Density-of-statesplotsgiveusanaturalwaytolookattheeffectofdisorder,thatis,what happensto the electron bands whena crystal is notperfectly periodic. As discussedin Section1.1, bands and band gaps appear wheneverthere is overlap of atomic orbitals, regardlessofperiodicity.
In the longwavelength limit (whenthe characteristic length of the disorderis much longerthan the atomic latticespacing), we canmodel disorder asregions withslightly largerorsmallerspacingbetweenatoms.Wecanthenapproximatetheeffectofthedisorder
byrecalculating the band energy for a largeror smaller lattice spacing in each region.
Larger spacing corresponds toless orbitaloverlap of adjacentatoms, whichmeans less bonding–antibondingsplitting.Thiscorrespondstoasmallerbandgap;inotherwords,the upper,antibondingstateswillhavelowerenergyandthelower,bondingstateswillhave higherenergy.Thismeansthatinaregionoflargerlatticespacing,therewillbeelectron statesinsidethenominalenergygap.
Intheabsenceofanyotherinformation,wecanassumethatthedisorderisdistributed randomly. In the longwavelength limit, we canview the disorderedcrystal asa set of perfectly ordered crystals with band gaps that are distributed according to a Gaussian distribution,accordingtothecentrallimittheorem,
P(Eg) = 1
√2π(±E)e−(Eg(0)−Eg)2/2(±E)2, (1.8.11) where Egisthebandgapforaperfectlyorderedcrystaland ±Egisacharacteristicrange ofenergyfluctuations.Thetotaldensityofstatesofthecrystal willthenbegivenbythe convolutionofthisdistributionwiththedensityofstatesforaperiodicstructure,
D(E) =à
dEgD(E − Eg)P(Eg). (1.8.12) Theeffectoftheconvolutionis tosmearoutthebandgapsofasolid.Disorder doesnot necessarilyeliminatetheexistenceofbandsandbandgaps,however.Figure1.23(b)illus- trateshowasmalldegreeofdisordersmearsthebands,whileleavingthemstillmuchthe same.In general,everyreal crystal hassome degreeof band smearingbecause thereis alwayssomedegreeofdisorder.
E D(E)
(a)
E D(E)
(b)
E D(E)
±Fig.1.23 Densityofstatesoftwoelectronbandsinthepresenceofdisorder.(c) (a)Nodisorder.(b)Lowdisorder.(c)Larger
disorder.Inthislastcase,thebandgapdisappears,althoughthedensityofstatesisstilllowerinthatregion.
43 1.8 DensityofStates
Localized states
Localized states Extended states
(a) (b)
E
E D(E)
x
±Fig.1.24 (a)Randompotentiallandscape,withboundstatesinthepotentialminimaatlowenergy.(b)Mobilityedgeinthe
densityofstates.
A large degree of disorder can cause the tail of one band to overlap with the tail ofanother band, leadingto a continuum of statesfilling the band gap, as illustrated in Figure1.23(c). The density-of-statesplot inthis casegives usmore information thana banddiagram,becauseoneoftencanseepeaksinthedensityofstatesthatareband-like, evenwhentherearenowell-definedbandgaps.
Nearaband edge,thedensityofstatestypicallylookslikeacontinuous functiondue tothesmearingfrom disorder,butthenatureofthestatescanbequite different.Asthe disordergivesavariationofthebandgap,theeffectofthedisorderistocreatelocalpeaks andvalleys inthepotential energyfeltbyanelectron,asillustratedinFig.1.24(a). The deepestvalleyswillingeneralbesofarapartthatthecouplingbetweenthemisnegligible.
Inthiscase,theelectronicstatesinthevalleysare localized stateswithdiscreteenergies, similartothestatesinasquarewell.Wethereforeexpectthatmanyofthelowest-energy statesforelectronswillcorrespondtolocalizedelectronsthatcannotmove.Onlargelength scales, the average over many of these localized states will give a continuous density ofstates.Empirically, thedensityoflocalized statesbelowtheband gapin manysolids isproportional to e−E/EU,where EU issome characteristicenergy.Thisis knownas an Urbach tail.
AswesawinSection1.1,thecouplingbetweenlocalizedstatesincreasescontinuously astheygetnearer toeach other, due totunnelingbetweenthe differentenergy minima.
Therefore, bands of localized statescanarise, just asthey didfor atomic states. These bandscan allowelectron motionacross thecrystal, buttheywillhave lowconductivity becauseoftheirlowdensityofstates.
Sincetrappedstatesbelowthebandedgecanconductelectrons,wemightexpectthatthe electricalconductivitydecreasescontinuouslytowardzeroastheenergyoftheelectrons decreases.Infact,itcanbeproventhatthereisasharpenergycutoff,calledthe mobility edge,belowwhichthestatesdonotcontributetoconductivityatall,asillustratedinFigure 1.24(b).AsdiscussedinSection9.11,quantum interferencecanimplyzeroconductivity inthepresenceofdisorder,aneffectknownas Anderson localization.
Exercise1.8.4 Ifthedistributionofenergygaps inadisordered,three-dimensional solid is givenby(1.8.11), and the densityof statesis givenby (1.8.7),use (1.8.12) to plot thedensity of statesof a disorderedthree-dimensional materialnear a band minimum.