1

NotesonBipolarThermalConductivity

MarkLundstromPurdueUniversityOctober26,2017

1. Electronicthermalconductivity………………………………..12. Lorenznumber…………………………………………………………..33. Physicalpictureofbipolarthermalconduction………..44. References………………………………………………………………….8

1.ElectronicthermalconductivityBasicequations:

σ = ′σ E( )dE

−∞

+∞

∫ =σ n +σ p (1a)

S = − 1qT

E − EF( ) ′σ E( )dE−∞

+∞

∫

′σ E( )dE−∞

+∞

∫ (1b)

ST = − 1

qTE − EF( ) ′σ E( )dE

−∞

+∞

∫ = STn + STp (1c)

κ 0 =

1q2T

E − EF( )2′σ E( )dE

−∞

+∞

∫ =κ 0n +κ 0 p (1d)

κ e =κ 0 −Tσ S 2 . (1e)Thequantities,σ , ST ,and κ 0 add;ifwehavetwobands,wejustaddthecontributionfromeachband.Thequantities, S and κ e donotadd.Itiseasytoshowthat

S =

σ nSn +σ pSp

σ n +σ p

. (2)

2

From(1e)wefind:

κ e =κ 0n +κ 0 p −T σ n +σ p( ) σ nSn +σ pSp

σ n +σ p

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

. (3)

Nowwritethisas

κ e = κ 0n −Tσ nSn( ) + κ 0 p −Tσ pSp( )+ T

σ n +σ p( ) − σ nSn +σ pSp( )2

+ σ n +σ p( )σ nSn + σ n +σ p( )σ pSp⎡⎣⎢

⎤⎦⎥. (4)

Theelectronicthermalconductivityfortheconductionbandaloneis

κ e =κ 0n −Tσ nSn , (5a)andforthevalencebandalone,

κ p =κ 0 p −Tσ pSp , (5b)

Using(5a)and(5b)in(4):

κ e =κ en +κ ep +T

σ n +σ p( ) ×

− σ nSn +σ pSp( )2

+ σ n +σ p( )σ nSn2 + σ n +σ p( )σ pSp

2⎡⎣⎢

⎤⎦⎥

(6)

κ e =κ en +κ ep +T

σ n +σ p( ) ×−σ n

2Sn2 −σ p

2Sp2 − 2σ nσ pSnSp +σ n

2Sn2 +σ pσ nSn

2 +σ p2Sp

2 +σ nσ pSp2⎡⎣ ⎤⎦

κ e =κ en +κ ep +

Tσ n +σ p( ) × −2σ nσ pSnSp +σ pσ nSn

2 +σ nσ pSp2⎡⎣ ⎤⎦

3

κ e =κ en +κ ep +T

σ nσ p

σ n +σ p( ) × −2SnSp + Sn2 + Sp

2⎡⎣ ⎤⎦

κ e =κ en +κ ep +Tσ nσ p

σ n +σ p( ) Sp − Sn( )2. (7)

Thisisouranswer.Theelectronicthermalconductivityconsistsofacontributionduetotheconductionbandalone,anothercontributionduetothevalencebandalone,andabipolarcontribution.Itiseasytoseethatthebipolarcontributionisimportantonlywhen

σ n ≈σ p .2.LorenzNumberTheLorenznumberwegetincludingbipolarconductionis:

L = κ e

σT (8)

L = κ en

σ nTσ n

σ n +σ p( ) +κ ep

σ pTσ p

σ n +σ p( ) +σ nσ p

Sp − Snσ n +σ p

⎛

⎝⎜⎞

⎠⎟

2

Thefirsttwotermsin(8)areeasytounderstand–theyrepresenttheweightedcontributionsfromeachofthetwobands.Thethirdtermrequiressomediscussion.Toestimatethemagnitudeofthebipolarcontributions,assumethat

σ n ≈σ p ,then

L = κ en

2σ nT+

κ ep

2σ pT+Sp − Sn( )24

Wealsoknowthatatmid-gap:

Sp ≈ −Sn = S ≈EG

2qT.

ThenormalizedLorenznumberis

′L = L

kB q( )2= κ en

2σ nT kB q( )2+

κ ep

2σ pT kB q( )2+ EG

2kBT⎛⎝⎜

⎞⎠⎟

2

4

SothebipolarcontributiontothenormalizedLorenznumberis

′Lbipolar =

EG

2kBT⎛⎝⎜

⎞⎠⎟

2

. (9)

3.Physicalpictureofbipolarthermalconduction

J =σ

dEF

dx−σ S

dTdx (10)

JQ = Tσ S

dEF

dx−κ 0

dTdx (11)

Case1:Holesonly.Underopen-circuitconditions:

J p =σ p

dEF

dx−σ pSp

dTdx

= 0 (12)

dEF

dx= Sp

dTdx (13)

Insert(13)in(11)

JQ = Tσ pSp

2 dTdx

−κ 0

dTdx

= Tσ pSp2 −κ 0

⎡⎣ ⎤⎦dTdx

JQ = − κ 0 −Tσ pSp

2⎡⎣ ⎤⎦dTdx

= −κ e

dTdx

. (14)

ThephysicalpictureisshowninFig.1below.Theelectricalcurrentiszero,buttheheatcurrentisnotzero.Hotterholesmovetotheleft,andcoolerholesmovetotheright.

5

Fig.1. Illustrationofholecurrentflowunderelectricallyopencircuitconditions

withatemperaturegradientapplied.HolediffusiondownthetemperaturegradientanduptheQFLgradientpreciselybalanceelectricallybutnotthermally.

Nowconsiderwhathappenswhentherearebothelectronsandholes.From(10):

J p + Jn = σ p +σ n( ) dEF

dx− σ pSp +σ nSn( ) dT

dx= 0 (15)

dEF

dx=

σ pSp +σ nSn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

(16)

ThisgradientoftheQFLdrivesboththeholeandelectroncurrents,anditdependsonboththeholeandelectronconductivities.Compare(16)to(13).Since Sn isnegative,thenumeratorof(6)isreduced,andthedenominatorisalsolarger,sothegradientoftheQFLisreduced.Accordingly,thefirsttermontheRHSof(12)isreduced,sonow

J p ≠ 0 ; J p < 0. Thereisanetholecurrenttotheleft.

Whatdirectionistheelectroncurrent?Fortheelectroncurrent,webeginwith(10)fortheconductionband:

Jn =σ n

dEF

dx−σ nSn

dTdx

(17)

Nowuse(16)in(17):

Jn =σ n

σ pSp +σ nSn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪−σ nSn

dTdx

(18)

TheelectroncurrentdependsontheholeconductivityandholeSoretcoefficient,becausetheydeterminethegradientoftheQFL.From(18)

6

Jn =

σ nσ pSp

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

+σ n

2Sn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

−σ pσ nSn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

−σ n

2Sn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

Jn =

σ nσ p

σ p +σ n

Sp − Sn( ) > 0 (19)

Theelectroncurrentispositive;electronsflowtotheleft.Nowuse(16)in(12)toexaminetheholecurrent.

J p =σ p

σ pSp +σ nSn

σ p +σ n

⎛

⎝⎜

⎞

⎠⎟

dTdx

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪−σ pSp

dTdx

J p = −

σ nσ p

σ p +σ n

Sp − Sn( ) dTdx

= −Jn (20)

Theholecurrentisnegative;holesalsoflowtotheleft.Thephysicalpictureisshownbelow–bothelectronsandholesflowtotheleftandrecombineatthecontact,butthecurrentsareinoppositedirections,sotheycancel.

Fig.2 Illustrationofholeandelectroncurrentflowunderelectricallyopencircuitconditionswithatemperaturegradientapplied.MinoritycarriershaveloweredtheQFLgradientssothatholediffusiondownthetemperaturegradientanduptheQFLgradientnolongerpreciselybalanceelectrically.Holesflowtotheleftcontactatpreciselythesamerateatwhichelectronsflowtotheleftcontact.Inthecontact,electronsandholesrecombine,givingupanenergyalittlelargerthanthebandgap.Thisprocessconstitutesthebipolarheatconduction.

7

Discussion:WearemostinterestedinextrinsicsamplesdopedtooperateatmaximumzT.Consideranextrinsic,p-typesample:

σ p >>σ n .Equation(20)becomes

J p = −σ n Sp − Sn( ) dT

dx= −Jn . (21)

Theminoritycarrierconductivitycontrolsbothcurrents.When σ n → 0 , J p → 0 ,as

consideredinCase1.ThephysicsofbipolarheatconductionwasillustratedinFig.2.Howcanweunderstandthisbetter?Recalltheelectronicthermalconductivityfrom(7):

κ e =κ en +κ ep +T

σ nσ p

σ n +σ p( ) Sp − Sn( )2. (22)

Theheatcarriedbythebipolarcomponentis

JQ

bip = −κ bip

dTdx

= −Tσ nσ p

σ n +σ p( ) Sp − Sn( )2 dTdx

. (23)

Howdoesthisrelatetotheelectricalcurrent?From(20),wefind

J p = −

σ nσ p

σ p +σ n

Sp − Sn( ) dTdx

, (24)

whichcanbeusedin(13)towrite

JQbip

J p

= T Sp − Sn( ) , (25)

whichhasanicephysicalinterpretationasdiscussedbelow.RecallthattheSeebeckcoefficientisrelatedtotheaverageenergyatwhichcurrentflows, EJ ,withrespecttotheFermilevel:

S = −

EJ − EF

qT. (26)

Fortheconductionband,

Sn = −

EC + Δn − EF

qT. (27a)

andforthevalenceband,

Sp = −

EV − Δ p − EF

qT. (27b)

8

Here, Δn ishowfarabovethebottomoftheconductionbandcurrentflows( Δn = 2kBT foranon-degeneratesemiconductor,and

Δ p ishowfarbelowthetopofthevalencebandcurrentflows.)Using(27a)and(27b)in(25),wefind

JQbip

J p

=EG + Δn + Δ p( )

q (28)

or

JQ

bip = EG + Δn + Δ p( ) J p q( ) (29)

Thephysicalinterpretationisthatthefluxofelectronsandholestothecontact(

Jn q = − J p q )timestheenergygivenupwhentheyrecombine(slightlylargerthanthebandgap)representsbipolarheatflow.

References

G.BuschandM.Schneider,“HeatConductioninSemiconductors,”PhysicaXX.No.11,p.1084,1954AmsterdamConferenceonSemiconductors,1954.H.FrohlichandC.Kittel.“RemarkonthePaperbyProf.G.Busch,”PhysicaXX.No.11,p.1086,1954AmsterdamConferenceonSemiconductors,1954.H.J.Goldsmid,“TheThermalConductivityofBismuthTelluride,”Proc.Phys.Soc.,LXIX,2-B,1955.AndrewF.May,EricS.Toberer,AliSaramat,andG.JeffreySnyder,“Characterizationandanalysisofthermoelectrictransportinn-typeBa8Ga16−xGe30+x,”PhysicalReviewB80,125205,2009.ShanyuWang1,JiongYang1,TrevorToll,JihuiYang,WenqingZhang,andXinfengTang,“Conductivity-limitingbipolarthermalconductivityinsemiconductors,”ScientificReports,5,10136,DOI:10.1038/srep10136,2015.LibinZhang,PenghaoXiao,LiShi,GraemeHenkelman,JohnB.Goodenough,andJianshiZhou,“SuppressingthebipolarcontributiontothethermoelectricpropertiesofMg2Si0.4Sn0.6byGesubstitution,”JournalofAppliedPhysics,117,155103,2015.MischaThesbergandHansKosina,“OntheLorenznumberofmultibandmaterials,”PhysicalReview,B,95,125206,2017.