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NotesontheWiedemann-FranzLawMarkLundstromPurdueUniversityJanuary3,2018
Historically,“TheWiedemann-FranzLawstatesthatformetalsatnottoolowtemperaturestheratioofthethermalconductivitytotheelectricalconductivityisdirectlyproportionaltothetemperaturewiththevalueoftheconstantofproportionalityindependentoftheparticularmetal.”[1].KittelwritestheWiedemann-Franz(WF)Lawas[1]
κ eσ
= L0T =π 2
3kBq
⎛
⎝⎜⎞
⎠⎟
2
T , (1)
where L0 istheLorenznumber(whichisactuallyaratio,notanumber).AsKittelnotes,thefactthat(1)canbederivedfromtheelectrongastheoryofmetalsandthatitappliestoawiderangeofmetalsunderawiderangeofconditions,wasamajorsuccessintheearlyhistoryofthetheoryofmetals[1].BothKittelandZiman[2]pointout,however,thattheLorenznumber(whichZimancallstheLorenzratio)canchangeformetalsatlowtemperature.ThisisusuallyattributedtoabreakdownoftheRelaxationTimeApproximation(RTA)–seeAshcroftandMerman[3],pp.322,323.)Itissometimesstatedthatthescatteringtimesintheelectricalandthermalconductivitiesmaybedifferent[1].Assummarizedineqns.(6),below,however,whensolvingtheBTEintheRTA,thereisonlyasinglescatteringtimethatdeterminestheelectricalandthermalconductivity(itappearsinthemean-free-path, λ E( ) in(6e)).AsdiscussedbyAshcroftandMerman,thebreakdownoftheWFlawatlowtemperaturesisnottheresultofdifferentscatteringtimesfortheelectricalandthermalconductivities,but,rather,theduetothefactthatthesamescatteringhasdifferenteffectsontheelectricalandthermalconductivities([3]footnoteonp.323).Moregenerally,wecanalwayswrite
κ eσ
≡ LT (2)
andsimplyregarditasthedefinitionoftheLorenznumber.AccordingtoZiman,weshouldonlyrefertothisrelationastheWiedemann-FranzLawwhen
L = L0 = π
2 3( ) kB q( )2 [3].Ontheotherhand,itiscommontoreferto(2)asthe“Wiedemann-FranzLaw”andapplyitnotonlytometalsbutalsotosemiconductorsforwhichLisrarely L0 .See,forexample[4],whichalsopointsoutthatweshouldregardtheWiedemenn-FranzLawtobea“ruleofthumb”andnotalawofnature.Evenformetallicconditions,violationsoftheWiedemann-FranzLawoccur(see[5]foroneexample),andforsemiconductors, L israrelyequalto L0 .ThecalculationoftheLorenznumberrequiresanaccuratebandstructureandknowledge
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oftheenergy-dependentscatteringprocesses[6].Inthesenotes,wediscusscalculationsforasimple,parabolicenergybandwithpower-lawscattering.Wewillconsider(2)tobethedefinitionoftheLorenznumber,whichcanbewrittenas([7],p.96)
L =kBq
⎛⎝⎜
⎞⎠⎟
2E − EFkBTL
⎛
⎝⎜⎞
⎠⎟
2
−E − EFkBTL
⎛
⎝⎜⎞
⎠⎟
2⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪, (3)
wheretheaverage, i ,isdefinedbyeqn.(5.51)in[7].Foramaterialwithasingleenergy
channelat E = E1 , M E( ) = M0δ E − E1( ) ,anditiseasytoshowmathematicallyfromeqn.(3)that L = 0 .Thephysicalreasonisclear.Ifthereisasinglechannel,thenwhenweopen-circuitittomeasure κ e ,noelectronsflow.Sincethereisnoflowofelectrons,therecanbenoflowofheatIn3D,foraconstantmean-free-pathandparabolicenergybands,wefindforanon-degeneratesemiconductor
L ≈ 2
kBq
⎛⎝⎜
⎞⎠⎟
2
(4)
andforadegeneratesemiconductor,
L ≈ π
2
3kBq
⎛⎝⎜
⎞⎠⎟
2
. (5)
Whatarethecorrespondingrelationsfor1Dand2Dsemiconductorswithparabolicenergybands?LorenzNumberin3D,2D,and1DTocomputeLfrom(2)wemustcalculatethethermoelectriccoefficients:
σ = ′σ E( )∫ dE (6a)
S = −
kBq
E − EFkBT
⎛⎝⎜
⎞⎠⎟
′σ E( )∫ dE ′σ E( )∫ dE (6b)
κ 0 = T
kBq
⎛⎝⎜
⎞⎠⎟
2E − EF
kBT⎛
⎝⎜⎞
⎠⎟
2
′σ E( )∫ dE (6c)
κ e =κ 0 − S2σT (6d)
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where
′σ E( ) = 2q
2
hλ E( ) M E( )A −
∂ f0∂E
⎛⎝⎜
⎞⎠⎟= q2Ξ E( ) − ∂ f0∂E
⎛⎝⎜
⎞⎠⎟. (6e)
From(2)and(6d),wefind
L =
κ eσT
=κ 0 − S
2σTσT
=κ 0σT
− S 2 . (7)
TofindL,wesimplyneedtocompute κ e andσ .In(6e), Ξ E( ) isknownasthe“transportfunction”or“transportdistribution”[8].Ingeneral,thecomputationofLrequiresnumericalcalculations[6],butforparabolicenergybandswithpowerlawscattering,analyticalresultsarepossible(seetheappendixof[7]).1) Lorenznumberin3DforparabolicbandsFromtheAppendixof[7]:
κ e = T
kBq
⎛⎝⎜
⎞⎠⎟
22q2
hλ0
m*kBT2π2
⎛
⎝⎜⎞
⎠⎟Γ r + 3( ) r + 3( )F r+2 ηF( )−
r + 2( )F r+12 ηF( )F r ηF( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪ (8)
σ = 2q
2
hλ0
m*kBT2π!2
⎛
⎝⎜⎞
⎠⎟Γ r + 2( )F r ηF( ) , (9)
wherepowerlawscatteringhasbeenassumed:
λ E( ) = λ0 E − ECkBT
⎛
⎝⎜⎞
⎠⎟
r
. (10)
TheresultingLorenznumberin3Dis
L =
κ eσT
=kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 3( )Γ r + 2( ) r + 3( )
F r+2 ηF( )F r ηF( )
−r + 2( )F r+12 ηF( )
F r2 ηF( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪. (11)
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Casei)Maxwell-BoltzmannStatistics
L =kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 3( )Γ r + 2( ) r + 3( )− r + 2( ){ }
=kBq
⎛⎝⎜
⎞⎠⎟
2 r + 2( )!r +1( )!
(12)
For r = 0 ,wefind
L = 2
kBq
⎛⎝⎜
⎞⎠⎟
2
(13)
asexpected.Equation(13)istheWFlawin3Dforparabolicenergybands,energy-independentscattering,andMaxwell-Boltzmannstatistics.Notethatthemean-free-pathisproportionaltovelocitytimesscatteringtime.ForAcousticDeformationPotential(ADP)scattering,thescatteringrate(oneoverthescatteringtime)isproportionaltothedensity-of-states, D E( ) .Forparabolicbands, D E( )∝ E − EC( )
d−2( ) 2 ,whered=1,2,or3for1D,2D,or3D.ForADPscattering,wefind
λ E( )∝υ E( )τ E( )∝υ E( ) D E( )∝ E − EC( )3−d( ) 2 . (14)
ForADPscatteringin3D, r = 0 ,forADPscatteringin2D, r = 1 2 ,andforADPscatteringin1D, r = 1 .Theassumptionofaconstantmean-free-pathinthe3DcalculationsaboveisequivalenttoassumingADPscattering.Caseii)StronglydegenerateconditionsInthiscase,wemustexpandtheFermi-Diracintegralsfor ηF >> 0 .FromJeongandLundstrom,p.101[7]
F r ηF( )→ ηF
r+1
Γ r + 2( ) +π 2
6ηF
r−1
Γ r( ) + ... Therestisleftasanexerciseforthereader.
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2) Lorenznumberin2DforparabolicbandsFromtheAppendixof[7]:
κ e = TkBq
⎛⎝⎜
⎞⎠⎟
22q2
hλ0
2m*kBTπ
⎛
⎝⎜⎜
⎞
⎠⎟⎟Γ r + 5
2⎛⎝⎜
⎞⎠⎟
r + 52
⎛⎝⎜
⎞⎠⎟F r+3/2 ηF( )−
r + 32
⎛⎝⎜
⎞⎠⎟F r+1/2
2 ηF( )F r−1/2 ηF( )
⎧
⎨⎪⎪
⎩⎪⎪
⎫
⎬⎪⎪
⎭⎪⎪
(15)
and
σ = 2q2
hλ0
2m*kBTπ
⎛
⎝⎜⎜
⎞
⎠⎟⎟Γ r + 3
2⎛⎝⎜
⎞⎠⎟F r−1/2 ηF( ) . (16)
TheLorenznumberin2Dis
L =κ eσT
=kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 52
⎛⎝⎜
⎞⎠⎟
Γ r + 32
⎛⎝⎜
⎞⎠⎟
r + 52
⎛⎝⎜
⎞⎠⎟F r+3/2 ηF( )F r−1/2 ηF( )
−r + 3
2⎛⎝⎜
⎞⎠⎟F r+1/2
2 ηF( )F r−1/2
2 ηF( )
⎧
⎨⎪⎪
⎩⎪⎪
⎫
⎬⎪⎪
⎭⎪⎪
. (17)
Casei)Maxwell-BoltzmannStatistics
L =kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 52
⎛⎝⎜
⎞⎠⎟
Γ r + 32
⎛⎝⎜
⎞⎠⎟
r + 52
⎛⎝⎜
⎞⎠⎟− r + 3
2⎛⎝⎜
⎞⎠⎟
⎧⎨⎩
⎫⎬⎭
=kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 52
⎛⎝⎜
⎞⎠⎟
Γ r + 32
⎛⎝⎜
⎞⎠⎟
(18)
For r = 0 (energy-independentmean-free-path),wefind
L =
kBq
⎛⎝⎜
⎞⎠⎟
2 Γ 5 2( )Γ 3 2( ) =
kBq
⎛⎝⎜
⎞⎠⎟
2 32Γ 3 2( )Γ 3 2( )
L = 3
2kBq
⎛⎝⎜
⎞⎠⎟
2
(2DconstantMFP) (19)
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Equation(19)istheLorenznumberin2Dforparabolicenergybands,energy-independentscattering,andMaxwell-Boltzmannstatistics.Weseethatthenumericalfactorof3/2isdifferentthanthefactorof2inthe3Dcase.If,however,weassumeADPscatteringin2D,then r = 1/ 2 and(18)gives
L =
kBq
⎛⎝⎜
⎞⎠⎟
2 Γ 3( )Γ 2( ) = 2
kBq
⎛⎝⎜
⎞⎠⎟
2
(2DADPscattering) (20)
whichisidenticaltotheresultfor3DADPscattering.Caseii)StronglydegenerateconditionsInthiscase,wemustexpandtheFermi-Diracintegralsfor ηF >> 0 .3) Lorenznumberin1DforparabolicbandsFromtheAppendixof[7]:
κ e = T
kBq
⎛⎝⎜
⎞⎠⎟
22q2
hλ0 Γ r + 2( ) r + 2( )F r+1 ηF( )−
r +1( )F r2 ηF( )F r−1 ηF( )
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪ (21)
σ = 2q
2
hλ0 Γ r +1( )F r−1 ηF( ) (22)
TheLorenznumberin1Dis
L =
κ eσT
=kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 2( )Γ r +1( ) r + 2( )
F r+1 ηF( )F r−1 ηF( )
−r +1( )F r2 ηF( )F r−1
2 ηF( )⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪ (23)
Casei)Maxwell-BoltzmannStatistics
L =kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 2( )Γ r +1( ) r + 2( )− r +1( ){ }
=kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 2( )Γ r +1( )
(24)
For r = 0 (energy-independentmean-free-path),wefind
L =
kBq
⎛⎝⎜
⎞⎠⎟
2 Γ 2( )Γ 1( ) =
kBq
⎛⎝⎜
⎞⎠⎟
2 1( )!0( )!
L = 1
kBq
⎛⎝⎜
⎞⎠⎟
2
(1DconstantMFP) (25)
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Equation(25)istheLorenznumberfor1Dwithparabolicenergybands,energy-independentscattering,andMaxwell-Boltzmannstatistics.Weseethatthenumericalfactorof1isdifferentthanthefactorof2inthe3Dcaseandthefactorof3/2in2D.If,however,weassumeADPscatteringin1D,then r = 1and(24)gives
L =
kBq
⎛⎝⎜
⎞⎠⎟
2 Γ r + 2( )Γ r +1( ) =
kBq
⎛⎝⎜
⎞⎠⎟
2 Γ 3( )Γ 2( ) = 2
kBq
⎛⎝⎜
⎞⎠⎟
2
(1DADPscattering) (26)
whichisidenticaltotheresultfor3DADPscattering.Caseii)StronglydegenerateconditionsInthiscase,wemustexpandtheFermi-Diracintegralsfor ηF >> 0 .Discussion:IfweconsidertheLorenznumberin3Dasgivenby(11),in2Dasgivenby(17),andin1Dby(23)andassumeADPscatteringineachcase,soin3D, r = 0 ,in2D, r = 1 2 ,and1D, r = 1 wefindthesameanswerin1D,2D,3D:
L = 2 3
F 2 ηF( )F 0 ηF( )
−2F 1
2 ηF( )F 0
2 ηF( )⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
kBq
⎛⎝⎜
⎞⎠⎟
2
. (27)
ForparabolicenergybandswithADPscattering,theLorenznumberisidenticalatanylevelofdegeneracy.Thisresultcouldhavebeenanticipated.Thetransportdistribution, Ξ E( ) in(6e)determinesallofthetransportcoefficients.Thetransportdistributionisproportionalto M E( )λ E( ) .Themean-free-pathisproportionaltovelocitytimesscatteringtime,andthescatteringtimeisinverselyproportionaltotheDOS.ThenumberofchannelsisproportionaltovelocitytimestheDOS[7],soforparabolicbands,wefind
Ξxx E( )∝ M E( ) A( )× λ E( )∝υ E( )D E( )×υ E( ) D E( )∝υ 2 E( )∝ E − EC( ) . (28)SincethetransportfunctionisindependentofdimensionforADPscatteringinparabolicbands,allthermoelectrictransportcoefficientsareindependentofdimension.
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References[1] C.Kittel,IntroductiontoSolidStatePhysics,4thed.,p.263,JohnWileyandSons,1971[2] J.Ziman,PrinciplesoftheTheoryofSolids,p.200,CambridgeUniversityPress,1964.[3] N.W.AshcroftandN.D.Mermin,SolidStatePhysics,2003.[4] G.D.MahanandM.Bartkowiak,"Wiedemann–Franzlawatboundaries,"Appl.Phys.
Lett.,74,p.953,1999.[5] A.Casian, "Violationof theWiedemann-Franz law inquasi-one-dimensional organic
crystals,"Phys.Rev.B,81,155415,2010.[6] XufengWang,JesseMaassen,MarkLundstrom,“OntheCalculationofLorenz
NumbersforComplexThermoelectricMaterials,”https://arxiv.org/abs/1710.03711,October13,2017.
[7] C.Jeong,R.Kim,M.Luisier,S.Datta,andM.Lundstrom,"OnLandauerversus
Boltzmannandfullbandversuseffectivemassevaluationofthermoelectrictransportcoefficients,"023707.J.ofAppl.Phys.,107,2010.
[8] G.D. Mahan and J.O. Sofo, "The best thermoelectric," Proc. National Academy of
Sciences,93,pp.7436-7439,1996.
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