergyofanelectroninthecrystalisThenwehaveThegapisequaltotheFouriercomponentofthecrystalpotential.BlochfunctionsTheBlochtheorem:Theeigenfunctionsofthewaveequationforaperiodicpotentialaretheproductofaplanewavetimesafunctionwiththeperiodicityofthecrystallattice.whereThesolutionsoftheSchr?dingerequationforaperiodicpotentialmustbeofaspecialform:Theone-electronwavefunction iscalledaBlochfunctionandcanbedecomposedintoasumoftravelingwaves.Blochfunctionscanbeassembledintolocalizedwavepacketstorepresentelectronsthatpropagatefreelythroughthepotentialfieldoftheioncores.TheproofoftheBlochtheoreminonedimensionSupposeNidenticallatticepointsofthelatticeconstanta.thepotentialenergyisperiodicina,U(x+sa)=U(x).Theeigenfunctionsofthewaveequationforaperiodicpotentialarek.

Duetothelatticetranslationsymmetry,wehavewhereCisaconstant.Considertheperiodicboundarycondition,ThenThereforewehavewithk=2s/Na.TheproofoftheBlochtheoremDefineatranslationoperationT,where(=1,2,3)istheprimitivetranslationvector.Onecanprove:andThereforeTandHhavesameeigenfunctions.Suppose(r)istheeigenfunctionofTandH.whereEandaretheeigenvaluesofHandTrespectively.Duetotheperiodicboundarycondition:ThenThusSetThenDefineafunction:ThenThereforeThecentralequationConsideralinearlatticewiththelatticeconstanta.ThepotentialenergyU(x)isinvariantunderacrystallatticetranslation.WecanexpandU(x)asaFourierseriesinthereciprocallatticevectorsG.Inone-electronapproximation,theSchr?dingerequationisThewavefunction

(x)canbeexpressesasthesumoverallvaluesoftheplanewavepermittedincrystal,wherek=2n/Lduetotheperiodicboundarycondition.SubstitutethewavefunctionintotheSchr?dingerequation:i.e.Therefore

withthenotationthecentralequationk(x)isthewavepacketwhichisalinearcombinationoftheplanewaveswiththewavecectors

k+G.Inprinciple,thereareaninfinitenumberofCktobedetermined.HoweverinpracticeasmallnumberofCkwillsuffice.ThecentralequationistheSchr?dingerequationexpressedinthereciprocalspace.Herewehaveasetofalgebraicequationsinsteadofthedifferentialequation.RestatementoftheBlochtheoremThewavefunctionisgivenaswherewedefineSupposeTisacrystallatticevector,TG=2s.DiscussionsabouttheBlochtheorem(1)TheBlochfunctiondoesnothavethesameperiodicityasthelattice,i.e.Asprovedbefore,GenerallykisnotareciprocallatticevectorG,ThereforeHoweverHence(2)TheBlochfunctionhasthesameperiodicityasthereciprocallattice:SetG”=G’GFromtheSchr?dingerequation,onehasandTherefore(3)Ifthelatticepotentialvanishes,U(x)=0.Thecentralequationreducesto(k)Ck=0,sothatallCkGarezeroexceptCk,anduk(r)isconstant.Thuswehave(4)ThecrystalmomentumofanelectronThequantitykentersintheconservationlawsthatgoverncollisionprocessesincrystal.ThecrystalmomentumofanelectronIfanelectronabsorbsinacollisionaphononofwavevector,theselectionruleisSolutionofthecentralequationForthepotentialenergythecentralequation representsasetofsimultaneouslinearequations.Theseequationsareconsistentifthedeterminantofthecoefficientsvanishes.Thedeterminantinprincipleisinfiniteinextent,butinpracticeasmallnumberofCkwillsuffice.ThevaluesofthecoefficientsUGforactualcrystalpotentialstendtodecreaserapidlywithincreasingmagnitudeofG.SupposethatthepotentialenergyU(x)containsonlyasingleFouriercomponentUg=Ug=U,wheregdenotestheshortestG.Takefivesuccessiveequationsofthecentralequation.ThenthedeterminantofthecoefficientsisgivenbyForagivenk,thesolutionofthedeterminantgivesasetofenergyeigenvalues

nk,whichlieondifferentbands.Kronig-PenneymodelAssumetheperiodicpotentialisthesquare-wellperiodicpotential.Asshowninthefigure,inoneperiodthesquare-wellpotentialisTheperiodicityofU(x)isa+b.ThewaveequationisIntheregion0<x<a,U(x)=0,theeigenfunctionisalinearcombinationofplanewaveswiththeenergyIntheregionb<x<0,U(x)=U0,theeigenfunctioniswithFromtheBlochtheorem,TheconstantsA,B,C,andDarechosensothatandd/dxarecontinuousatx=0andx=a.Atx=0Atx=aThedeterminantofthecoefficients:IfwerepresentthepotentialbytheperiodicdeltafunctionandsetP=Q2ba/2,InthelimitQ>>KandQb<<1,theequationreducestoNote:Kisnotthewavevector

koftheBlochfunction.Kronig-PenneymodelinreciprocalspaceWeusetheKronig-Penneymodelofaperiodicdeltafunctionpotential:WhereAisaconstantandathelatticespacing.Thesumofsisoverallatomsinaunitlength,whichmeansover1/aatoms.ThusWehavethecentralequation:hereG=2n/a.DefineThenSumbothsideovernThenwehavewherewewriteThenthefinalresultisOtherapproximationsEmptylatticeapproximationActualbandstructuresareusuallyexhibitedasplotsofenergyversuswavevectorinthefirstBrillouinzone.Whenwavevectorshappentobegivenoutsidethefirstzone,theyarecarriedbackintothefirstzonebysubtractingasuitablereciprocallatticevector.Consideranemptylattice,U(r)=0.Theenergiesareapproximatedbythefreeelectronenergies .Howevertheplanewaveismodulatedbythelattice.TheelectronenergyinaemptylatticeiswithkinthefirstBrillouinzoneandGallowedtorunoverappropriatereciprocallatticepoints.e.g.thelow-lyingfreeelectronbansofasimplecubiclatticealong[100]direction.ApproximatesolutionnearazoneboundarySupposeForawavevectorexactlyattheBrillouinzoneboundary,sothatatthezoneboundarythekineticenergyofthetwocomponentwavesk=G/2areequal,i.e.G/2=G/2.SupposeCG/2areimportantcoefficientsatthezoneboundaryandneglectallothers.Thenwehaveonlytwoequationswithk=G/2:HereForanontrivialsolutionwhenceTheradiooftheC:ThenthewavefunctionatthezoneboundaryisHereonesolutiongivesthewavefunctionatthebottomoftheenergygap,andtheothergivesthewavefunctionatthetopoftheenergygap.SolutionsnearthezoneboundaryG/2Withthesametwocomponentapproximation,thewavefunctionis:FromthecentralequationwehaveThedeterminantequalszero:TheenergyIfweexpandtheenergyintermsofaquantity intheregion :AttheBrillouinzoneboundaryThereforeNumberoforbitalsinabandEachprimitivecellcontributionsexactlyoneindependentvalueofktoeachenergyband.Withaccounttakenofthetwoindependentorientationsoftheelectronspin,th