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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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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.