Heat generation due to spin transport in spin valves

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by  Xiao-Xue Zhang

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:[email protected](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 <z<d),wehave
dominated by the spin-flip scattering in the AP configuration,
∆µ(z)=erAPαAPJcosh(z/lN)/coshξ (A.9)
andbythespin-conservingscatteringinthePconfiguration. N N sf
Weprovedingeneralthatthespin-dependentheatgeneration µ¯±(z)=eE0Nz+K1B±∆µ (A.10)
isequaltothe‘Jouleheating’ofthespin-coupledinterfacere- J 1 ∂∆µ
J (z)= ± (A.11)
sistance(rSI)onlyinsomespecialsegment. TheconceptofrSI ± 2 2eρ∗ ∂z
N
hasanotherlimitation: itmaybenegativeinsomecaseswhen
F(z)= EN (A.12)
it is defined in an individual layer. Therefore, J2r should be 0
SI
interpreted as the work done by the extra field in the FM lay- whereEN =ρ∗J. IntheFM2layer(z>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
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s
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∂z ∂z ∂z
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ΣC = J∆VC+Σspin,C (C.3)
heat 0 heat
where∆VCandΣspin,CaredefinedinEqs.(20)and(28),respec-
0 heat
tively.
10