Heat generation due to spin transport in spin valves Xiao-XueZhang,Pei-SongHe,Bao-HeLi,Yao-HuiZhu∗ PhysicsDepartment,BeijingTechnologyandBusinessUniversity,Beijing100048,China Abstract Using a macroscopic approach, we studied theoretically the heat generation due to spin transport in a typical spin valve with 7 nonmagnetic spacer layer of finite thickness. Our analysis shows that the spin-dependent heat generation can also be caused by 1 another mechanism, the spin-conserving scattering in the presence of spin accumulation gradient, in addition to the well-known 0 2spin-flipscattering. Thetwomechanismshaveequalcontributionsinsemi-infinitelayers,suchastheferromagneticlayersofthe spinvalve. However,inthenonmagneticlayerofathicknessmuchsmallerthanitsspin-diffusionlength,thespin-dependentheat n generation is dominated by the spin-flip scattering in the antiparallel configuration, and by the spin-conserving scattering in the a Jparallel configuration. We also proved that the spin-dependent heat generation cannot be interpreted as the Joule heating of the 5 spin-coupled interface resistance in each individual layer. An effective resistance is proposed as an alternative so that the heat 1generationcanstillbedescribedsimplybyapplyingJoule’slawtoanequivalentcircuit. ]Keywords: spinvalve,heatgeneration,spin-flipscattering,spin-conservingscattering,spin-coupledinterfaceresistance i c s - l1. Introduction finitethickness,suchasthenonmagnetic(NM)spacerlayerin r t aspinvalve.Thevariationofheatproductionwiththethickness m Heat generation is a serious issue even for spintronic de- oftheNMlayerisalsoanimportantproblem. .vices.[1,2,3]Forexample,largecurrentisusuallyrequiredfor t atheoperationofaspin-transfer-torquemagneticrandom-access From a macroscopic viewpoint, Ref. [6] also showed that mmemory.[4]Thereductionoftheworkingcurrentandtheasso- Joule heating of the spin-coupled interface resistance (r ) is SI -ciatedheatingisstillachallengingproblem. Moreover,heating equaltotheadditionalheatgenerationduetospintransportin d inspintronicdevicesleadstotemperaturegradient,whichmay aspinvalvewithoutaNMspacerlayer. Itisnecessarytostudy n inversely have remarkable influence on the spin transport via o whether this conclusion holds generally for spin valves with cthespin-dependentSeebeckeffect.[5] NM spacer layers of finite thickness. This question is closely [ Recenttheoreticalinvestigationshaveshownthatthereisstill related to the nonlocal character of the heat generation, which dissipationevenifapurespincurrentispresent.[6,7,8,9,10] 1 issimilartothesituationofspinpumpingdiscussedinRef.[9]. vMeanwhile,experimentalstudieshavealsodemonstratedvari- Itiswell-knownthattheNMlayerinaspinvalvehasnocon- 4ous spin-dependent heating effects. [11, 12, 13, 14, 15, 16] It tributiontothespin-coupledinterfaceresistancebecauseofthe 8hasbeenwellestablishedthatheatgenerationinspintronicde- absenceoftheextrafield.[17,18]However,theNMlayercon- 9 vices is dependent on the spin accumulation and spin current tributestotheadditionalheatgenerationduetospintransport. 3 incomparisontothatinconventionalelectronicdevices. How- 0 ThustheheatgenerationdoesnotobeyJoule’slawineachin- .ever,previousworksdidnotpaymuchattentiontothedistinc- dividuallayer,althoughitdoesthroughoutthewholestructure. 1 tionsandcharacteristicsofthevariousmechanismscausingthe 0 Thischaracterneedstobeinterpretedproperly.Inviewofthese heat generation, especially the one due to the spin-conserving 7 openquestions,westudiedanalyticallytheheatgenerationdue 1scattering. This mechanism can also lead to spin-dependent tospintransportinaspinvalveusingamacroscopicapproach :heat generation when electrons flow to positions with lower v basedontheBoltzmannequation.[6] chemical potential without spin flip. We will show that this i X mechanismcanbeidentifiediftheequationoftheheatgener- Thispaperisorganizedasfollows. InSec.2, wederivethe rationiswritteninamoreappropriateform, inwhichallterms basicequationsfortheheatgenerationinmagneticmultilayers a are exactly the products of current-force pairs. Unfortunately, and find the general relation between the spin-dependent heat notallofthecurrent-forcepairsarecorrectlychoseninthepre- generationandthespin-coupledinterfaceresistance. Thenthe vious papers. Furthermore, it is worthwhile to investigate in basicequationsareappliedtoaspinvalveinSec. 3. Wealso depth the relative magnitude of heat generation due to spin- compare the contribution from the spin-conserving and spin- conserving and spin-flip scattering, especially in a layer with flip scattering. In Sec.4, we discuss in depth the validity of thespin-coupledinterfaceresistanceandintroducetheeffective ∗Correspondingauthor resistanceasanalternative. Finally, ourmainresultsaresum- Emailaddress:yaohuizhu@gmail.com(Yao-HuiZhu) marizedinSec.5. PreprintsubmittedtoElsevier January17,2017 2. Basicequations exactly the current corresponding to the force ∂∆µ/∂z, which hasonlyexponentialterms. Onecanseethiseasilybysubtract- We are mainly concerned with the heat generation in a ingthe‘±’componentsofEq.(A.4) spin valve driven only by a constant current of density J, which flows in the z-direction and perpendicular to the layer 1 ∂∆µ J =∓βJ+ (4) plane.[17]Insteadystate,thetimerateofheatgenerationcan spin eρ∗ ∂z F be calculated by using a macroscopic equation like Eq. (6) in wheretheFMlayeralsohas‘up’(‘down’)magnetization. The Ref.[6] totalspincurrentinEq.(4), J , hasbeenwrittenasthesum spin σ = J+∂µ¯+ + J−∂µ¯− + 4(∆µ)2G (1) ofabulktermandanexponentialone heat e ∂z e ∂z e2 mix Jbulk =∓βJ (5) spin where we use σ to denote the heat-generation rate follow- heat 1 ∂∆µ ing Ref. [19]. This kind of equation can be derived by using Jexp = (6) spin eρ∗ ∂z theBoltzmannequation[6]aswellasnonequilibriumthermo- F dynamics[19]inthelinearregime. InEq.(1), J+ (J−)andµ¯+ Therefore, it is necessary to transform the first two terms of (µ¯ ) are the current density and the electro-chemical potential − σ inEq.(2)intoexactcurrent-forcepairs. Fortunately,this heat in spin-up (down) channel, respectively. The electron-number desired form can be achieved by simply substituting Eqs. (3) currentsaregivenby−J /e,where−eisthechargeofanelec- ± and(4)intoEq.(2) tron. Moreover,∆µ = (µ¯+−µ¯−)/2describesthespinaccumu- lation, and Gmix defined by Eq. (5) of Ref. [6] stands for the σbulk+σsc = JEF+ Jsepxipn∂∆µ (7) associatedspin-fliprate. heat heat 0 e ∂z ThefirsttwotermsofEq.(1)canberegardedasthedecrease wherewehaveintroduced in the (electrochemical) potential energy current or the heat generation,ineachspinchannel.[20]Theheat-generationrates Jexp ∂∆µ of the two channels are unequal in ferromagnetic (FM) layers σsc = spin (8) heat e ∂z andatspin-selectiveinterfaces. Ifthetwochannelscannotex- changeheateffectivelywitheachotherorotherheatreservoir, The first term, σbulk = JEF, in Eq. (7) has become the prod- theymayhavedifferenttemperatures[21]. However,thisisbe- uctofacurrent-foheractepaira0nditmeansthebulkJouleheating. yondthescopeofthepresentworkandweneglectthiseffectby Thesecondterm,σsc ,hasalsobeenwrittenastheproductof assumingthatthetwospinchannelscanexchangeenergyeffec- acurrentanditscorhreeastpondingforce,anditcanberegardedas tively(thethermalizedregime[21]). Thenitismoremeaning- the heat generation due to the spin-conserving (sc) scattering. fultowriteEq.(1)intermsof J = J++J− and Jspin = J+−J− The reason is that, the gradient of spin accumulation drives a likeEq.(12c)ofRef.[6] spinfromapositiontoanotheronewithlowerchemicalpoten- tialviathespin-conservingscattering,andtheexcesschemical J∂µ¯ J ∂∆µ ∂J ∆µ σ = + spin + spin (2) potentialofthespinleadstoheatgeneration. Thiscanalsobe heat e ∂z e ∂z ∂z e understoodbywritingthistermasthesumofthecontributions whereµ¯ =(µ¯++µ¯−)/2istheaverageelectrochemicalpotential. from the two spin channels (see Appendix B). The heat gen- Althougheachtermofσ hasbeenwrittenastheproductof eration due to spin-conserving scattering σsc did not attract heat heat generalized force and current, which is consistent with the re- muchattentioninthepreviousstudies,suchasRefs.[6,9].This quirementofnonequilibriumthermodynamics,[19]thecurrent- mechanismisespeciallyimportantfortheheatgenerationdue force pairs in the first two terms of Eq. (2) are not exact pairs totheinterfaceresistancebecausethespin-flipscatteringisusu- andneedtoberewritteninamoreappropriateform. ally neglected at interfaces. We will look into σsc in Sec. 3, heat InFMlayers,thetotalcurrentdensity J doesnotdependon andcompareitwithσsf (seebelow)intypicalspinvalves. heat position z and thus it is not exactly the current corresponding Nextwewillrewritethelasttermofσ ofEq.(2)inaform heat totheforce∂µ¯/∂z,whichhasexponentialtermsofz. Thisbe- withmoreobviousphysicalinterpretation. Subtractingthe‘±’ comesobviousifwesumthe‘±’componentsofEq.(A.4) componentsofEq.(A.3)intwodifferentways,onecanderive 1∂µ¯ β∂∆µ ∂J 4eN ∆µ ∆µ F = = EF± (3) spin = s = (9) e ∂z 0 e ∂z ∂z τF(N) er lF(N) sf F(N) sf inanFMlayerwith‘up’(‘down’)magnetizationandbulkspin asymmetry coefficient β. Here, β and the spin-dependent re- where rF = ρ∗FlsFf and rN = ρ∗NlsNf. In Eq. (9), Ns is the density sistivityρ↑(↓) [orconductivityσ↑(↓)]satisfytherelationρ↑(↓) = ofstatesforspin sandτFsf (τNsf)thespin-fliprelaxationtimeof 1/σ = 2ρ∗[1−(+)β].[17]Notethatthesubscript“↑”(“↓”) theFM(NM)layer. ThenusingEq.(9),wecanrewritethelast ↑(↓) F termofEq.(2)as denotes the majority (minority) spin direction. In Eq. (3), the effectivefieldF(z)oftheFMlayershasbeenseparatedintothe σshfeat =αspinµm (10) bulk term EF = J(1−β2)ρ∗ and the exponential term. Mean- where α = µ N /τ and µ = 2∆µ. Here α describes 0 F spin m s sf m spin while, the spin current J has a bulk term and thus it is not thedampingrateofthespinaccumulation,thatis,thenumber spin 2 ofelectronsflippingfromspin-uptospin-downstatesperunit where∆VCand∆VCaredefinedas 0 I timeperunitvolume,whichiscalledthespinfluxψ˙ inRef.[7]. ∆VC = J(1−γ2)r∗ (20) Thusαspinandµmcanalsoberegardedasageneralizedcurrent- 0 b force pair, and σsf stands for the heat generation due to the ∆µ(z+)−∆µ(z−) heat ∆VC = JrC =±γ C C (21) spin-flip(sf)scattering. I SI e Collecting all the terms, we can then write the rate of heat In Eq. (21), rC is defined as the spin-coupled interface resis- generationinFM(NM)layerintheformofJoule’slaw SI tance of the interface. The sign ‘+’ (‘−’) in Eq. (21) corre- σ =σbulk+σsc +σsf sponds to the configuration where the spin-up channel is the heat heat heat heat =(1−β2)ρ∗ J2+ρ∗ (cid:16)Jexp(cid:17)2+ρ∗ (cid:34) ∆µ (cid:35)2 (11) mhaivneority(majority)one. SubstitutingEq.(19)intoEq.(13),we F(N) F(N) spin F(N) erF(N) Σ = J(cid:16)∆VL+∆VC+∆VR(cid:17)+J(cid:16)∆VL+∆VC+∆VR(cid:17) heat 0 0 0 I I I wherewehaverewrittenσsc andσsf usingEqs.(6)and(9). (cid:104) (cid:105) (22) Thefirsttermσbulk isduethoeatthenomheiantalresistanceandexists + Jspin(zR)∆µ(zR)−Jspin(zL)∆µ(zL) /e heat nomatterwhetherthecurrentisspin-polarizedornot.However, wherethefirsttermontheright-handsideisthenominalJoule σsc and σsf rely on the gradients of the spin accumulation heat heat heatingandthesecondthe‘Jouleheating’ofthespin-coupled andspincurrentaccordingtoEqs.(6)and(9),respectively. interfaceresistance. ThelasttermofEq.(22)disappearsonce Finally,wewilldiscussthegeneralrelationbetweenthespin- J ∆µhasthesamevalueatz andz . spin L R dependent heat generation and the ‘Joule heating’ of the spin- On the other hand, by using Eqs. (3) and (4), one can also coupled interface resistance. [6, 17] To this purpose, we write rewriteEq.(2)as Eq.(2)as σheat = JF+ 1e∂∂z(cid:16)Jspin∆µ(cid:17) (12) σheat = JE0F(N)+ 1e∂∂z(cid:16)Jsepxipn∆µ(cid:17) (23) by using F = (1/e)∂µ¯/∂z. Without loss of generality, we will wherethelasttermisthesumofσsc andσsf . Thenthetotal heat heat considerasegmentofamultilayerfromzLtozR,whichincludes heat-generation rate in the segment zL < z < zR can also be aninterfaceatzC (zL < zC < zR). IntegratingEq.(12)fromzL writtenas tozR,wecanwritethetotalheatgenerationinthissegmentas Σ = J(cid:16)∆VL+∆VC+∆VR(cid:17)+Σspin (24) Σ =ΣL +ΣC +ΣR (13) heat 0 0 0 heat heat heat heat heat where where ΣLheat = J∆V0L+J∆VIL+ 1e (cid:16)Jspin∆µ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)z−C (14) Σshpeiant = 1e (cid:16)Jsepxipn∆µ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)zz−CL +Σshpeiant,C+ 1e (cid:16)Jsepxipn∆µ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)zzR+C (25) zL standsforthespin-dependentpartofthetotalheatgeneration. ΣCheat = eJ (cid:104)µ¯(z+C)−µ¯(z−C)(cid:105)+ 1e (cid:16)Jspin∆µ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)zz+C− (15) Thecontributionofthe1interface(cid:104)canbewrittenas(cid:105) ΣRheat = J∆V0R+J∆VIR+ 1e (cid:16)Jspin∆µ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)zzR+ C (16) whereJsa (zΣsh)peiianst,Cde=fineeJdsspabiny(zC) ∆µ(z+C)−∆µ(z−C) (26) C spin C aredefinedastheintegralheat-generationratefortheleftlayer, J (z )=∓γJ+Jsa (z ) (27) theinterface,andtherightlayer,respectively.InEqs.(14),∆VL spin C spin C 0 and∆VLaredefinedas OnecanseethatEq.(27)issimilartoEq.(4). Itisalsoeasyto I (cid:90) zC (cid:16) (cid:17) verifyΣCheat = J∆V0C+Σshpeiant,C. UsingEq.(A.24),wecanrewrite ∆V0L = E0Ldz= J 1−β2 ρ∗L(zC−zL) (17) Σshpeiant,CintheformofJoule’slaw zL ∆VL = JrL =(cid:90) zC(cid:16)F−EL(cid:17)dz=±β∆µ(z−C)−∆µ(zL) (18) Σshpeiant,C =rb∗(cid:104)Jsspain(zC)(cid:105)2 (28) I SI 0 e zL whichwillbediscussedfurtherinSec.3.2. where we have used Eq. (3). Similarly, ∆VR and ∆VR in (16) ComparingEqs.(22)and(24),onecanseethatΣspinisequal 0 I heat are defined in the regime z < z < z . In Eq. (18), we have tothe‘Jouleheating’ofthespin-coupledinterfaceresistance C R defined rL as the spin-coupled interface resistance of the left (cid:104) (cid:105) SI Σspin = J ∆VL+∆VC+∆VR (29) layerbyextendingthedefinitioninRef.[17]. heat I I I The interface has been treated as a layer with an infinites- ifthelasttermofEq.(22)vanishes,thatis, imal thickness in Eq. (15). This approach is equivalent to the J (z )∆µ(z )= J (z )∆µ(z ) (30) methodusedinRef.[6],whichisproveninAppendix C.Using spin L L spin R R Eqs.(A.21),(A.24),and(A.25),wecanrewriteEq.(15)as Infact,thisrequirementcanbesatisfiedeasilybecauseamag- ∆µ(z+)−∆µ(z−) neticmultilayerusuallyhasseveralpositions,where Jspin∆µis ΣCheat = J∆V0C+J∆VIC+Jspin(zC) C e C (19) zero. WewilldiscussrSIinmoredetailinSec.4. 3 3. Heatgenerationinspinvalves 1 .0 In this section, we will apply the basic equations derived in tion S Nh e,saft,A P o r S Nh e,sact,P Sec.2tospinvalveswithfiniteNMlayer. Theinsertionofthe ra 0 .8 e NinMhelaatyegreneenraabtlieosnubsettowesetundypamraollreelr(ePa)lisatnicdalalnytitphaeradlilffeler(eAnPce) t gen 0 .6 a alignmentsofthetwoFMsaswellastheinfluenceoftheNM- l he 0 .4 layerthicknessontheheatgeneration. Tobespecific,weplace ra theoriginofthez-axisatthecenteroftheNMlayer. Theleft teg 0 .2 S N ,s c ,A P o r S N ,s f,P and right FM/NM interfaces are located at z = −d and z = In h e a t h e a t 0 .0 d, respectively. The two semi-infinite FM layers are made of the same material with collinear magnetization. The various 0 .0 0 1 0 .0 1 0 .1 1 1 0 quantities in Eq. (11) can be derived according to Valet-Fert N M -la y e r th ic k n e s s 2x theory [17] and the detailed results are listed in Appendix A. SubstitutingtheseresultsintoEq.(11),onecangetthetimerate Figure1:Integralheatgenerationrate(inunitofΣNhe,aAtPandΣNhe,aPtforAPandP ofheatgenerationintheleft(‘+’)andright(‘−’)FMlayers alignments,respectively)inhalfoftheNMlayerasafunctionoftheNM-layer thickness(inunitoflN). Thered-solidcurvecorrespondstotheintegralheat sf   generationduetothespin-flip(spin-conserving)scatteringintheAP(P)config- σPhe(AatP) =(cid:16)1−β2(cid:17)ρ∗FJ2+ΣFhe,Pa(tAP)l2F exp±2(zlF±d) (31) u(srpaitnio-nfl,ipa)nsdcathtteerbinlagckin-dtahsehAedPc(uPr)vceosntfiangdusraftoiornt.hatduetothespin-conserving sf sf where where ΣFhe,Pa(tAP) =(cid:90)−∞−d(cid:104)σshce,aPt(AP)+σhsfe,aPt(AP)(cid:105)dz=rF(cid:104)αPF(AP)J(cid:105)2 (32) ΣNhe,aPt =(cid:90) 0(cid:16)σshce,aPt+σshfe,aPt(cid:17)dz=rNP (cid:16)αPNJ(cid:17)2 (38) −d isthespin-dependentpartoftheintegralheatgenerationinthe isthespin-dependentpartoftheintegralheatgenerationinthe left FM layer. The right FM layer has the same contribution becausebothσsc,P(AP)andσsf,P(AP)areevenfunctionsaboutthe lefthalfoftheNMlayer. Similarly,therighthalfhasthesame heat heat contribution due to symmetry. In Eq. (38), rP is defined as origin. ThedimensionlessparameterαP(AP) inEq.(32)willbe N F rP = r tanhξ and the dimensionless parameter αP(AP) will be determinedbyboundaryconditionsinSec.3.2. N N N determinedbyboundaryconditionsinSec.3.2. TheheatgenerationintheNMlayerismorecomplicatedand hastobetreatedtermbyterm.Thebulktermσbulkhasthesame heat 3.1. Comparisonofspin-conservingandspin-flipscattering valueσbulk =ρ∗J2foreitherAPorPalignment,anditsintegral over theheNatM laNyer (−d < z < d) can be regarded as the Joule Now we are ready to compare the magnitude of σshceat and heatingoftheNM-layernominalresistance2dρ∗. FortheAP σshfeat intwotypicalsituations: asemi-infinitelayerandafinite N layer. Thespinvalvecontainsbothtypesoflayers,namelythe alignment,thetwospin-dependenttermscanbewrittenas semi-infinite FM layers and the NM layer with a finite thick- σhsce,aAtP = lN2sΣinNhhe,aA(t2Pξ)sinh2lzN (33) ntieasllsy. IonnththeeFsMcallaeyoefrst,htehespsipni-ndaiffcucusimonulaletinognthdelscfaaysseshxopwonnenin- sf sf Eqs.(A.5)and(A.13). Withoutlossofgenerality,wewillcon- σshfe,aAtP = lN2sΣinNhhe,aA(t2Pξ)cosh2lzN (34) seiadseilryovnelyriftyheleftFMlayer(z < −d). UsingEq.(11),onecan sf sf   whereΣN,AP =(cid:90) 0(cid:16)σsc,AP+σsf,AP(cid:17)dz=rAP(cid:16)αAPJ(cid:17)2 (35) σshce,aPt(AP) =σshfe,aPt(AP) =ΣFhe,Pa(tAP)l1sFf exp2(zlsF+f d) (39) heat −d heat heat N N which shows that the spin-conserving scattering has the same contribution tothe heatgeneration atany positionas thespin- isthespin-dependentpartoftheintegralheatgenerationinthe flipscatteringinsemi-infinitelayers. Thustheirintegralsfrom lefthalfoftheNMlayer.Thecontributionfromtherighthalfis equaltothatfromthelefthalfsincebothσsc,AP andσsf,AP are −∞to−darealsoequaltoeachother[seeEq.(32)]. even functions. In Eq. (35), rAP is defined ahesatrAP = rhecatothξ, AsforthefiniteNMlayer,itismoremeaningfultocompare where we have ξ = d/lN. OnNthe other hand, fNor theNP align- theintegralheatgenerationduetothespin-conservingandspin- sf flip scattering, denoted by ΣN,sc and ΣN,sf, respectively. In the ment,wehave heat heat APconfiguration,wehave σshce,aPt = lN2siΣnNhhe,(aP2tξ)cosh2lzN (36) ΣN,sc,AP =(cid:90) 0σsc,APdz= 1−ηΣN,AP (40) sf sf heat heat 2 heat σshfe,aPt = lN2siΣnNhhe,(aP2tξ)sinh2lzN (37) ΣNhe,asft,AP =(cid:90)−d0σshfe,aAtPdz= 1+2ηΣNhe,aAtP (41) sf sf −d 4 wherethedimensionlessparameterη = 2ξ/sinh(2ξ)describes (a) the asymmetry between ΣN,sc,AP and ΣN,sf,AP. This becomes heat heat moreobviousifonelooksattheirrelativedifference(ΣN,sf,AP− heat ΣN,sc,AP)/ΣN,AP = η, which decreases from 1 to 0 as ξ varies heat heat from 0 to ∞. Figure 1 shows their variation with the NM layer thickness 2ξ (in unit of lN). In the regime of 2ξ = sf 2d/lN (cid:28) 1, which is practical for experiments, ΣN,sc,AP/ΣN,AP and sΣfN,sc,AP/ΣN,AP approach 0 and 1, respectivelyh.eaTtherefhoeraet, (b) heat heat the spin-dependent heat generation of the NM layer is domi- natedbythespin-flipscatteringintheAPalignment. Thisfea- turecanbeinterpretedasfollows.Wehavez/lN (cid:28)1intheNM sf layer(−d < z < d)if2d/lN (cid:28) 1. Thisallowsustoexpandthe sf spinaccumulation∆µ[seeEq.(A.9)]intermsofz/lN andkeep sf up to the first-order term. The result is a term independent of (c) position because the first-order term also vanishes. Then Jexp spin approacheszerobecauseitisproportionaltothegradientof∆µ accordingtoEq.(4). Therefore,σsc,AP approacheszerointhis heat regimeaccordingtoEq.(8). InthePconfiguration,theintegralheatgeneration,ΣNhe,asct and Figure 2: Effective circuits for the spin-dependent heat generation in a spin ΣN,sf,canbewrittensimilarlyas valvewithAPandPconfigurationneglectingtheinterfaceresistance. (a)In heat theAPconfiguration,theresistancerFsf andrFschavethesamevalue2rF. The ΣN,sc,P =(cid:90) 0σsc,Pdz= 1+ηΣN,P (42) vηa),riraebslpeercetisvisetlayn.cTesheryNsc,vAaPryanwditrhNsf,tAhPeathreicdkenfienssedofasth2erNANPM(1l−ayηe)r.an(db)2IrnNAPth(1e+P heat −d heat 2 heat configuration,rsf andrscarethesameasthosein(a),whilersc,Pandrsf,Pare (cid:90) 0 1−η definedas2rP(1F+η)anFd2rP(1−η),respectively.(c)TheequiNvalentcirNcuitof ΣN,sf,P = σsf,Pdz= ΣN,P (43) (a)and(b).TNhevariableeffeNctiveresistancesaredefinedbyEqs.(71)and(72). heat heat 2 heat −d However,theirrelativemagnitudeisswitchedincomparisonto whichsatisfytheidentityαP(AP)+αP(AP) =β. Theseparameters the AP alignment and the spin-conserving scattering becomes F N canbeinterpretedbytheeffectivecircuitsinFigs.2(a)and(b). dominantintheregime2d/lN (cid:28) 1(seeFig.1). Thisbehavior sf can be understood in a similar way to the AP configuration. In the AP configuration shown by Fig. 2(a), the current of Thelow-orderexpansionofspincurrent J [seeEq.(A.20)] density βJ/2 flows from the spin-down to the spin-up channel spin is independent of position. Then ∆µ approaches zero because ineitherhalfofthespinvalve. Itmodelstheelectron-number it is proportional to the gradient of J according to Eq. (9). current (spin flux) flowing inversely due to the spin-flip scat- spin Therefore, σsf,P approaches zero in this regime according to tering in the FM and NM layers. We have the current density heat JAP = JαAP/2and JAP = JαAP/2byusingsimplecircuitthe- Eq.(10). F F N N In the opposite regime, 2ξ = 2d/lN (cid:29) 1, the integral heat orem. Theheatgenerationduetothespin-flipscatteringinthe sf FM layer and either half of the NM layer is modeled by the generation ΣN,sc and ΣN,sf approach the same value for both P heat heat Joule heating of resistances rsf and rsf,AP, respectively. Sim- andAPalignmentsasshowninFig.1.Thismeansthatthespin- F N ilarly, the resistances rsc and rsc,AP model the heat generation conservingscatteringhasthesamecontributionasthespin-flip F N due to the spin-conserving scattering. One can easily recover onewhentheNM-layerthicknessbecomesmuchlargerthanthe spin-diffusionlength. Thesemi-infinitecasediscussedaboveis Eqs.(32),(40),and(41)byapplyingJoule’slawtothecircuit. In the P configuration as shown by Fig. 2(b), the current of recoveredinthelimitofthicklayer. density βJ/2 flows from the spin-down to the spin-up channel in the left half of the spin valve, which is similar to the AP 3.2. Boundaryconditionsandheatgenerationatinterfaces configuration. However, the current of density βJ/2 flows in- If the FM/NM interface is an Ohmic contact, the interface verselyintherighthalfofthespinvalvebecausethesignofthe resistanceisusuallynegligibleincomparisontothebulkterm. spinaccumulationandassociatedspinrelaxationisswitchedin Using the general solutions and boundary conditions without this half. One can find the current density JP = JαP/2 and F F interfaceresistancein Appendix A,wecandeterminethedi- JP = JαP/2fromthecircuit. ThenonecanrecoverEqs.(32), N N mensionlessparameters (42),and(43)byapplyingJoule’slawtothecircuit. Moreover, wealsoconstructedanequivalentcircuit,showninFig.2(c),for βrP(AP) thecircuitsinFigs.2(a)and(b). Theequivalentcircuithasthe αP(AP) = N (44) F r +rP(AP) sameheatgenerationastheoriginalone. However,thecurrent F N densitypassingtheequivalentcircuitisthetotalcurrentdensity βr αP(AP) = F (45) J insteadsothatitcanbeconnectedinserieswiththenominal N r +rP(AP) F N resistances(notshowninthefigure). 5 If the interface resistance has to be taken into account, the dimensionlessparameterscanbedeterminedsimilarly 50 25 (cid:104) (cid:105) β rP(AP)+r∗ −γr∗ αP(AP) = N b b (46) 40 FM1 N FM2 20 F r +r∗+rP(AP) F b N m) βr +γr∗ Ω αP(AP) = F b (47) n 30 15m) N r +r∗+rP(AP) 2 ( Ω F b N J n which still satisfy the identity αPF(AP) +αPN(AP) = β. Equations +sf,AP/at20 10F/J ( (44) and (45) can be recovered if rb∗ is neglected in Eqs. (46) scσhe and(47). 10 5 AccordingtoEq.(28),wecanwritethespin-dependentheat generationduetotheleftinterfaceresistanceas 0 0 ΣC,P(AP)(−d)=r∗(cid:104)Jsa,P(AP)(−d)(cid:105)2 (48) -60 -40 -20 0 20 40 60 heat b spin z (nm) wherethespin-upelectronsareintheminoritychannelforboth PandAPconfiguration,andJsa,P(AP)(−d)isdefinedbyEq.(27). Figure3: Thespin-dependentheatgeneration(blue-solidcurve)andtheextra spin field(red-dashedcurve)asfunctionsofpositionzinaspinvalvewithAPcon- Toshowthemeaningofthisresult,weconsiderthecurrentden- figuration. ThetwoFMlayersaremadeofCo(ρ∗ = 86nΩm,lF = 59nm, sitydrivenbytheelectricfield(ef)alone β=0.5)andtheNMlayerofCu(rN=7nΩm,lsNf =F 450nm).Thestfhicknessof theNMlayeris20nmandtheinterfaceresistanceisneglectedforsimplicity. 1∓γ Jef(−d)= J (49) ± 2 where J passesthroughr+ = 2rb∗(1+γ)andr− = 2rb∗(1−γ)in vscarlivbeisn.gWtheehweialltsgheonweratthieonv,aalinddittyheanndinltirmodituactieoenffoefctriSvIeirnesdies-- parallel. Onecaneasilyverifythatthenominalheatgeneration due to the interface resistance, [J+ef(−d)]2r+ +[J−ef(−d)]2r−, is tanceasanalternative. givenby J∆V [seeEq.(20)]. Thespincurrentresultingfrom 0 4.1. Validityofthespin-coupledinterfaceresistance theelectricfieldaloneis Jef,P(AP) = Jef(−d)− Jef(−d) = −γJ. spin + − The spin-coupled interface resistances can be calculated by The spin current due to the change of spin accumulation (sa) across the interface is the difference between JP(AP)(−d) and using Eqs. (18) and (21). Substituting Eqs. (A.8), (A.12), and spin (A.16)intoEq.(18),wehave Jef,P(AP)(−d), that is, Jsa,P(AP)(−d) = JP(AP)(−d) + γJ already spin spin spin usedinEq.(48). Thespin-dependentheatgenerationissolely rF,P(AP) =βr αP(AP) (53) SI F F causedbythespin-conservingscatteringinthepresenceofspin rN,P(AP) =0 (54) accumulationchangeacrosstheinterfacebecausethespin-flip SI scatteringisneglected. whererF,P(AP)isforeitherofthetwoFMlayersandrN,P(AP)for Similarly,thespin-dependentheatgenerationattherightin- SI SI theNMlayer. Substituting(A.24)intoEq.(21)andusingEq. terfacecanbewrittenas (52),wecanwriter duetotheinterfaceas SI ΣChe,Pat(AP)(d)=rb∗(cid:104)JsPp(iAnP)(d)+(−)γJ(cid:105)2 (50) rC,P(AP) =γr∗αP(AP) (55) SI b C where the spin-up channel is the minority (majority) one in P whichhasthesamevalueatthetwointerfacesofthespinvalve. (AP)alignment.SubstitutingJP(AP)(±d)intoEqs.(48)and(50), spin One may be tempted to interpret the spin-dependent heat we can also write the spin-dependent heat generation at each generationastheJouleheatingofthespin-coupledinterfacere- interfaceas ΣC,P(AP)(±d)=r∗(cid:104)αP(AP)J(cid:105)2 (51) sistance. AccordingtoEq.(30),thisispossibleonlyif Jspin∆µ heat b C has the same value at the two ends of the segment under con- where sideration. The spin valve has at least three points satisfying (cid:104) (cid:105) this requirement: z = ±∞ and z = 0. [17] Thus Joule’s law is γ r +rP(AP) −βr αP(AP) =γ−αP(AP) = F N F (52) valid (at least) in the following three segments: −∞ < z < 0, C N r +r∗+rP(AP) 0 < z < ∞, and of course −∞ < z < ∞. To be specific, we F b N willconsiderthelefthalf(−∞ < z < 0)ofthespinvalve. By isalsoadimensionlessparameter. using Eq. (29), we can write formally the spin-dependent heat generationastheJouleheatingofthecorrespondingr SI 4. Relationtothespin-coupledinterfaceresistance (cid:104) (cid:105) Σspin,P(AP) = J2 rF,P(AP)+rC,P(AP) (56) heat SI SI Thissectionwillgiveaspecificdiscussionontherelationbe- tween the spin-dependent heat generation and the ‘Joule heat- where we have used rN,P(AP) = 0. Note that this result does SI ing’ of the spin-coupled interface resistance r in the spin notholdforarbitrarysegmentandr J2 cannotberegardedas SI SI 6 theheatgenerationeitherbecauseitmaybenegativeincertain whereαP maybenegativeifthevariousparametersarechosen C situation. Itsrealmeaningwillbediscussedinthefollowing. properly. Then rC,P = ∆VC/J also becomes negative. There- SI I Wewillfocusonthelefthalfofthespinvalveandconsider fore, J(∆VF +∆VC) should be regarded as the work done by I I only the P alignment without loss of generality. Substituting the extra field instead of the Joule heating of the spin-coupled Jspin =−βJ+JsepxipnintoEq.(13),wehave interfaceresistancealthoughtheyareequalinsomespecialseg- (cid:16) (cid:17) ments, forexample, Σspin,P = J2rF,P + J2rC,P inthelefthalfof Σheat = J ∆V0F+∆V0C+∆V0N thespinvalve. heat SI SI +J(cid:16)∆VF+∆VC(cid:17)−∆E +Σspin,P (57) Toshowthemeaningof∆Ecp,werewriteitsfirsttermas I I cp heat wherewehaveintroduced βJ Jbulk (cid:104) (cid:105) Jbulk (cid:104) (cid:105) βJ γJ (cid:104) (cid:105) e ∆µ(z−C)= −+e µ+(z−C)−µ0 + −−e µ−(z−C)−µ0 (66) ∆E = ∆µ(z−)+ ∆µ(z+)−∆µ(z−) (58) cp e C e C C whereµ istheequilibriumchemicalpotentialandµ thechem- 0 ± and ical potential for spin s = ±, respectively. We have used the Jexp(z−) Jexp(z+) quasi-neutralityapproximation,µ+(z−C)+µ−(z−C)−2µ0 =0,[18] Σspin,P = spin C ∆µ(z−)− spin C ∆µ(z+) whenderivingEq.(66).Thetwotermsontheright-handsideof heat e C e C (59) Eq.(66)standforthechangeofenergystoredinthechemical- γJ+J (z )(cid:104) (cid:105) + spin C ∆µ(z+)−∆µ(z−) potentialimbalanceperunittimeinthetwospinchannels, re- e C C spectively, when the bulk electron-number current flows from Using Eqs. (A.21), (A.22), and (A.24), one can easily verify z−C to −∞. The reasonable source of the net energy change is that Eq. (59) is the spin-dependent heat generation defined in the work done by the extra field in the FM layer according to Eqs.(25)and(56). Notethat Jexp isdiscontinuousattheinter- Eq.(61). Similarly,thesecondtermof∆Ecpcanbewrittenas spin facelocatedatz = z although J iscontinuous. Comparing C spin Eqs.(24)and(57),on(cid:16)ehas (cid:17) γeJ (cid:104)∆µ(z+C)−∆µ(z−C)(cid:105)= J+e−f(ezC)δµ++ J−e−f(ezC)δµ− (67) J ∆VF+∆VC =∆E (60) I I cp wherewehaveintroducedδµ = µ (z+)−µ (z−)andusedthe ± ± C ± C which is valid in arbitrary segment of the spin valve. More quasi-neutrality approximation. The two terms on the right- specifically,wealsohave handsideofEq.(67)canberegardedasthechangeofenergy stored in chemical-potential imbalance of the two spin chan- βJ J∆VF = ∆µ(z−) (61) nels, respectively, whenthecurrentdrivenbytheelectricfield I e C alonetraversestheinterface. Thisenergychangeissuppliedby γJ (cid:104) (cid:105) J∆VC = ∆µ(z+)−∆µ(z−) (62) theextrafieldattheinterfaceaccordingtoEq.(62). I e C C The physical process can be summarized as follows. The where Eqs. (18) and (21) have been used. On the other hand, (cid:16) (cid:17) (cid:104) (cid:105) spin-dependentheatgenerationduetothespin-conservingand using Eq. (56) and J ∆VF+∆VC = J2 rF,P+rC,P , we also spin-flip scattering leads to the dissipation of energy stored in I I SI SI have thechemical-potentialsplittingineverylayerofthespinvalve Σshpeiant,P =∆Ecp (63) includingtheFMlayers,theinterface,andtheNMlayer. Then However,thisrelationisvalidonlyifJ ∆µhasthesamevalue thischangeofthechemical-potentialenergyiscompensatedby spin atthetwoterminalsofthesegment. Therefore, J(∆VF+∆VC) the work of the extra electric field in the FM layer and at the I I ismorecloselyrelatedto∆E thanΣspin,P.Itiscrucialtofigure interface. However,thecompensationprocessdoesnothappen outthemeaningofJ(∆VF+c∆pVC)andhea∆tE . intheNMlayersincethereisnoextrafieldinthislayer. I I cp According to the definition of ∆VF [see Eq. (18)], J∆VF I I should be regarded as the work done by the extra field, which 4.2. Effectiveresistance is dominated by the electrostatic field. [18] We stress that this workmaybenegativewhentheinterfaceresistanceisincluded. Thespin-dependentheatgenerationineachindividuallayer Toshowthisfeature,werewriteJ∆VFas cannotbeinterpretedasJouleheatingofr inthislayer.Thisis I SI easytoseeifweconsidertheNMlayer.Ithasaspin-dependent J∆VF =βr αPJ2 (64) I F F heatgenerationbutnocontributiontor accordingtoEq.(54). SI whereαP isgivenbyEq.(46). Bychoosingthevariousparam- Similaranalysisshowsthatthisresultisalsotrueinotherlay- F ersandatinterfaces. Moreover,rF,P(AP)andrC,P(AP)inEqs.(53) etersproperly, onecanmakeαP negative. ThenrF,P = ∆VF/J SI SI alsobecomesnegativeandsoitFisill-defined. SimSiIlarly, J∆IVC and(55)canbenegativewhenαPF(AP)andαPC(AP)arenegativeac- I cordingtoEqs.(46)and(52). Therefore,itisnecessarytofind standsfortheworkdonebytheextrafieldacrosstheinterface analternativetor ifonehopestodescribetheheatgeneration anditmayalsobenegative. UsingEq.(52),wehave SI ofasinglelayerwithJoule’slaw. UsingEqs.(32), (35), (38), J∆VC =γr∗αPJ2 (65) and (51), we can write the spin-dependent part of the integral I b C 7 3 .0 1 .0 S F ,A P S C ,A P h e a t 2 .5 h e a t tion 0 .8 tion ra ra 2 .0 e e n n e 0 .6 e t g t g 1 .5 a a e e l h 0 .4 S N ,P l h ra h e a t ra 1 .0 S N ,A P Integ 0 .2 S N ,A P Integ S Fh e,Aa Pt h e a t h e a t 0 .5 S F ,P 0 .0 h e a t 0 .0 0 .0 0 1 0 .0 1 0 .1 1 1 0 0 .0 0 1 0 .0 1 0 .1 1 1 0 x x N M -la y e r th ic k n e s s 2 N M -la y e r th ic k n e s s 2 Figure4: Thespin-dependentpartoftheintegralspin-dependentheatgenera- Figure5: Thevariationoftheintegralspin-dependentheatgeneration(inunit tion(inunitofΣF,AP)intheFMandNMlayersasfunctionsoftheNM-layer ofΣF,AP)withtheNM-layerthickness(inunitoflN)fortheAPalignment.The heat heat sf thickness(inunitoflN).WeusedthesameparametersasthoseofFig.3. solidanddash-dotlinesarefortheFM1layerandthelefthalfoftheNMlayer sf asinFig.4.Thecontributionoftheinterfaceresistance(rb∗=2rFandγ=0.6) isshownbythedash-dot-dotline. heatgenerationineachlayeras ΣF,P(AP) = J2r∗ (68) the limit of 2ξ → ∞, the difference between P and AP align- heat F,P(AP) mentsdisappears. ThelimitsoftheheatgenerationintheFM ΣN,P(AP) = J2r∗ (69) heat N,P(AP) and NM layers depend on rF and rN. If rF is equal to rN, all ΣC,P(AP) = J2r∗ (70) thecurvesapproachthesamelimit. Moreover,underthecondi- heat C,P(AP) tionr =r ,wealsohaveΣN,AP =ΣN,P forarbitraryNM-layer wherewehaveintroducedtheeffectiveresistances thicknFessaNccordingtoEqs.(h3ea5t)andh(e3a8t). r∗ =r (cid:104)αP(AP)(cid:105)2 (71) Whentheinterfaceresistanceistakenintoaccount,thespin- F,P(AP) F F dependent part of the integral heat generation exhibits several r∗ =rP(AP)(cid:104)αP(AP)(cid:105)2 (72) importantfeaturesasshowninFig.5. Thecontributionofthe N,P(AP) N N FM layer may decrease to zero if the NM-layer thickness in- r∗ =r∗(cid:104)αP(AP)(cid:105)2 (73) creasestoacertainvalue. ThenwehaveαAP = 0accordingto C,P(AP) b C F Eq.(32)andthespinaccumulationdisappearstogetherwiththe for one FM layer, half of the NM layer, and one FM/NM additionalfieldintheFMlayer.Inthiscase,thelossofpotential interface, respectively. The effective resistances r∗ and F,P(AP) energyduetoheatgenerationcanonlybecompensatedbythe r∗ canbeunderstoodbyconstructinganequivalentcircuit N,P(AP) additional field at the interface. Moreover, the contribution of as shown in Fig. 2(c). Comparing Eqs. (56) with the sum of theNMlayermayexceedthatfromtheFMlayerevenwhen2ξ Eqs.(68),(69),and(70),wehave issmallenoughthatthemagneto-heatingeffectisstillremark- able. Thisindicatesthepossibilityofallocatingheatgeneration rP(AP) =r∗ +r∗ +r∗ (74) SI F,P(AP) N,P(AP) C,P(AP) indifferentlayersbyengineeringtheinterfaceresistance. wherewealsohaverP(AP) =rF,P(AP)+rC,P(AP). Tworesistances SI SI SI have been introduced for each layer: the effective resistance 5. Conclusions and the spin-coupled interface resistance. In general, they are notequaltoeachotherinasinglelayerorataninterface. The Ouranalyticalresultsshowthatthespin-dependentheatgen- most obvious example is the NM layer, where r∗ has a N,P(AP) eration is due to two mechanisms: the spin-flip and spin- finitevaluewheneverd (cid:44)0whilerN isalwayszero. conserving scattering. In the presence of spin accumulation, SI Figure 4 shows the quantitative results for the integral heat heat is generated when electrons undergo transitions between generation without interface resistance. The curves can also thetwospinchannelswithdifferentchemicalpotentialviathe beregardedasthevariationoftheeffectiveresistancesdefined spin-flip scattering. On the other hand, with the existence of inEqs.(71)and(72)sincetheyareproportionaltotheintegral spinaccumulationgradient,spin-conservingscatteringcanalso heatgeneration.Intheregime2ξ (cid:28)1,ΣF,APapproachesaposi- lead to heat generation when electrons move to positions with heat tiveconstantJ2β2r ,whileΣF,P,ΣN,AP,andΣN,P approachzero. lower chemical potential in the same spin channel. The two F heat heat heat This behavior indicates the existence of the magneto-heating mechanisms have equal contributions in semi-infinite layers. effect: differentheatgenerationinPandAPconfigurations. In However, in the NM layer of a thickness much shorter than 8 itsspin-diffusionlength, thespin-dependentheatgenerationis IntheNMlayer(−d d),wehave 0 N ers and at interfaces instead of Joule heating. It converts into (cid:104) (cid:105) the energy stored in the chemical-potential splitting, which in ∆µ(z)=erFαAFPJexp −(z−d)/lsFf (A.13) turn compensates the spin-dependent energy dissipation in all µ¯ (z)=eEFz+KC ±(1∓β)∆µ (A.14) thelayersincludingtheNMlayer. Effectiveresistancesandas- ± 0 1 J ∆µ sociatedcircuitsarealsointroducedtoovercomethelimitation J (z)=(1±β) ∓ (A.15) ± 2 2er ofr indescribingheatgeneration. F SI β∆µ F(z)= EF+ (A.16) 0 elF 6. Acknowledgments sf Usingtheseequations,onecaneasilyderive We thank Prof. Suzuki and Prof. Tulapurkar for fruitful discussions. This work was supported by National Natu- J =∓βJ± ∆µ (A.17) ral Science Foundation of China [grant numbers 11404013, spin er F 11605003, 61405003, 11174020, 11474012]; Beijing Natural fortheFM1(FM2)layer,and ScienceFoundation[grantnumber1112007];and2016Gradu- 1 ∂∆µ ateResearchProgramofBTBU. J = =αAPJsinh(z/lN)/sinhξ (A.18) spin eρ∗ ∂z N sf N Appendix A. GeneralExpression fortheNMlayer. InthePconfiguration,bothFM1andFM2layershave“up” The current density J and electrochemical potential µ¯ = s s magnetization. Theexpressionsofthevariousquantitiescanbe µ −eV ofthetwospinchannels, s = ±, satisfythefollowing s derivedsimilarlyandthusweonlylistsomeresultsthatwillbe equations referredtointheprevioussections. IntheNMlayer,wehave e ∂J µ¯ −µ¯ s = s −s (A.1) ∆µ(z)=−erPαPJsinh(z/lN)/sinhξ (A.19) σ ∂z l2 N N sf s s J =−αPJcosh(z/lN)/coshξ (A.20) σ ∂µ¯ spin N sf J = s s (A.2) s e ∂z Note that J has only exponential part in the NM layer. spin where σ and l denote the conductivity and spin-diffusion The dimensionless parameters αP(AP) and αP(AP) are given in s s F N length for spin s, respectively. [17] These two equations can Sec.3.2. betransformedinto Thecurrentdensityandelectrochemicalpotentialsatisfythe e ∂J ∆µ boundaryconditionsataninterfacelocatedatz=zC ± =±2 (A.3) (cid:16) (cid:17) (cid:16) (cid:17) σ± ∂z l±2 Js z+C −Js z−C =0 (A.21) J± =σ±(cid:32)F± 1∂∆µ(cid:33) (A.4) δµ¯s(zC)=µ¯s(cid:16)z+C(cid:17)−µ¯s(cid:16)z−C(cid:17)=ersJs(zC) (A.22) e ∂z wherethespin-dependentinterfaceresistancer canbewritten where ∆µ = (µ¯+ −µ¯−)/2 and F = (1/e)∂µ¯/∂z. Solving these as s equationsforthespinvalveconsideredinSec.3,wecanwrite r =2r∗(cid:2)1−(+)γ(cid:3) (A.23) µ¯ (z),J (z)andF(z)intermsof∆µ.FortheAPconfiguration, ↑(↓) b ± ± Here, γ is the interfacial asymmetry coefficient and ↑ (↓) de- themagnetizationdirectionis“up”intheleftFMlayer(FM1) notesthemajority(minority)spinchannel. Subtractingthe‘±’ and “down” in the right FM layer (FM2). In the FM1 layer (z<−d),wehave componentsofEq.(A.22),wehave ∆µ(z)=erFαAFPJexp(cid:104)(z+d)/lsFf(cid:105) (A.5) ∆µ(z+C)−∆µ(z−C)=eJsapcin(zC)rb∗ =e(cid:104)Jspin(zC)±γJ(cid:105)rb∗ (A.24) µ¯ (z)=eEFz+KA±(1±β)∆µ (A.6) wherewehaveusedEq.(27).Thesign‘+’(‘−’)correspondsto ± 0 1 J ∆µ the configuration in which the spin-up channel is the minority J±(z)=(1∓β) ± (A.7) (majority) one. On the other hand, summing the ‘±’ compo- 2 2er F nentsofEq.(A.22),wehave β∆µ F(z)= EF+ (A.8) (cid:104) (cid:105) 0 elsFf µ¯(z+C)−µ¯(z−C)=eJ(1−γ2)rb∗±eγ Jspin(zC)±γJ rb∗ (A.25) 9 Appendix B. Heatgenerationduetospin-conservingscat- References tering References The first two terms of Eq. (1) can also be rewritten in the [1] A.Fert,Nobellecture: Origin,development,andfutureofspintronics, followingway. 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[14] F.L.Bakker,A.Slachter,J.-P.Adam,B.J.vanWees,Interplayofpeltier andseebeckeffectsinnanoscalenonlocalspinvalves, Phys.Rev.Lett. Appendix C. Heatgenerationduetointerfaceresistance 105(2010)136601.doi:10.1103/PhysRevLett.105.136601. [15] F. K. Dejene, J. Flipse, B. J. van Wees, Verification of the thomson- onsagerreciprocityrelationforspincaloritronics,Phys.Rev.B90(2014) Theheatgenerationduetointerfaceresistanceisonlycaused 180402.doi:10.1103/PhysRevB.90.180402. thespin-conservingscatteringandcanbecalculatedbysimply [16] J.Flipse, F.L.Bakker, A.Slachter, F.K.Dejene, B.J.vanWees, Di- rectobservationofthespin-dependentpeltiereffect,Nat.Nanotechnol.7 summingthetwospinchannels[6] (2012)166–168.doi:10.1038/nnano.2012.2. ΣCheat =(cid:2)J+(zC)δµ¯++J−(zC)δµ¯−(cid:3)/e, (C.1) [17] Tne.tVicalmetu,lAti.laFyeerrts,,TPhheyosr.yRoefvt.heBp4e8rp(e1n9d9ic3u)l7ar09m9a–g7n1e1to3r.esdisotia:n1ce0.in11m0a3g/- PhysRevB.48.7099. where δµ¯ = µ¯ (z+)−µ¯ (z−) is the change in electrochemical [18] Y.-H. Zhu, D.-H. Xu, A.-C. Geng, Charge accumulation due to spin s s C s C transportinmagneticmultilayers, PhysicaB446(2014)43–48. doi: potentialacrossthe(infinitesimallythin)interface.Substituting 10.1016/j.physb.2014.04.036. theboundaryconditions,Eq.(A.22),intoEq.(C.1),wehave [19] D.Kondepudi,I.Prigogine,ModernThermodynamics: FromHeatEn- ginetoDissipativeStructures,Wiley,NewYork,1998. doi:10.1002/ ΣCheat = J+2(zC)r++J−2(zC)r− (C.2) [20] H97.8B1.1C1a8l6le9n8,7T2h3e.rmodynamicsandanIntroductiontoThermostatistics, 2ndEdition,Wiley,NewYork,1985. whichisjustJoule’slawinthetwo-channelmanner. Itismore [21] M.Hatami,G.E.W.Bauer,Q.Zhang,P.J.Kelly,Thermalspin-transfer meaningfultowriteitintermsofJandJspin torqueinmagnetoelectroicdevices,Phys.Rev.Lett.99(2007)066603. doi:10.1103/PhysRevLett.99.066603. ΣC = J∆VC+Σspin,C (C.3) heat 0 heat where∆VCandΣspin,CaredefinedinEqs.(20)and(28),respec- 0 heat tively. 10

Heat generation due to spin transport in spin valves

Similar Heat generation due to spin transport in spin valves