258 CHAPTER16. LIVEWIRESANDDEADSTARS
level,butatthispointwemustinvoketheExclusionPrinciple:Therecanbenomore
thanoneelectroninanygivenquantumstate. Thustherecanamaximumoftwo
electrons(spinupandspindown)atanyallowedenergyinanenergyband.
Atthelowestpossibletemperature(T= 0 K),theelectrons’configurationisthe
lowestpossibleenergyconsistentwiththeExclusionPrinciple. AperfectInsulator
isacrystalinwhichtheelectronscompletelyfilloneormoreenergybands,andthere
isagapinenergyfromthemostenergeticelectrontothenextunoccupiedenergy
level.InaConductor,thehighestenergybandcontainingelectronsisonlypartially
filled.
Inanappliedelectricfieldtheelectronsinacrystalwilltendtoaccellerate,and
increasetheirenergy. But...theycanonlyincreasetheirenergyifthereare(nearby)
higherenergystatesavailable,forelectronstooccupy. Iftherearenonearbyhigher
energystates,asinaninsulator,nocurrentwillflow(unlesstheappliedfieldisso
enormousthat electronscan”jump”acrosstheenergygap). Inaconductor,there
areanenormousnumberofnearbyenergystatesforelectronstomoveinto.Electrons
arethereforefreetoaccellerate,andacurrentflowsthroughthematerial.
Theactualphysicsofconduction,inarealsolid,isofcoursefarmorecomplex
than thislittle calculation would indicate. Still, the Kronig-Pennymodel doesa
remarkablejobofisolatingtheessentialeffect,namely, theformationof separated
energybands,whichisduetotheperiodicityofthepotential.
16.2 The Free Electron Gas
IntheKronig-Penneymodel,theelectronwavefunctionshaveafree-particleformin
theintervalbetweentheatoms;thereisjustadiscontinuityinslopeatpreciselythe
positionoftheatoms. Inpassingtothe three-dimensionalcase,we’llsimplify the
situationjustabitmore,byignoringeventhediscontinuityinslope. Theelectron
wavefunctionsarethenentirelyofthefreeparticleform,withonlysomeboundary
conditionsthat needto beimposedat thesurface of thesolid. Tossingawaythe
atomic potential meanslosing the energy gaps; thereis only one”band,” whose
energies aredetermined entirely by the boundary conditions. For some purposes
(suchasthermodynamicsofsolids,orcomputingthebulkmodulus),thisisnotsuch
aterribleapproximation.
WeconsiderthecaseofNelectronsinacubicalsolidoflengthLonaside.Since
theelectronsareconstrainedtostaywithinthesolid,butweareotherwiseignoring
atomicpotentialsandinter-electronforces,theproblemmapsdirectlyintoagasof
non-interactingelectronsinacubicalbox. Insidethebox,theSchrodingerequation
foreachelectronhasthefreeparticleform
−
̄h^2
2 m∇^2 ψ(x,y,z)=Eψ(x,y,z) (16.27)