Multilayer Thermionic Refrigerator and Generator

Generator

G.D.Mahan,J.O.Sofo[*],andM.BartkowiakDepartmentofPhysicsandAstronomy

UniversityofTennessee,Knoxville,TN,37996-1200,and

SolidStateDivision

OakRidgeNationalLaboratoryOakRidge,TN37831-6030

February1,2008

Abstract

Anewmethodofrefrigerationisproposed.Coolingisobtainedbytherm-ionicemissionofelectronsoverperiodicbarriersinamultilayergeometry.ThesecouldbeeitherSchottkybarriersbetweenmetalsandsemiconduc-torsorelsebarriersinasemiconductorsuperlattice.Thesamedeviceisanefficientpowergenerator.Acompletetheoryisprovided.

1

Anewmethodisproposedforrefrigerationandpowergeneration.Thedevicesarecomposedofmultilayersofperiodicbarriers.Thecurrentsareperpendiculartothebarriers.Suchdevicesarepresentlyunderconsiderationforthermoelectriccoolingandpowergeneration.Hereweproposethatthesamedevicescanbeusedforefficientthermioniccooling,orpowergeneration.Whatisthedifferencebetweenathermoelectricdevice[1,2]andatherm-ionicone[3,10]?Theirdescriptionsareremarkablysimilar.Inbothcasesoneputsatemperaturegradientonasemiconductor.Bothdevicesarebasedupontheideathatelectronmotioniselectricity.Buttheelectronmotionalsocarriesenergy.Forcingacurrenttransportsenergyforboththermionicandthermoelectricdevice.Thebasicdifferenceseemstobewhetherthecurrentflowisballisticordiffusive.Inthermionicmotion,thedevicehasrelativelyhighefficiencyiftheelectronsballisticallygooverandacrossthebarrier.Theycarryalloftheirkineticenergyfromoneelectrodetotheother.Inthermoelectricdevices,themotionofelectronsisquasiequilibriumanddiffusive.OnecandescribetheenergytransportbyaSeebeckcoefficient,whichisanequilibriumparameter.Therehavealsobeenrecenttheoriesofnonequilibriumthermoelectriceffects[11].Theyseemtobemidwaybetweenthethermionic(ballistic)andthermoelectic(quasiequilibrium)regimes.Wenotethatmanysemiconductorsuperlattices,withshortperiods,arebeingmade[12]andmeasuredalongthec-axis.Theseperiodsaresoshortthatelectronmotionoverthebarrierisprobablyballistic.Wesuggestthatthesedevicescoolbythermioniceffectsratherthanthermoelectricones.

Earlierwesuggested[3]thatthermionicdevicescouldbeusedasrefrig-erators.Fortheusualdeviceoftwometalplatesseparatedbyanairgap,coolingatroomtemperaturerequiresalowworkfunction.Nometalshavevaluesthatlow(φ∼0.3eV).Numerousgroupshavesuggested[4]-[10]thatsuchsmallbarriersareeasilyattainableinsemiconductorsystems.Herethebarriersaresemiconductors,whiletheelectrodescouldbeeithermetalsorothersemiconductors.Weconsideredthisgeometryinouroriginalpaper,butthoughtthatthethermalconductivityofthesolidwouldbeamajorobstacle.Hereweshowthatthethermaleffectscanbedealtwithbygoingtoamultilayergeometry.Weshowthatsuchdeviceshaveefficiencieswhichcouldbetwicethoseofthermoelectricones.

Thephysicsbehindthermioniccoolingissimple.Mostphysicistsarefamiliarwiththetechniqueofcoolingliquidhelium-4bypumpingthevaporfromthecryostat.Themostenergeticheliumatomsleavetheliquidand

2

becomegasmolecules.Pumpingthemawayremovestheseenergeticatoms,therebycoolingtheliquid.Inthermionicrefrigeration,oneusesavoltagetosweepawaythemostenergeticelectronsfromthesurfaceofaconductor.Thoseelectronswithsufficientenergytoovercometheworkfunctionaretakenawaytothehotsideofthejunction.Removingtheenergeticelectronsfromthecoldsidecoolsit.Chargeneutralityismaintainedatthecoldsidebyaddingelectronsadiabaticallythroughanohmiccontact.

Thermionicdevicesmusthavetheelectronsballisticallytraversethebar-rierinordertohaveahighefficiency.Thisrequiresthatthemean-free-pathλoftheelectroninthebarrierbelongerthanthewidthLofthebarrier.ThisconstrainsthebarrierwidthLtoberathersmall.Thisfactisakeyfeatureoftheanalysis.Thegeneralconstraintsare

λ>L>LtLt=

h¯

eφ

(1)

R1=2RI+

R1

L

Ke

(3)(4)

Thefirsttermintheenergycurrentistheelectronpartfromthermionicemis-sion.Itisgivenbelow.Thesecondtermisthephononpart,whichcontainsthethermalresistanceR1foronebarrier.Thethermalresistancedependsuponthethickness(Le)andthermalconductivity(Ke)oftheelectrodes,aswellastwointerface(’Kapitza’)termsRI[13,14].Thelargesttermwillusu-allybeL/Kforthesemiconductorbarrier.TheproblemisthatLissmallwhichmakesR1smallwhichmakesδT/R1big.Thisistheproblemwiththe

3

thermalconductivity.ItcanbeovercomebyhavingδTbesmall.Thisisthereasonforthemultilayergeometry.EachbarriercansupportonlyasmalltemperaturedifferenceδTi.Amacroscopictemperature∆TisobtainedbyhavingNlayerssothat∆T=N󰀉δTi󰀊.MostofourmodelingassumesthatδTihasanaveragevalueof1-2◦C.Belowweshowthatadequatecoolingpowerisavailableforthesevalues.

1

1.1

Refrigeration

SingleBarrier

ThefirststepinthederivationistosolveforthecurrentsoverasinglebarrierofwidthL.Weassumethebarrierisaconstantatzeroappliedvoltage,whichmeansithasasquareshape.AtanonzerovoltageδVthebarrierhastheshapeofatrapezoid.Theformulasfortheelectrical(J)andheat(JQ)currentsaregivenintermsofthehotandcoldtemperatures(Th,Tc)[3]

JRj=ATj2e−eφ/kBTjA=

2emkB

(5)

R1

δT=Th−Tc

(8)(9)

Theseequationsareforasinglebarrier.Eqn.(5)isthestandardRichardson’sequation[15]forthethermioniccurrentoveraworkfunctioneφwhichinthiscaseistheSchottkybarrierheightbetweenthemetalandsemiconductor.Alternately,itisthebarrierinasemiconductorquantumwell.ThefactorofTdenotesthefractionofelectronstransmittedfromthemetaltothesemiconductor.Itiscalculatedusingquantummechanicalmatchingofthewavefunctions.TheformulasforJandJQassumethebiaseδVistolowertheFermilevelonthehotside,sothatthenetflowofelectronsisfromcoldtohot.

Forelectronsthechargeeisnegativewhichmakesφ,Jalsonegative.Wefindthisconfusingtotreat,sowetakeeandφaspositiveasifthesystem

4

wereaholeconductor.ThenJ>0forparticleflowtotheright,whichweareassuming.

WeshowthatforasinglelayertheoptimalvalueofappliedbiaseδV∝δTwhichisalsosmall.DenoteasTthemeantemperatureofthelayer,andthenTc=T−δT/2,Th=T+δT/2.Thenweexpandtheaboveformulasforthecurrentsinthesmallquantities(δT/T,eδV/kBT)andfind,aftersomealgebra

J=

eJR

kBT

eVJ=kBδT[b+2]eVQ=kBδTqq=b+2+u

2+Z

u=

kBR1JR

ekB2π2h¯3

=Z0eb

T

󰀇2

(12)(13)

(14)(15)

(17)(18)

Z0=(kBTR)

2

=

=

δVJ

kBT(b+2)

δV(δV−VJ)5

(20)(21)

WenowvaryδVintheaboveequationtofindthevalueofδV=Vmwhichgivesthemaximumefficiencyηmwhichis

Vm=VQ+

󰀄

ηˆ(φ)=

δT󰀇ηˆ

b+2q+

√(23)

theelectronballisticallycrossesthebarrierregion,andthenlosesanamountofenergyeδViintheelectrode.Theheatgeneratedateachelectrodemustflowoutofthesampleaccordingtotheequation

d

Li

(25)

whereLiistheeffectivewidthoneonebarrierplusonemetalelectrode.WetakeLitobeaconstant,althoughitcouldvarywithi.ThewidthofthedeviceisD=NLi.Innormalusagetherewouldbeanegativesignontheright-handsideofthisequation.ItisabsentsincewechangedthesignofJ.ThefunctionschangeslowlywithiandwetreatthemascontinuousvariablesδTi/Li=dT/dxandδVi/Li=dV/dx.ThevariationofδViuponiisunknown,butweguessthatitisproportionaltoδTi.Introducethedimensionlessfunctionv(x)

eδVi=kBδTiv

δTi

J=JR

(26)

uJR

I=dx

=

Iu

(29)

ITyZ

󰀉

RN

e

=NR1.

RN(32)(33)

ThevariableI(x)isintroducedineqn.(29,30).Aftertryingvariousalterna-tives,wedecideditistheconvenientvariableforthepresentproblem.Theparameteryisadimensionlesscurrent.Eventuallywevarythisparametertofindtheoptimalefficiency.Thethermalresistanceoftheentiredeviceis

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denotedasRN.Itsvalueisirrelevantsinceitocursinywhichisavariationalparameter.Useeqn.(29)toevaluateδViin(26)andthisresultisinsertedintoeqn.(25)

JQc

=JQ=dx

(I)=dx

=

J∆J

󰀈VkBTc

J(φ+2kBT

−yZyITZ

iZ(Ti)

eN

󰀉

󰀂

Ti

(37)

comparabletothethermionicresultswhenTR=500K.ThusonewantstooperatethethermionicdevicewithvaluesofTRlessthan500K.

ThefactorofTRisdeterminedbythethermalresistivity.Reducingthethermalresistivitybyafactorof25onlyreducestheefficiencybyafactorof5.ThustheefficiencyscaleswithTRratherthanwithR1.Thisisquitedifferentthaninthermoelectricdevices,wheretheresultisalmostdirectlyproportionaltothethermalresistivity.Noteineqn.(35)thatZentersintheformulainthecombinationofZ/(2+Z)whichsaturatesatlargevaluesofZ.TheincreaseisZ0isoffsetbyalowervalueofbarrierheightφ.

Theexactresultsshowthatthethermionicrefrigeratorhasabout30%oftheidealCarnotefficiency∆T/TcforvaluesofTR=200-300K.OurcalculationsshowthattheefficiencyslowlydeclineswithincreasingvaluesofTR.ThevalueofφwherethemaximumefficiencyoccursalsodeclineswithincreasingvalueofTR.

2MultilayerThermionicPowerGeneration

Powergenerationcanbeunderstoodbyconsideringasimpledevicewithonebarrierbetweentwoelectrodes.Iftheelectrodesareatdifferenttempera-tures,andifthereisnoinitialvoltagebetweentheelectrodes,thenelectronsarethermallyexcitedoverthebarrier.Anetelectronflowgoesfromhottocold.Iftheelectrodesareinsulated,thentheelectrodesbecomechargedandthesystemdevelopsanopencircuitvoltage,whichopposestheflowofelectronsandreducesittozero.Connectingtheelectrodestoanexternalcircuitcausescurrenttoflow.Powercanbeextractedfromthedevice.Thebehaviorisidenticalinconcepttoasolarcell.Heretheefficiencyiscalcu-latedusingtheusualdefinition:itistheexternalpowerJ∆VdividedbytheheatextractedfromthehotelectrodeJQh.

2.1OneBarrier

Againthebarriersareassumedtobethinnerthanthemean-free-pathoftheelectrons,sothatonecanapplytheformulasofthermionicemission.ThereisasmalltemperaturedropδTandvoltagedropδVacrossthebarrier.The

9

efficiencyofasinglebarrierusingeqns.(10)–(14)is

η=

e

δV−VQ

(40)

ThevoltageδVisvariedtofindthevalueδVmwhichgivesthemaximumefficiencyηm

δVm=ηm=

KBδTδT

q(√

u)

(41)

(eφ+2kBTh)(1−Ih)

(43)

Theseequationsweresolvedonthecomputer.WeassumedthatTc=300KandTh=400K,sothat∆T=100K.Figure3showstheefficiencyasafunctionofbarrierheightforseveralvaluesofTR.Thisshouldbecomparedwithathermoelectricgeneratorwhichhasanefficiencyofη=0.048forthesameoperatingtemperatures.ClearlythethermionicdeviceismoreefficientforsmallvaluesofTR.Figure4showsthetemperatureandδViprofilesalongthethermionicdevice,assumingthatN=100.

10

3ThermoelectricAnalogy

Thermionicdevicesarenotthermoelectricdevices.However,itisusefultoexaminetheanalogybetweenthetwokindsofdevices.Oncewelinearizetheeqns.(10,11,25)forsmallvoltagedrops,andthenmakethemcontinuousinamultilayergeometry,theybecomeidenticalinformtotheequationsofathermoelectric.Thisanalogygivestheeffectiveformulasfortheconductivityσ,theSeebeckcoefficientS,andthethermalconductivityK.

σ=

eJRLi

(45)(46)(47)

(b+2)ekB

K=[]Li

R1

σS2T

Z=

u

ThefirstterminthethermalconductivityistheelectroniccontributionKe.NotethatitobeysamodifiedWiedemann-FranzLaw

Ke=σT

󰀈

kB

Z0

=

2mkBR1

∆T(γ+1)11

(50)

∆T(γ−1)ηg=

1+Z

(52)

areremarkablyaccuratecomparedtothecomputersolutions.TheEqs.(50)and(51)estimatethecorrectefficiencywithanaccuracyofabout1%.Theseformulasprovideanalyticalestimatesoftheefficiency.Onecanalsousethethermoelectricanalogytofindthecurrentdensitiesatmaximumefficiencyaswellasotherparameters.

InthermoelectricdevicesitishighlydesirablebutratherraretohavevaluesofZ>1.However,inmodelingthermionicdeviceswefindvaluesofZfectivemuchSeebecklargerthancoefficientoneforinreasonableeqn.(45)isvaluesaboutof250-300thermalµV/Kresistance.sincevaluesTheef-ofbarebetweenoneandtwo.Byusingthelowvaluesofthermalconductivityreportedalongthec-axisofasuperlattice,weestimatethatTR=200-400K.ForthisrangeofparameterstheeffectivedimensionlessfigureofmeritZisbetween2and5.Theefficienciesofthethermionicdevicesarecorrespond-inglymuchhigherthaninthermoelectricdevices.Ballistictransportcarriesmoreheatthandiffusiveflow.

4Discussion

Adetailedtheoryispresentedofthepropertiesofamultilayerthermionicrefrigeratorandpowergenerator.Itisshownsthatifthethermalresistanceishigh,thatthedevicescanbetwicetheefficiencyoftheequivalentther-moelectricdevices.Thevaluesofthermalresistiviityneededtomakethemworkwellareintherangeofreportedvaluesformultiplequantumwells.Itappearsthesedevicesarepractical.

OneoftheinterestingquestionsistoselectthevalueofN.Thisdeter-minesthenumberofmultilayers.Aslongasthisnumberislargerthanaboutteninrefrigerators,itdoesnotchangetheefficiency.ThechoiceofNmayaffectthethermalresistance.Forafixedtemperaturedrop∆T,thevalueofNaffectstheaveragetemperaturedropperlayer󰀉δT󰀊=∆T/N.Thecoolingpoweristheenergycurrentfromthecoldside.Anestimateis

JAT

¯φ∆TQc=

¯istheaveragetemperatureofthedevice.ForarefrigeratortakeAwhereT

¯=280K,φ=0.050eV,and󰀉δT󰀊=1K.=120A/(cm2K2)alongwithT

ThentheresultisJQc=212W/cm2.Thisisalargecoolingpower.Thushavingasmalltemperaturedroppermitsamplecooling.Itdoesrequireasmallbarrierheight.Incidently,thenumericalsolutionsallowanaccuratecalculationofJQcandtheaboveformulaisanunderestimatebyaboutafactoroftwoorthree.OnecouldincreaseNbyafactoroftenandstillhaveamplecoolingpower.ThemanufacturingcostswoulddependonN.Anotherissueismechanicalstrength.Athickerdevice(largerN)isstronger.Thustherearetradeoffsbetweencoolingpower,manufacturingcosts,andmechanicalstrength.Suchengineeringissuesarebeyondthescopeofthepresentdiscussion.

WethankL.Woodsforhelpfuldiscussions.Researchsupportisacknowl-edgedfromtheUniversityofTennessee,andfromOakRidgeNationalLab-oratorymanagedbyLockheedMartinEnergyResearchCorp.fortheU.S.DepartmentofEnergyundercontractDE-AC05-96OR22464.

Appendix

Hereweexaminethetransportofelectricityandheatusingthedrift-diffusioneqn.(1).Wewillshowthattheamountofheatisnegligiblysmallwhenoneusesthisequation.Unlessthelayerthicknessisintheregimewherethermionicemissionisvalid,thesemiconductorbarrierdoesnottransportsignificantamountsofheat.

Weconsiderasquarebarrierforthesemiconductor.AssumethereisasmallappliedpotentialandasmalltemperaturedifferenceδT=Th−Tc.Thentheformulaforthecurrentshouldbe

J=eµn

δV

L

󰀃

(54)(55)(56)(57)

n(0)=n0e−eφ/kBTcn(L)=n0e−eφ/kBTh

n=n0e−eφ/kBT

WeexpandthedensityexponentsusingTh,c=T±δT/2andfind

J=

eµn

VD=bkBδT(59)

Theaboveformularesembleseqn.(10).ThetwoconstantvoltagesVJandVDaresimilar.Theprefactorsareverydifferent.Denotebyrtheratioofthetwoprefactors,whichgivestheratioofthemagnitudesofthecurrentspredictedbydrift-diffusioncomparedtothecurrentspredictedbythermionicemission

r=

σ0

eµn

󰀅

L0

(62)

2π2h¯3

whereσistheconductivityofthesemiconductorandL0isacharacteristic

lengthwhichwetaketobeonemicron.Atroomtemperaturewefindthatσ0=8.1kS/m.InSbhasamobilityof8m2/(Vs)atn=1020/m3forσ=130S/m.Evenforb=4thispredictsaratioofr≈10−3.Sothecurrentduetodrift-diffusionisonethousandtimessmallerthanthecurrentduetothermionicemission.Theenergycurrentshavethesameratioofproportionality.Thustheflowofheatisnegligiblewhenthethicknessofthesemiconductorisgreaterthanthemeanfreepathoftheelectron,sothattheparticlesdiffuse.

14

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FigureCaptions

1.ThereductionintheCarnotefficiencyηˆforasinglebarrierasafunctionofbarrierheightφforvaluesofTR=100,200,300,400,and500K.2.EfficiencyηofamultilayerthermionicrefrigeratorwithTc=260KandTh=300K.ResultsareplottedasafunctionofbarrierheightfordifferentvaluesofTR.3.Exactefficiencyofamultilayerthermionicpowergeneratorasafunc-tionofbarrierheightφforseveralvaluesofTR.WeassumedTc=300KandTh=400K.4.ChangeinthetemperatureTiandvoltageViasafunctionofpositionalongthemultilayerpowergenerator.WeassumedthatTc=300K,Th=400K,φ=0.054V,andTR=200K.

17

420400380360340320300280

0

20

40

Ti

i

6080100

0.30.280.260.240.220.20.180.16

0

20

40

dVi

i

6080100

0.5

TR=100K

0.4

η

0.3

200K

300K

0.2

0.1

500K

00

0.02

0.04

0.06

φ

0.08(eV)

0.10.12

0.12

η

0.1

TR=200K

300K400K

0.08

0.06

500K0.04

0.02

00

0.02

0.04

0.06

0.08

φ(eV)

0.10.120.14

3

2.5

TR=100K

η

2

1.5

300K

1

500K

0.5

200K

00

0.02

0.04

0.06

0.08

φ(eV)

0.10.120.14