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1、5.10Metals,Insulators,andSemiconductorsSolidsaredividedintotwomajorclasses:1.Metals:areconductorsbecauseanelectriccurrentflowswhenanelectricfieldisapplied;2.Insulators:nocurrentflowswhenanelectricfieldisapplied.Acriterionfordistinguishingbetweenthetwoclases:•Abandwhichiscompletely
2、filledcarriesnoelectriccurrent,eveninthepresenceofanelectricfield.Applyingthisstatementtothesodiumatom:•innerbands1s,2s,2parefullyoccupied=>theycarrynocurrent;•the3svalencebandcanaccomodate2Ncelectrons,Nc=#ofunitscells;•inNa,aBravaisbcclattice,eachcellhasoneatomcontributingoneva
3、lenceelectron:total#ofvalenceelectrons=Nc=>bandisonlyhalffilled(Fig.5.21(a):•sodiumisametalbecauseitsvalencebandisonlypartiallyfilled.Bysimilararguments,theotheralkalis:•Li,K,etc.,arealsometalsbecausetheirvalencebandsareonlypartiallyfull.Diamond(carbon)isaninsulator:•topbandorig
4、inatesfromhybridizationofthe2sand2patomicstates(§A.8):•=>twobandssplitbyanenergygap(Fig.5.21(b)),andsince•unitcellhastwoatoms=>eachofthesebandscanaccomodate8Ncelectrons=>eachdiamondatomcontributes4electrons=>8valenceelectronspercell:•diamondisaninsulatorbecausevalencebandiscom
5、pletelyfull.Semiconductor:ifgapbetweenvalenceband(normallyfilled)andbandimmedi-atelyaboveislessthanabout2eV=>electronscanbethermallyexcited=>bothbandspartiallyfilled=>smallconductivityresults[Fig.5.21(c)].285.11DensityofStatesThedensityofstatesforelectronsinabandisdefinedas:g(E)d
6、E=#ofelectronsstatesperunitvolumeintheenergyrange(E,E+dE).Toevaluateg(E):•drawashellink-spacewithinnersurfacedeterminedbyE(k)=EandoutersurfacedeterminedbyE(k)=E+dEasshowninFig.5.22.Then:•g(E)=#ofk-valueslyinginsidethisshelldividedbytheshellthicknessdE.Evidently:•g(E)dependsont
7、heshapeoftheenergycontours,andhenceonbandstructrure.Ifdispersionrelationhasthestandardform:h¯2k2E=(60)2m∗thenenergycontoursareconcentricspheressurroundingtheorigin(seeshellAinFig.5.22),andhence[usingEq.(60)]:!2m∗"3/221/24πkdk=4πEdE=sphericalshellvolume.2h¯Since#ofallowedk-valu
8、esperunitvolumeofk-spaceis1/(2π)3,wehave:4πk2dk1!2m∗"3/21/2#o
