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PrinciplesofSemiconductorDevicesTitlePageTableofContentsHelp B.VanZeghbroeck,2011

Chapter2:SemiconductorFundamentals

2.4Densityofstates2.4.1.Calculationofthedensityofstates2.4.2.Calculationofthedensityofstatesin1,2and3dimensions

Beforewecancalculatethedensityofcarriersinasemiconductor,wehavetofindthenumberofavailablestatesateachenergy.Thenumberofelectronsateachenergyisthenobtainedbymultiplyingthenumberofstateswiththeprobabilitythatastateisoccupiedbyanelectron.Sincethenumberofenergylevelsisverylargeanddependentonthesizeofthesemiconductor,wewillcalculatethenumberofstatesperunitenergyandperunitvolume.

2.4.1Calculationofthedensityofstates

ThedensityofstatesinasemiconductorequalsthedensityperunitvolumeandenergyofthenumberofsolutionstoSchrdinger'sequation.Wewillassumethatthesemiconductorcanbemodeledasaninfinitequantumwellinwhichelectronswitheffectivemass,m*,arefreetomove.Theenergyinthewellissettozero.ThesemiconductorisassumedacubewithsideL.Thisassumptiondoesnotaffecttheresultsincethedensityofstatesperunitvolumeshouldnotdependontheactualsizeorshapeofthesemiconductor.Thesolutionstothewaveequation(equation1.2.14)whereV(x)=0aresineandcosinefunctions:

(2.4.1)WhereAandBaretobedetermined.Thewavefunctionmustbezeroattheinfinitebarriersofthewell.Atx=0thewavefunctionmustbezerosothatonlysinefunctionscanbevalidsolutionsorBmustequalzero.Atx=L,thewavefunctionmustalsobezeroyieldingthefollowingpossiblevaluesforthewavenumber,kx.

(2.4.2)

Thisanalysiscannowberepeatedintheyandzdirection.Eachpossiblesolutionthencorrespondstoacubeinkspacewithsizen/LasindicatedonFigure2.4.1.

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Figure2.4.1: Calculationofthenumberofstateswithwavenumberlessthank.Thetotalnumberofsolutionswithadifferentvalueforkx,kyandkzandwithamagnitudeofthewavevectorlessthankisobtainedbycalculatingthevolumeofoneeighthofaspherewithradiuskanddividingitbythe

volumecorrespondingtoasinglesolution, ,yielding:

(2.4.3)

Afactoroftwoisaddedtoaccountforthetwopossiblespinsofeachsolution.Thedensityperunitenergyisthenobtainedusingthechainrule:

(2.4.4)

ThekineticenergyEofaparticlewithmassm*isrelatedtothewavenumber,k,by:

(2.4.5)

Andthedensityofstatesperunitvolumeandperunitenergy,g(E),becomes:

(2.4.6)

Thedensityofstatesiszeroatthebottomofthewellaswellasfornegativeenergies.Thesameanalysisalsoappliestoelectronsinasemiconductor.Theeffectivemasstakesintoaccounttheeffectoftheperiodicpotentialontheelectron.Theminimumenergyoftheelectronistheenergyatthebottomoftheconductionband,Ec,sothatthedensityofstatesforelectronsintheconductionbandisgivenby:

(2.4.7)

Example2.3 Calculatethenumberofstatesperunitenergyina100by100by10nmpieceofsilicon(m*=1.08m0)100meVabovetheconductionbandedge.WritetheresultinunitsofeV1.

Solution Thedensityofstatesequals:

Sothatthetotalnumberofstatesperunitenergyequals:

2.4.2Calculationofthedensityofstatesin1,2and3dimensions

Wewillherepostulatethatthedensityofelectronsinkspaceisconstantandequalsthephysicallengthofthesampledividedby2andthatforeachdimension.Thenumberofstatesbetweenkandk+dkin3,2and1dimensionthenequals:

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(2.4.8)

Wenowassumethattheelectronsinasemiconductorareclosetoabandminimum,Eminandcanbedescribedasfreeparticleswithaconstanteffectivemass,or:

(2.4.9)

EliminationofkusingtheE(k)relationabovethenyieldsthedesireddensityofstatesfunctions,namely:

(2.4.10)

forathreedimensionalsemiconductor,

(2.4.11)

Foratwodimensionalsemiconductorsuchasaquantumwellinwhichparticlesareconfinedtoaplane,and

(2.4.12)

Foraonedimensionalsemiconductorsuchasaquantumwireinwhichparticlesareconfinedalongaline.Anexampleofthedensityofstatesin3,2and1dimensionisshowninthefigurebelow:

Figure2.4.2: Densityofstatesperunitvolumeandenergyfora3Dsemiconductor(bluecurve),a10nmquantumwellwithinfinitebarriers(redcurve)anda10nmby10nmquantumwirewithinfinitebarriers(greencurve).m*/m0=0.8.

Theabovefigureillustratestheaddedcomplexityofthequantumwellandquantumwire:Eventhoughthedensityintwodimensionsisconstant,thedensityofstatesforaquantumwellisastepfunctionwithstepsoccurringattheenergyofeachquantizedlevel.Thecaseforthequantumwireisfurthercomplicatedbythedegeneracyoftheenergylevels:forinstanceatwofolddegeneracyincreasesthedensityofstatesassociatedwiththatenergylevelbyafactoroftwo.Alistofthedegeneracy(notincludingspin)forthe10lowestenergiesinaquantumwell,aquantumwireandaquantumbox,allwithinfinitebarriers,isprovidedinthetablebelow:

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Figure2.4.3: Degeneracy(notincludingspin)ofthelowest10energylevelsinaquantumwell,aquantumwirewithsquarecrosssectionandaquantumcubewithinfinitebarriers.TheenergyE0equalsthelowestenergyinaquantumwell,whichhasthesamesize

Next,wecomparetheactualdensityofstatesinthreedimensionswithequation(2.4.10).Whilesomewhattedious,theexactnumberofstatescanbecalculatedaswellasthemaximumenergy.TheresultisshowninFigure2.4.4.Thenumberofstatesinanenergyrangeof20E0areplottedasafunctionofthenormalizedenergyE/E0.Adottedlineisaddedtoguidetheeye.Thesolidlineiscalculatedusingequation(2.4.10).Acleardifferencecanbeobservedbetweenthetwo,whiletheyareexpectedtomergeforlargevaluesofE/E0.

Figure2.4.4: NumberofstateswithinarangeE=20E0asafunctionofthenormalizedenergyE/E0.(E0isthelowestenergyina1dimensionalquantumwell).Seetextformoredetail.

AcomparisonofthetotalnumberofstatesillustratesthesametrendasshowninFigure2.4.5.Herethesolidlineindicatestheactualnumberofstates,whilethedottedlineisobtainedbyintegratingequation(2.4.10).

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Figure2.4.5: NumberofstateswithenergylessthanorequaltoEasafunctionofE0(E0isthelowestenergyinan1dimensionalquantumwell).Actualnumber(solidline)iscomparedwiththeintegralofequation(2.4.10)(dottedline).

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