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Intrinsiccarrierconcentrationinsemiconductors - Galileo

Intrinsiccarrierconcentrationinsemicondu ctorsMelissinos,eq.( ),givestheformula,validatthermalequilibr ium,ni=Nsexp Eg2kBT (1)where,-niistheintrinsiccarrierconcent ration, ,thenumberofelectronsintheconductionband (andalsothenumberofholesinthevalenceband )perunitvolumeinasemiconductorthatiscomp letelyfreeofimpuritiesanddefects-Nsisthe numberperunitvolumeofeffectivelyavailabl estates;itsprecisevaluedependsonthemater ial,butitisoforder1019cm 3atroomtemperatureandincreaseswithtemper ature-Egistheenergygap(betweenthebottomo ftheconductionbandandthetopofthevalenceb and)-kBisBoltzmann'sconstant,kB=1:381 10 23 Joules/Kelvin-Tistheabsolutetemperaturei nKelvin; :Thephysicalbasisofeq.(1)canbeunderstood asfollows:conductionbandTheprobabilityof excitinganelectronfromthetopofthe"valenc ebandtothebottomoftheconductionbandisEgp roportionaltotheBoltzmannfactorexp EgkBT :# (seebelow) , ,theelectron-holerecombinationrateisprop ortionaltotheproductnp, ,weconcludethatnp=Kexp EgkBT (2) ,bydefinition,n=p=ni:Theneq.

The density of states in a solid is similar to that of free particles near the bottom of a band, but decreases back to zero at the top of a band. Here is a sketch of density of states dn=dE vs.

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Transcription of Intrinsiccarrierconcentrationinsemiconductors - Galileo

1 Intrinsiccarrierconcentrationinsemicondu ctorsMelissinos,eq.( ),givestheformula,validatthermalequilibr ium,ni=Nsexp Eg2kBT (1)where,-niistheintrinsiccarrierconcent ration, ,thenumberofelectronsintheconductionband (andalsothenumberofholesinthevalenceband )perunitvolumeinasemiconductorthatiscomp letelyfreeofimpuritiesanddefects-Nsisthe numberperunitvolumeofeffectivelyavailabl estates;itsprecisevaluedependsonthemater ial,butitisoforder1019cm 3atroomtemperatureandincreaseswithtemper ature-Egistheenergygap(betweenthebottomo ftheconductionbandandthetopofthevalenceb and)-kBisBoltzmann'sconstant,kB=1:381 10 23 Joules/Kelvin-Tistheabsolutetemperaturei nKelvin; :Thephysicalbasisofeq.(1)canbeunderstood asfollows:conductionbandTheprobabilityof excitinganelectronfromthetopofthe"valenc ebandtothebottomoftheconductionbandisEgp roportionaltotheBoltzmannfactorexp EgkBT :# (seebelow) , ,theelectron-holerecombinationrateisprop ortionaltotheproductnp, ,weconcludethatnp=Kexp EgkBT (2) ,bydefinition,n=p=ni:Theneq.

2 (2)isequivalenttoeq.(1)withK= ; , ,toagoodapproximation,Ns=2 m kBT2 ~2 3=2= m m 3=2 T300K 3=22:5 1019cm3(3)wherem isaneffectivemassthatisoftheorderoftheel ectronmassm(forSi,m =m=0:543).Nsincreaseswithtemperaturebeca usehigherstatesintheconductionband,andde eperstatesinthevalenceband, ,aroundroomtemperatureitisnotbadtoregard Nsasaconstant,becausethedominantTdepende nceineq.(1) ,oneshouldalsokeepthedependenceofEgonT, (3)thatNsdependsonlyonfundamentalconstan tsandonT,apartfromthefactthatthefreeelec tronmassmshouldbereplacedbym :Ifweaskwhichcombinationofm ;kBT;and~hasthedimensionsofaninverseleng th,wefindthatitis(m kBT)1=2~ ,wegeteq.(3)fromdimensionalanalysisalone , ,butontheaverageitsenergyis12kBT:Theunce rtaintyinmomentumisthen p=(m kBT)1= 'sprinciple,theminimumuncertaintyinposit ionissuchthat p x= ~.Ineffect,everyelectronstateoccupiesani ntervalofwidth ,perunitlength,is1 x= p ~=(m kBT)1=2 ~(4)Formotioninthreedimensions,wemustsim ilarlyconsiderthateachelectronstateoccup iesthevolume( x) (4),anddoublingittoaccountforspinupandsp indown,wegetthenumberofeffectivelyavaila blestatestoafreeelectronattemperatureTas 2 m kBT= 2~2 3=2;veryclosetothecorrectNsofeq.

3 (3).TherigorouscalculationofniTheprobabi lityoffindinganelectroninastateofenergyE ;attemperatureT;isgivenbytheFermifunctio nf=1exp E EFkBT +1(5) , (dn=dE)dEthenumberofavailablestatesperun itvolumewithenergybetweenEandE+dE;andwec alldn= (6) , (ineV)foratypicalsemiconductor,showingav alenceband(dashed)between0and24,abandgap between4and5,andpartofaconductionband(do tted) ,weassumeatfirstthatthedensityofstatesne arthetopofthevalencebandisthemirrorimage ofthedensityofstatesnearthebottomoftheco nductionband; ,alongwiththeFermilevelatEF=4:5eV(dash-d otline),andtheFermifunctionf(solidline), forkBT=0:1eV:(Thiscorrespondstothetemper atureof1160K,toohighforpracticaldevices, butgoodforacleardrawing.) (dotted),alongwithf(thinsolidline).These arethefactorsintheintegrandofeq.(6).Also shownistheproductofthesetwofactors,which isthedensityofoccupiedstates(thickline). , (3), ,Ec;isfarabovetheFermienergy,comparedtok BT(inthegraph,Ec=5eVandEc EF=0:5eV);hencewecanusef'exp E EFkBT (7)Physically,thismeansthattheFermifacto rfcanbeapproximatedbyaBoltzmannfactor, (2m )3=22 2~3(E Ec)1=2(8)ClearlyitisconvenienttouseE Ecasavariabletodealwithstatesintheconduc tionband,sowewriteeq.

4 (7)intheformf'exp Ec EFkBT exp E EckBT andsubstitutingbackineq.(6)weobtainn=exp Ec EFkBT (2m )3=22 2~31 ZEc(E Ec)1=2exp E EckBT dETheintegralgives(kBT)3=2p =2:Intheintrinsicsemiconductorthatwecons ider,Ec EF=Eg=2,whereEg=Ec m kBT2 ~2 3=2exp Eg2kBT (9)4asanticipatedineqs.(1)and(3).Wemusts tilljustifyeq.(8).Wecanstartfromeq.( )inMelissinos,whichgivesarelation,forfre eelectrons,betweentheenergyEandthenumber ofstatesperunitvolumehavingenergylesstha nE:E=~22m 3 2n 2=3 InvertingthisrelationbetweennandEwefind3 2n= 2mE~2 3=2 Differentiating,weobtainthedensityofstat esforfreeelectrons:dndE=(2m)3=22 2~3E1=2(10)Statesnearthebottomofthecondu ctionbandrepresentelectronsthatarenearly freetomovefromatomtoatomwithaneffectivem assm ; Ecandmwithm ineq.(10),weobtaineq.(8).Ingeneral,theel ectronsnearthebottomoftheconductionbandm oveaboutwithaneffectivemassmcwhilethehol esnearthetopofthevalencemoveaboutwithadi fferenteffectivemassmv:Asaresult,theintr insicFermilevelisnotexactlyinthemiddleof thegap,buteq.

5 (9)isstilltrue,providedthatweusem =(mcmv)1=2:Therelevantformulasaregivenin Melissinos,page11,withNc=2 mckBT2 ~2 3=2Nv=2 mvkBT2 ~2 3=2andNs=(NcNv)1=2:5

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