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Theory of thermal spin-charge coupling in electronic systems. B. Scharf,1 A. Matos-Abiague,1 I. Zˇutić,2 and J. Fabian1 1Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 2Department of Physics, State University of New York at Buffalo, NY 14260, USA (Dated: January 30, 2012) Theinterplaybetweenspintransportandthermoelectricityoffersseveralnovelwaysofgenerating, manipulating,anddetectingnonequilibriumspininawiderangeofmaterials. Hereweformulatea phenomenologicalmodelinthespiritofthestandardmodelofelectricalspininjectiontodescribethe electronic mechanism coupling charge, spin, and heat transport and employ the model to analyze several different geometries containing ferromagnetic (F) and nonmagnetic (N) regions: F, F/N, 2 and F/N/F junctions which are subject to thermal gradients. We present analytical formulas for 1 the spin accumulation and spin current profiles in those junctions that are valid for both tunnel 0 and transparent (as well as intermediate) contacts. For F/N junctions we calculate the thermal 2 spin injection efficiency and the spin accumulation induced nonequilibrium thermopower. We find conditions for countering thermal spin effects in the N region with electrical spin injection. This n a compensatingeffectshouldbeparticularlyusefulfordistinguishingelectronicfromothermechanisms J of spin injection by thermal gradients. For F/N/F junctions we analyze the differences in the nonequilibrium thermopower (and chemical potentials) for parallel and antiparallel orientations 7 of the F magnetizations, as evidence and a quantitative measure of the spin accumulation in N. 2 Furthermore,westudythePeltierandspinPeltiereffectsinF/NandF/N/Fjunctionsandpresent ] analytical formulas for the heat evolution at the interfaces of isothermal junctions. l l a PACSnumbers: 72.15.Jf,72.25.-b,85.75.-d h Keywords: spintronics,spincaloritronics,spinSeebeckeffect,spinPeltiereffect,spin-chargecoupling - s e m I. INTRODUCTION improvements over the past few years have made its ap- plication in the context of generating and transporting . t spin appear possible.22–26 a The central theme in spintronics is the generation and m control of nonequilibrium electron spin in solids.1–4 Un- tilrecentlythespingenerationhasbeendonebyoptical, - d magnetic,and,mostimportantfordeviceprospects,elec- n trical means.2,5 In a typical device spin-polarized elec- o trons from a ferromagnetic conductor are driven by elec- c tromagnetic force to a nonmagnetic conductor. There [ the spin accumulates, with the steady state facilitated 2 by spin relaxation. (There are also novel ways to gen- v erate pure spin currents, without accompanying charge 8 currents.6–10) The concept of electrical spin injection 0 8 wasfirstproposedbyAronov,11 andexperimentallycon- 1 firmed by Johnson and Silsbee,12 who also formulated FIG. 1: (Color online) Schematic illustrations of the Seebeck 2. theproblemfromanonequilibriumthermodynamicsand (a)andspinSeebeck(b)effects. Here∆T isthetemperature 1 drift-diffusion view.13,14 An equivalent description in difference, V the voltage, j the charge current, js the spin 1 termsofquasichemicalpotentials,convenienttotreatdis- current,andverticalarrowsdenoteup/downspinprojections. 1 crete (junction) systems was formulated systematically : by Rashba.15 This model, which we call the standard At the heart of spin caloritronics is the spin Seebeck v i model of spin injection, is widely used to describe elec- effect (see Fig. 1).27–29 The conventional Seebeck effect, X trical spin injection into metals and semiconductors1,2,5 alsocalledthermopower,20 describesthegenerationofan r and can also be extended to ac currents.16 electric voltage if a thermal gradient is applied to a con- a Untilrecentlyoneparticularlyinterestingpossibilityof ductor. In analogy, the spin Seebeck effect describes the generating spin, by spin-heat coupling, has been largely generationofspinaccumulationinferromagnetsbyther- neglected. Thegenerationofnonequilibriumspinbyheat mal gradients. The effect was originally observed in the currentsandtheoppositeprocessofgeneratingheatcur- ferromagnetic conductor NiFe,27,30 where indication of rents by spin accumulation has already been proposed spin accumulation over large length scales (millimeters), by Johnson and Silsbee13 based on nonequilibrium ther- independentofthespinrelaxationscalesintheferromag- modynamics concepts (see also Ref. 17). The spin-heat net,wasfound. Sinceitalsoexistsatroomtemperature, coupling is now the central point of spin caloritronics thespinSeebeckphenomenonmayhavesometechnolog- (or spin calorics).18,19 Although the theory of thermo- ical applications.31 electricity has long been known,20,21 only experimental However, the spin Seebeck effect is not limited to 2 loy(Ni Fe )(PY)/copper/PYvalvestack.38,39Thespin 80 20 Peltier effect describes the heating or cooling at the interface between a ferromagnetic and normal conductor driven by a spin current (see Fig. 2). Another fascinating discovery is that of the ther- mally driven spin injection from a ferromagnet to a normal conductor.40 In this experiment thermal currents in permalloy drive spin accumulation into copper, detected in a non-local geometry.2,5 The structures were of submicron sizes, so it is plausible that the effects are electronic in nature, although magnon contributions to such thermal spin injection setups could also be sizable. A practical model was introduced in Refs. 38,41 to find, with a finite elements numerical scheme, the profiles of temperature and spin accumulation in the experimental devices. Recently, yetanotherformofthermalspinflow, coinedSeebeckspintunneling,hasbeendemonstratedin ferromagnet-oxide-silicontunneljunctions.42 Hereatem- perature difference between the ferromagnet and silicon causes a transfer of spin angular momentum across the interface between both materials. An important goal for both theory and experiment of the spin Seebeck phenomena is to decipher the roles of FIG. 2: (Color online) Schematic illustrations of the Peltier (a) and spin Peltier (b) effects, where j and j denote the the electronic and non-electronic contributions. It is yet s charge and spin currents. The thermal current j is different unclear under which circumstances the electronic contri- q in each region. Small vertical arrows denote up/down spin butionmaydominate. Itseemslikelythatwhengoingto projections. smaller, submicron structures in which the spin accumulation will be a bulk effect, the spin phenomena carried byelectronswillbecomeimportant. Similarly,inmateri- metals. It has also been observed in ferromagnetic alswithstrongmagnondamping,suchthatmagnonsare insulators29 as well as in the ferromagnetic semiconduc- in local equilibrium with the given temperature profile, tor (Ga,Mn)As.32 This suggests that the spin Seebeck electrons may ultimately carry the entire spin Seebeck effectdoesnotneedtobeconnectedwithchargeflow. In effect. Itisthusimportanttosetthebenchmarksforthe (Ga,Mn)As the sample was even cut preventing charge electroniccontributionsinusefuldevicegeometries. This redistribution over the whole slab; the spin Seebeck sig- is what this paper does: we explore the role of the elec- nals were unaffected and in both cases, of compact and tronic contributions in F/N and F/N/F junctions which disconnected samples, the Pt stripes pick up the same are subjected to thermal gradients and derive useful an- inverse spin Hall signals.33,34 The evidence points to a alytical formulas for various spin injection efficiencies. mechanism of magnon-assisted spin pumping from the Ourpurposeistwofold: First,weusethedrift-diffusion ferromagnet into the Pt, producing spin currents there. framework of the standard model of spin injection pre- A theory for this spin pumping from a ferromagnetic sentedinRefs.1,2,5andgeneralizeittoincludeelectronic insulator was suggested in Ref. 35. It was predicted heat transport and thereby derive a theory for charge, that phonons can play an important role in the spin spin, and heat transport in electronic materials. Sec- Seebeck effect, leading to its huge enhancement.36 Re- ondly, we apply this theory to describe F/N and F/N/F cent measurements of the spin Seebeck effect in multiple junctions placed in thermal gradients. While the Peltier (Ga,Mn)As samples also suggest that the spin Seebeck and Seebeck effects in such structures have been inves- effect can be driven by phonons.37 In order to explain tigated in Ref. 43, we focus here on the description of the main trends of the observed temperature and spatial thermal spin injection and the investigation of the cor- dependence of the spin Seebeck effect in (Ga,Mn)As, a responding spin accumulation. We also look at the spin phenomenological model involving phonon-magnon cou- injectioninthepresenceofbothelectricandthermalcur- pling was introduced.37 rents, and find the conditions under which the resulting In addition to the Seebeck effect, there is also another spin current in N vanishes. In all junctions studied we thermoelectric effect, the Peltier effect, which refers to present,asgeneralaspossible,analyticalformulasforthe theevolutionofheatacrossanisothermaljunctionoftwo spinaccumulationandspincurrentprofiles,aswellasfor differentmaterialsduetoanelectriccurrentbeingpassed the thermal spin injection efficiency and the nonequilib- through the junction.20,21 Recently a spin caloritronics rium(spinaccumulationdriven)spinSeebeckcoefficient. analogtothePeltiereffect,termedspinPeltiereffect,has Moreover,welookatseveraldifferentsetupsofthePeltier beenpredictedandexperimentallyobservedinapermal- andspinPeltiereffectsandcalculatetheirrespectivecon- 3 tributionstotheheating/coolingattheinterfacesinF/N Assuming the local nonequilibrium distribution function and F/N/F junctions. to be only energy dependent because momentum relax- The manuscript is organized as follows: Following the ation happens on length scales much smaller compared introduction of the formalism and the basic equations in tothevariationoftheelectricpotentialϕ(x),oneobtains Sec. II, the electronic contribution to the spin Seebeck (cid:26) (cid:27) effect in a ferromagnetic metal is discussed within the f (ε,x)=f ε−η[T(x)]−eµλ(x)−eϕ(x) . (3) framework of this formalism in Sec. III, while Secs. IV λ 0 k T(x) B andVaredevotedtothediscussionofthermalspininjec- Therefore, the nonequilibrium electron and energy den- tionandrelatedthermoelectriceffectsinF/NandF/N/F sities read junctions respectively. A short summary concludes the manuscript. (cid:90) n (x)= dεg (ε)f (ε,x) λ λ λ (4) =n0{η[T(x)]+eµ (x)+eϕ(x),T(x)}, II. SPIN-POLARIZED TRANSPORT IN THE λ λ PRESENCE OF THERMAL FLUCTUATIONS: CONCEPTS AND DEFINITIONS (cid:90) e (x)= dεεg (ε)f (ε,x) λ λ λ (5) A. Spin-unpolarized transport equations =e0{η[T(x)]+eµ (x)+eϕ(x),T(x)}. λ λ As a first step we will restrict ourselves to the descrip- The electrostatic field gives rise to an electric current. tion of transport in an electronic system which consists This charge current consists of two parts: the drift cur- only of electrons of one species, that is, either of spin up rent, proportional to the electric field E(x) = −∇ϕ(x) orspindownelectrons(denotedbythesubscriptλ=↑/↓ andthediffusioncurrent, proportionaltothegradientof throughout this manuscript). The derivation presented the local electron density. here is a textbook matter20,44 and is given here to intro- Since the proportionality factor of the diffusion current, duce the terminology needed for the spin-polarized case the diffusivity D (ε), is energy dependent, it is conve- λ and to match the concepts from the standard spin injec- nienttotreatelectronswithdifferentenergiesseparately. tion model of Ref. 2. The spectral diffusion current density reads If this system is in thermodynamic equilibrium, the temperature T and the chemical potential η(T) are uni- j (x,ε)dε=eD (ε)∇[g (ε)f (ε,x)]dε, (6) Dλ λ λ λ form throughout the system. Knowing the chemical potential,53 one can calculate the density of the respec- from which the complete diffusion current can be ob- tive electron species under consideration from tained by integrating over the entire energy spectrum. The total charge current for electrons of spin λ is given n0[η(T),T]=(cid:90) dεg (ε)f (cid:20)ε−η(T)(cid:21), (1) by λ λ 0 k T B (cid:90) j (x)=−σ ∇ϕ(x)+e dεD (ε)g (ε)∇f (ε,x), (7) where k denotes the Boltzmann constant, g (ε) the λ λ λ λ λ B λ electronic density of states at the energy ε, and f the 0 equilibriumFermi-Diracdistributionfunction. Similarly, where σλ is the conductivity. Inserting Eq. (3) into the equilibrium energy density is given by Eq. (7), using the Einstein relation,55 and keeping only terms linear in the nonequilibrium quantities µ (x) and λ (cid:90) (cid:20)ε−η(T)(cid:21) ϕ(x), we find e0[η(T),T]= dεεg (ε)f . (2) λ λ 0 k T B (cid:26) (cid:27) η[T(x)] j (x)=σ ∇ +µ (x) −S σ ∇T(x). (8) The system is not in equilibrium if an electric field λ λ e λ λ λ −∇ϕ(x) is present in its bulk. In this case the chemical potential becomes space dependent. This is taken Here the conductivity is given by the Einstein relation intoaccountbyreplacingη(T)withη(T)+eµλ(x),where (cid:90) (cid:18) ∂f (cid:19) dtheepeqnudaesnicchee.5m4icSailnpceotwenetiwalanµtλ(txo)innocworcpoonrtaatiensththeeesffpeacctes σλ =e2 dεDλ(ε)gλ(ε) − ∂ε0 ≈e2Dλ(εF)gλ(εF) (9) of thermal gradients into our formalism, we furthermore and the Seebeck coefficient by allow for different local equilibrium temperatures by re- placing the constant temperature T by a space depen- e (cid:90) (cid:18) ∂f (cid:19)ε−η[T(x)] dent temperature T(x). As a consequence there is an S =− dεD (ε)g (ε) − 0 λ σ λ λ ∂ε T(x) additional position dependence of the chemical potential λ due to the temperature, that is, η(T) has to be replaced ≈−LeT(x)(cid:20)gλ(cid:48)(εF) + Dλ(cid:48)(εF)(cid:21). by η[T(x)]. Thus, the total chemical potential is given g (ε ) D (ε ) λ F λ F by η[T(x)]+eµ (x). (10) λ 4 In both cases the integrals are calculated to the first forthecompletelocalelectrondensityofthesystem. Ex- non-vanishingorderintheSommerfeldexpansion.20 The panding the electron density up to the first order in the Lorenz number is L = (π2/3)(k /e)2 and g(cid:48)(ε ) and local nonequilibrium quantities, µ (x), µ (x), and ϕ(x), B λ F ↑ ↓ D(cid:48)(ε ) are the derivatives of the density of states and andusingtheSommerfeldexpansionsubsequentlytocal- λ F thediffusivitywithrespecttotheenergyevaluatedatthe culatetheintegralswhichenterviaEq.(1), wecanwrite Fermi level ε . the electron density as F In addition to the charge current, there is a heat current in nonequilibrium. A treatment similar to that of n(x)=n +δn(x). (15) 0 the charge current above yields Here we have introduced the local equilibrium electron (cid:26) (cid:27) η[T(x)] density, n = n0{η[T(x)],T(x)} + n0{η[T(x)],T(x)}, j (x)=S σ T(x)∇ +µ (x) 0 ↑ ↓ q,λ λ λ e λ (11) and the local nonequilibrium electron density fluctua- −Lσ T(x)∇T(x). tions, λ If the charge and heat currents are defined as in Eqs. (8) δn(x)=eg[µ(x)+ϕ(x)]+egsµs(x). (16) and(11),currentsj (x)>0andj (x)>0flowparallel λ q,λ to the x direction. Additionally, we have introduced the quasichemical po- At sharp contacts the chemical potential and the tem- tential, µ = (µ↑ +µ↓)/2, the spin accumulation, µs = perature are generally not continuous. Thus, instead of (µ↑ − µ↓)/2, as well as the densities of states g = Eqs. (8) and (11), discretized versions of these equations g↑(εF)+g↓(εF) and gs = g↑(εF)−g↓(εF) at the Fermi are used. The charge current at the contact (C) is given level. We further assume that there is no accumulation by of charge inside the conductor under bias ϕ(x). This as- sumptionoflocalchargeneutralityisvalidformetalsand (cid:18) (cid:19) 1 highly doped semiconductors and requires n(x) = n .56 j =Σ ∆η +∆µ −S Σ ∆T (12) 0 λc λc e c λc λc λc c Hence, Eq. (15) yields the condition and the heat current by δn(x)=0. (17) (cid:18) (cid:19) 1 j =TS Σ ∆η +∆µ −LTΣ ∆T , (13) The local spin density, qλc λc λc e c λc λc c s(x)=n0{η[T(x)]+eµ (x)+eϕ(x),T(x)} where ∆η +e∆µ and ∆T denote the drops of the to- ↑ ↑ c λc c (18) talchemicalpotentialandthetemperatureatthecontact −n0{η[T(x)]+eµ (x)+eϕ(x),T(x)}, ↓ ↓ respectively. The(effective)contactconductanceandthe contact thermopower are given by Σ and S respec- can be evaluated analogously to the local electron den- λc λc tively, while T is the average temperature of the system. sity: First, Eq. (18) is expanded in the local nonequilibrium quantities up to the first order. The resulting integrals are performed employing the Sommerfeld ex- B. Spin-polarized transport equations pansionuptothefirstnon-vanishingorderand,asafinal step, thechargeneutralitycondition, Eq.(17), isusedto We now consider spin-polarized systems, which we simplify the result. This procedure yields treatasconsistingoftwosubsystems, oneofspinupand one of spin down electrons; each subsystem is described s(x)=s0(x)+δs(x), (19) by the equations from Sec. IIA. Energy as well as particles can be exchanged between with the local equilibrium spin density, s0(x) = the two spin pools (by collisions and spin-flip processes n0↑{η[T(x)],T(x)} − n0↓{η[T(x)],T(x)} and the local respectively). As energy relaxation (tens of femtosec- nonequilibrium spin density, onds) happens usually on much shorter time scales than g2−g2 spinrelaxation(picosecondstonanoseconds), weassume δs(x)=e sµ (x). (20) thatalocalequilibriumexistsateachpositionx. Conse- g s quently, both subsystems share a common local equilibrium chemical potential η[T(x)] and temperature T(x). It is important to note that s0(x) is determined by the On the other hand, the local nonequilibrium quasichem- local temperature T(x), as a result of the rapid energy ical potentials µ (x) can be different for each spin sub- relaxation as compared to the spin relaxation. λ system. The same procedure can be applied to calculate the From Eq. (4) we obtain energy density from Eq. (5), n(x)=n0{η[T(x)]+eµ (x)+eϕ(x),T(x)} e(x)=e0{η[T(x)]+eµ (x)+eϕ(x),T(x)} ↑ ↑ ↑ ↑ (14) (21) +n0{η[T(x)]+eµ (x)+eϕ(x),T(x)} +e0{η[T(x)]+eµ (x)+eϕ(x),T(x)}, ↓ ↓ ↓ ↓ 5 which can be split in a local equilibrium energy density, spin relaxation mechanisms. We stress that spin relax- e (x) = e0{η[T(x)],T(x)}+e0{η[T(x)],T(x)}, and lo- ationprocessesbringthenonequilibriumspins(x)tothe 0 ↑ ↓ cal energy density fluctuations δe(x), that is, (quasi)equilibrium value s0(x), defined locally by T(x). Here we deviate from the treatment given in Ref. 30. e(x)=e (x)+δe(x). (22) The heat current, 0 Calculating δe(x) in the same way as δs(x), we find that jq(x)=jq,↑(x)+jq,↓(x) (cid:26) (cid:27) T (Sσ+S σ ) η[T(x)] δe(x)=0, (23) = s s ∇ +µ(x) 2 e (28) consistent with our assumption of fast energy relaxation + T (Ssσ+Sσs)∇µ (x)−LTσ∇T(x), to the local quasiequilibrium. 2 s Next,weconsiderthecurrentsflowingthroughthesys- istheheatcarriedthroughthesystembytheelectronsof tem. Sinceourgoalistocalculatethequasichemicaland both spin species. Closely related is the energy current, spinquasichemicalpotentials,aswellasthetemperature profile, we not only derive transport equations based on (cid:26)η[T(x)] (cid:27) j (x)=j (x)− +µ(x) j−µ (x)j (x). (29) Eqs. (8) and (11), but also continuity equations for each u q e s s ofthecurrentsconsidered,thatis,charge,spin,andheat currents. Inserting Eqs. (24), (26), and (28) and using that the di- Thechargecurrentconsistsoftheelectriccurrentscar- vergenceofthechargecurrentvanishesinasteadystate, ried by spin up and spin down electrons, that is, Eq. (25), we find j(x)=j (x)+j (x) T(x) ↑ ↓ ∇j (x)= ∇[Sj+S j (x)]−µ (x)∇j (x) (cid:26) (cid:27) u 2 s s s s η[T(x)] =σ∇ e +µ(x) +σs∇µs(x) (24) −∇(cid:20)LσT(x)(cid:18)1− S2+Ss2+2SSsPσ(cid:19)∇T(x)(cid:21) 4L 1 − 2(Sσ+Ssσs)∇T(x), − j↑2(x) − j↓2(x), σ σ where the conductivities are given by σ = σ +σ and ↑ ↓ ↑ ↓ (30) σ =σ −σ ,andtheSeebeckcoefficientsbyS =S +S s ↑ ↓ ↑ ↓ where P = σ /σ is the conductivity spin polarization. and S =S −S . In nonmagnetic materials σ =0 and σ s s ↑ ↓ s TheaboveformulacontainsThomson(firstterm)aswell S = 0. In our model we consider a steady state, which s as Joule heating (final two terms). Equation (23) can be requires used to formulate the continuity equation for the energy current by enforcing the energy conservation, ∇j(x)=0, (25) ∇j (x)=0. (31) that is, a uniform electric current, j(x)=j. u The spin current is the difference between the electric Thus, if j is treated as an external parameter, the currents of spin up and spin down electrons, transport equation for the charge current, Eq. (24), as wellasthetransportandcontinuityequationsforthespin j (x)=j (x)−j (x) s ↑ ↓ andheatcurrents, Eqs.(26), (27), (28), and(31), forma (cid:26) (cid:27) η[T(x)] complete set of inhomogeneous differential equations to =σ ∇ +µ(x) +σ∇µ (x) s e s (26) determine the quasichemical potentials µ(x) and µs(x), thetemperatureprofileT(x),aswellasthecurrentsj (x) 1 s − 2(Ssσ+Sσs)∇T(x). and jq(x). The solution to this set of differential equations, that couple charge, spin, and heat transport, will As we have seen, the spin density s(x) deviates from its be discussed in the next section. local equilibrium value s (x). Unlike charge, spin is not 0 conservedandspinrelaxationprocessesleadtoadecrease of the local nonequilibrium spin to s (x). Therefore, the C. Spin diffusion equation and its general solution 0 continuity equation for the spin current is given by In the following the general solutions to the equa- δs(x) tions introduced in Sec. IIB will be discussed. Insert- ∇j (x)=e , (27) s τ ing Eq. (26) into the spin current continuity equation, s Eq. (27), and using Eqs. (20), (24), and (25) generalizes where τs is the spin relaxation time. We will not dis- the standard45,46 spin diffusion equation, tinguish between different spin relaxation mechanisms in our model. Instead, we treat τ as an effective µ (x) 1 s ∇2µ (x)= s + ∇·[S ∇T(x)]. (32) spin relaxation time which incorporates all the different s λ2 2 s s 6 Here we have introduced the spin diffusion length1,2 Asbefore,A,B,C,D,andEareintegrationconstantsto bespecifiedbyboundaryconditions. However, assuming (cid:112) λs = τsgσ(1−Pσ2)/[e2(g2−gs2)]. (33) a constant temperature gradient in ferromagnets is not consistentwithEq.(31)andthereforethisapproximation As we are primarily interested in linear effects, we ne- cannot be used in situations which depend crucially on glect the position-dependence of the spin Seebeck coeffi- the heat current profile (see next section). cientS ,whichentersviaT(x),andarriveatasimplified s The spin and heat currents can be obtained by insert- diffusion equation for the spin accumulation, ing the solutions found above into Eqs. (26) and (28). µ (x) S ∇2µ (x)= s + s∇2T(x), (34) s λ2 2 s D. Contact properties where S is evaluated at the mean temperature T. In s order to solve this equation, we need the temperature To find the specific solution for a system consisting of profile which can be determined from Eq. (31). If only different materials, such as a F/N junction, we have to first order effects are taken into account, Eq. (31) gives know the behavior of the currents at the interfaces be- the differential equation tween two different materials. The currents at a contact can be obtained by applying Eqs. (12) and (13), giving 2S (1−P2) ∇2T(x)= λ2s(4L−S2s−Ss2−σ 2SSsPσ)µs(x), (35) jc =j↑c+j↓c =Σc(cid:18)1e∆ηc+∆µc(cid:19)+Σsc∆µsc (43) deforming the typically linear profile of T(x). The solu- 1 − (S Σ +S Σ )∆T , tion to the coupled differential Eqs. (34) and (35) reads 2 c c sc sc c (cid:18) (cid:19) (cid:18) (cid:19) x x µs(x)=Aexp λ˜ +Bexp −λ˜ , (36) j =j −j =Σ (cid:18)1∆η +∆µ (cid:19)+Σ ∆µ s s sc ↑c ↓c sc e c c c sc (44) 1 T(x)= 2Ss(1−Pσ2) µ (x)+Cx+D, (37) − 2(SscΣc+ScΣsc)∆Tc, 4L−(S+S P )2 s s σ (cid:18) (cid:19) with the modified spin diffusion length T 1 j =j +j = (S Σ +S Σ ) ∆η +∆µ qc q↑c q↓c 2 c c sc sc e c c (cid:115) 4L−S2−S2−2SS P T λ˜ =λ s s σ. (38) + (S Σ +S Σ )∆µ −LTΣ ∆T , s s 4L−(S+S P )2 2 sc c c sc sc c c s σ (45) IntegrationofEq.(24)yieldsthetotalchemicalpotential, where ∆Tc is the temperature drop at the contact, and ∆η ,∆µ ,and∆µ arethedropsofthelocalequilibrium c c sc η[T(x)] j S+S P chemical, quasichemical and spin quasichemical poten- +µ(x)= x−P µ (x)+ s σT(x)+E. e σ σ s 2 tials. Moreover,thecontactconductancesΣc =Σ↑c+Σ↓c (39) and Σc =Σ −Σ as well as the contact thermopowers s ↑c ↓c The integration constants A, B, C, D, and E have to be S = S +S and S = S −S have been intro- c ↑c ↓c sc ↑c ↓c determined by including the respective boundary condi- duced. tions of the√system under consideration. Equations (43)-(45) will be used in Secs. IV and V to If Sλ (cid:28) L (see the next section), it is often possible fix the integration constants of the general solutions, to assume a uniform temperature gradient, that is, Eqs. (40)-(42) and Eqs. (36)-(42) found in Sec. IIC. T(x)=Cx+D. (40) III. FERROMAGNET PLACED IN A THERMAL Then Eq. (34) reduces to the standard spin diffusion GRADIENT equation and its solution is given by (cid:18) x (cid:19) (cid:18) x (cid:19) AsafirstexampleweconsideraferromagneticmetalF µs(x)=Aexp λ +Bexp −λ , (41) of length L (−L/2<x<L/2) subject to a thermal gra- s s dient under open-circuit conditions, that is, j = 0. The while integration of Eq. (24) yields the total chemical gradient is applied by creating a temperature difference potential, ∆T =T2−T1 between both ends of the metal which are heldattemperaturesT andT respectively, asshownin (cid:18) (cid:19) 1 2 η[T(x)] +µ(x)= j + S+SsPσC x−P µ (x)+E. Fig. 3. e σ 2 σ s At the ends of the ferromagnet we impose the bound- (42) ary conditions T(−L/2) = T , T(L/2) = T , and set 1 2 7 10 (a) 10 (b) simplified model V] full model -70 5 1 5 [0 m+ /e-5 V] h-10 -80 0 -40 -20 x [0nm] 20 40 1 3 [ (c) m s 2A/m]2 -5 70 j [1s1 0 -40 -20 0 20 40 x [nm] -10 -40 -20 0 20 40 x [nm] FIG. 4: (Color online) Profiles of the spin accumulation (a), FIG.3: (Coloronline)Aschematicillustrationofaferromag- the total chemical potential (b), and the spin current (c) for net metal placed in a thermal gradient which leads to the Ni Fe at T = 300 K with L = 100 nm and ∆T = 100 81 19 generation of a spin current. mK. The solid lines show the results obtained if a constant temperature gradient ∇T = ∆T/L is assumed, while the F dashed lines (fully overlapping with the solid ones) show the j (±L/2)=0. Since we consider only first order effects, results obtained if the temperature profile is determined by s the Seebeck coefficients are assumed to be constant over ∇ju =0. the length of the ferromagnet and are evaluated at the mean temperature T = (T +T )/2. Using the above 1 2 √ boundary conditions and Eqs. (36)-(39) yields the spin FormetalsS (cid:28) LandEqs.(46)and(47)reduceto λ accumulation Eqs. (49) and (50), that is, the assumption of a uniform temperature gradient ∇T = ∆T/L is justified. Only at S ∆T sinh(x/λ˜ ) 4L−(S+S P )2 µ (x)= s λ˜ s s σ , the boundaries of the sample both temperature profiles s 2 s L cosh(L/2λ˜ ) N(L) differ (insignificantly) as there is a small exponential de- s (46) cay within the spin diffusion length λ˜ ≈ λ if the full s s and the spin current model is used compared to a perfectly linear temperature profile of the reduced model. (cid:34) (cid:35) S λ˜ ∆T cosh(x/λ˜ ) j (x)=− s s 1− s Equations (49) and (50) from the reduced model cor- s 2 R˜ L cosh(L/2λ˜ ) respondtotheprofilesofthespinaccumulationandspin s (47) current found in Ref. 47, where a Boltzmann equation 4L−S2−S2−2SS P × s s σ, approach has been used to describe thermoelectric spin N(L) diffusion in a ferromagnetic metal. where R˜ =λ˜ /(cid:2)σ(1−P2)(cid:3) and In Fig. 4 the results calculated for a model Ni81Fe19 s σ filmwithrealisticparameters30 [λ =5nm,σ =2.9×106 s N(L)=4L−S2−S2−2SS P 1/Ωm,S0 =(S↑σ↑+S↓σ↓)/(σ↑+σ↓)=−2.0×10−5 V/K s s σ with P = 0.7 and P = (S − S )/(S + S ) = 3.0] +S2(cid:0)1−P2(cid:1)tanh(L/2λ˜s). (48) at a meσan temperaturSe T = 3↑00 K↓are d↑isplay↓ed. The s σ L/2λ˜ length of the sample is L=100 nm and the temperature s differenceis∆T =100mK.AscanbeseeninFig.4, the If a constant temperature gradient is assumed and the agreement between both solutions is very good. reduced model given by Eqs. (40)-(42) is used, the spin Figure 4 (b) shows an almost linear drop of the total accumulation reads chemical potential between both ends of the ferromag- S ∆T sinh(x/λ ) net. Only at the contacts this linear drop is superim- µs(x)= 2s λs L cosh(L/2λs ), (49) posed by an exponential decay. It is also at the contacts s that nonequilibrium spin accumulates and decays within and the spin current thespindiffusionlength[seeFigs.4(a)and 4(c)]. Thus, onlynearthecontactsthereisanelectroniccontribution (cid:20) (cid:21) S λ ∆T cosh(x/λ ) to the spin voltage and our electronic model does not re- j (x)=− s s 1− s , (50) s 2 R L cosh(L/2λ ) produce the linear inverse spin Hall voltage observed in s this system,27 which suggests that a mechanism differ- where R = λ /(cid:2)σ(1−P2)(cid:3) is the effective resistance of ent from electronic spin diffusion is responsible for the s σ the ferromagnet. detected spin Hall voltage.47 Also, the “entropic” terms 8 tor, denoted by the additional subscripts F and N in the quantities defined in the previous sections. The ex- tension of the ferromagnet is given by −L < x < 0, F whereas the nonmagnetic conductor is described by val- ues 0 < x < L . We also assume that the properties of N the contact region C, located at x = 0, are known. By coupling the F and N regions to reservoirs with different temperatures, T and T respectively, a temperature 2 1 gradientiscreatedacrossthejunction. Themodelinves- tigated in the following is summarized in Fig. 5. Like in the previous section, we can assume uniform (but for each region different) temperature gradients ∇T and∇T andusethesimplifiedspindiffusionequa- F N tion,Eq.(34),andthecorrespondingsolutions,Eqs.(40)- (42), to describe the total chemical potential, the spin accumulation,andthetemperatureprofileineachregion FIG.5: (Coloronline)AschematicillustrationofaF/Njunc- separately. The integration constants are solved invok- tion placed in a thermal gradient. ing the following boundary conditions: T(−L ) = T , F 1 T(L ) = T , and j (−L ) = j (L ) = 0. Furthermore, N 2 s F s N weuseEqs.(43)-(45)andassume,asinthestandardspin in the spin accumulation as introduced in Ref. 30, which injectionmodel,2 thatthecharge,spin,andheatcurrents would lead to a uniform decay of the spin accumulation are continuous at the interface, giving us five additional across the whole sample, not just at the distances of the equations for the integration constants. From this set of spindiffusionlengthsoffoftheedges, donotariseinour equations the integration constants, including the gradi- theory. ents ∇T and ∇T , can be obtained. Depending on the F N choiceofthedirectionofthegradient,onefindsthatspin is either injected from the F region into the N region or IV. F/N JUNCTIONS extracted from the N region by a pure spin current, that is, a spin current without accompanying charge current. A. F/N junctions placed in thermal gradients In order to measure the efficiency of the thermal spin injection [j (0)<0] and extraction [j (0)>0] at the in- s s In this section we investigate an open (j = 0) F/N terface,wecalculatethethermal spin injection efficiency junction under a thermal gradient. The F/N junction κ=j (x=0)/∇T ,whichcorrespondstoaspinthermal s N consists of a ferromagnet and a nonmagnetic conduc- conductivity. Our model gives σ tanh(L /λ )(cid:8)tanh(L /λ )S R (cid:0)1−P2(cid:1)+(cid:2)1−cosh−1(L /λ )(cid:3)S R (cid:0)1−P2 (cid:1)(cid:9) κ=− N N sN F sF sc c Σ F sF sF F σF , (51) 2 R tanh(L /λ )+R tanh(L /λ )tanh(L /λ )+R tanh(L /λ ) F N sN c N sN F sF N F sF ∆T with the effective resistances for the F, N, and contact ∇T = , (57) regions, N σNRFN where R = λ /σ , (52) N sN N R = λ /(cid:2)σ (1−P2 )(cid:3), (53) R = LF + 1 + LN. (58) F sF F σF FN σ Σ σ R = 1/(cid:2)Σ (1−P2)(cid:3), (54) F c N c c Σ If the sample sizes are large, that is, if L (cid:29)λ and F sF and the contact conductance spin polarization LN (cid:29) λsN, as is usually the case (but not in Figs. 6 and 7 where L <λ ), the situation at the interface is N sN P =Σ /Σ . (55) not sensitive to the boundary conditions far away from Σ sc c the interface and Eq. (51) reduces to Equation(51)hasbeenderivedinthelimitofS (cid:28) √ λF/N/c σ S R (cid:0)1−P2(cid:1)+S R (cid:0)1−P2 (cid:1) L, in which the temperature gradients are given by κ=− N sc c Σ sF F σF 2 RF +Rc+RN (59) ∆T σ ∇TF = σFRFN, (56) =− 2N(cid:104)Ss(1−Pσ2)(cid:105)R, 9 where (cid:104)...(cid:105) denotes an average over the effective resis- 2 R (a) tances. The above expressions for the spin injection ef- F N ficiency and the gradients, Eqs. (51)-(59), could have also been obtained by using Eqs. (36)-(39) to calcu- 1 √ 0 Etelahffiqteeuciawettnheiolcelny-k.p1nr(,2oo5wfi9)lnesifsoarntmhdeutlaaspkfiionnr-ghtethaheteecllieomcutpirtliicnSagλlFse/pqNiun/icvian(cid:28)ljeencttioLonf. -7 [10V]s0 -6m/e+ [10V]-1 (b) Usingthespininjectionefficiency,Eq.(51)[orEq.(59) m h-2 -40 -20 0 20 40 -1 x [nm] for large devices], the profiles of the spin current and ac- R=10-16W m2 cumulation in the N region (0<x<L ) can be written c N R=10-14W m2 compactly as c -2 -50 -40 -30 -20 -10 0 10 20 30 40 50 j (x)=−κ∇T sinh[(x−LN)/λsN] (60) x [nm] s N sinh(L /λ ) N sN FIG. 6: (Color online) Profiles of the spin accumulation (a) and and the total chemical potential (b) for a Ni Fe /Cu junc- 81 19 µ (x)=−R κ∇T cosh[(x−LN)/λsN], (61) tion at T =300 K with LF =LN =50 nm and ∆T =−100 s N N sinh(LN/λsN) mK.ThesolidlinesshowtheresultsforRc =1×10−16 Ωm2, the dashed lines for R =1×10−14 Ωm2. c which reduce to 0 j (x)=κ∇T exp(−x/λ ) (62) (a) s N sN F N -1 and 4.5 µs(x)=−RNκ∇TNexp(−x/λsN) (63) 2m]-2 2m] 4 (b) for LN (cid:29) λsN. In particular, at the contact the spin 7A/-3 70W/3.5 alactceudmauslation in the nonmagnetic material can be calcu- j [10s-4 j [1q2.53 -40 -20 0 20 40 x [nm] µs(0+)=−RNκ∇TNcoth(LN/λsN). (64) -5 R=10-16W m2 c R=10-14W m2 Equation (51) also makes it clear that whether there is -6 c spininjectionorextractiondependsnotonlyonthedirec- -50 -40 -30 -20 -10 0 10 20 30 40 50 tionofthetemperaturegradient, butalsoonthespecific x [nm] materials chosen. Another quantity of interest is the total drop of the FIG.7: (Coloronline)Profilesofthespincurrent(a)andthe chemical potential across the F/N junction, heat current (b) for a Ni Fe /Cu junction at T = 300 K 81 19 withL =L =50nmand∆T =−100mK.Thesolidlines F N ∆(η/e+µ)=[η(T2)−η(T1)]/e+µ(LN)−µ(−LF), showtheresultsforRc =1×10−16 Ωm2,thedashedlinesfor (65) R =1×10−14 Ωm2. c because—in analogy to the calculation of the total resistance of the F/N junction in the case of the electrical spin injection2—it allows us to define the total Seebeck is the nonequilibrium contribution to the Seebeck coef- coefficient S of the device, which can be separated into ficient due to spin accumulation. If the extensions of an equilibrium and a nonequilibrium contribution: the F/N junction are much larger than the spin diffusion lengths, the nonequilibrium Seebeck coefficient can ∆(η/e+µ)≡S∆T ≡(S0+δS)∆T. (66) be expressed as Here SsFλsF(PΣ−2PσF) + κ[(PΣ−PσF)RF+PΣRN] δS = 2σF σN . (69) S = (SF +SsFPσF)LσFF +(Sc+SscPΣ)Σ1c +SNLσNN RFN 0 2R FN For illustration, the profiles of the total chemical po- (67) tential and the spin accumulation are displayed in Fig. 6 denotestheSeebeckcoefficientoftheF/Njunctioninthe for a junction consisting of Ni Fe (see Sec. III for absence of spin accumulation, whereas 81 19 the corresponding parameters) and Cu (λ = 350 nm, sN P [µ (−L )−µ (0−)]+P [µ (0−)−µ (0+)] σ = 5.88×107 1/Ωm, S = 1.84×10−6 V/K) with δS = σF s F s Σ s s N N ∆T a temperature difference ∆T = T2 − T1 = −100 mK (68) betweenbothendsofthejunctionandthemeantemper- 10 atureT =300K.30,40,48 Figure7showsthespinandheat currents for the same system. In Figs. 6 and 7 we have chosen R = 1×10−16 Ωm2 and R = 1×10−14 Ωm2, c c as well as P = 0.5, S = −1.0 × 10−6 V/K, and Σ c S = 0.5S .48 There is a drop of the total chemical sc c potential across the junction [see Fig. 6 (b)]. For the chosenparametersspinisinjectedfromtheFregioninto the N region, where nonequilibrium spin accumulates at the F/N interface and decays within the spin diffusion length [see Figs. 6 (a) and 7 (a) where L < λ ]. By N sN applying the temperature difference ∆T into the oppo- site direction, that is, by choosing T <T , the situation 1 2 reverses and spin would be extracted from the N region. Figure6(a)alsoillustratesthatthespinaccumulationin theNregiondecreaseswithincreasingcontactresistance. The heat current flows from the hot to the cold end of the junction [jq(x) > 0], as can be seen in Fig. (7) (b). FIG.8: (Coloronline)AschematicillustrationofaF/Njunc- Furthermore, one can observe that in the F region the tionplacedinathermalgradientwithachargecurrentbeing heat current is not perfectly constant and decreases at simultaneously driven through the junction. x = −L as well as at the contact,57 while in the N F region the heat current remains constant. We now discuss two important cases: transparent and B. Interplay between thermal gradients and tunnel contacts in large F/N junctions where L (cid:29)λ simultaneous charge currents F sF and L (cid:29)λ . For transparent contacts R (cid:28)R ,R N sN c F N and the spin injection efficiency reduces to Another interesting effect is the interplay between a σ S R (cid:0)1−P2 (cid:1) thermal gradient across the F/N junction and a simulta- κ=− N sF F σF . (70) neouschargecurrent(seeFig.8). Toanalyzethisprocess, 2 RF +RN wetakeEqs.(40)-(42),thistimewithafinitechargecur- Thermal electronic spin injection from a ferromagnetic rent j, and replace the boundary condition for the spin metal to a semiconductor, that is, the case of R (cid:29) current at x = −L by j (−L ) = P j while leaving N F s F σF R ,wouldsufferfromthesame”conductivity/resistance the boundary conditions for the temperature unchanged F mismatch problem”1,13,49,50 as the usual electrical spin and also taking j (L ) = 0 as before. By choosing the s N injection does. The nonequilibrium Seebeck coefficient charge current j = j appropriately, the effects of the com can then be written as charge current and the thermal gradient, each by itself S λ P (cid:18) R (cid:19) applicableforinjectingspinintotheNregionorextract- δS =− sF sF σF 1+ N . ing spin from it, can cancel each other out. As a result 2σ (L /σ +L /σ ) R +R F F F N N F N we find that for L (cid:29)λ a charge current (71) F sF In this case κ and δS are restricted only by the indi- R (cid:0)1−P2 (cid:1)S +R (cid:0)1−P2(cid:1)Sc vidual effective resistances R and R of the F and N j = F σF sF c Σ s∆T (74) F N com 2R (R P +R P ) regions. Moreover, the spin accumulation µ is contin- FN F σF c Σ s uous at transparent contacts, that is, µ (0+) = µ (0−) s s extracts (injects) the spin injected (extracted) through a and Eq. (64) yields the expression found in Ref. 40 for giventemperaturedifference∆T withnonetspincurrent µ (0)/∇T .58 s F in the N region. Tunnel contacts, on the other hand, have very large This effect is shown in Figs. 9 and 10 for the effective resistances R (cid:29) R ,R for which Eqs. (59) c F N Ni Fe /Cu junction investigated in this section (see 81 19 and (69) reduce to above). Wefindthatacurrentdensityofj =7.6×107 com κ=−σN S (cid:0)1−P2(cid:1) (72) A/m2 (jcom = 1.9 × 107 A/m2) is needed to compen- 2 sc Σ sate a temperature difference of ∆T = −100 mK if and Rc =1×10−16 Ωm2 (Rc =1×10−14 Ωm2). Figures9(a) and 10 (a) show that there is no spin accumulation and SsFλsF(PΣ−2PσF) + Ssc(1−PΣ2)[PσFRF−PΣ(RF+RN)] no spin current in the nonmagnetic material under the δS = 2σF 2 . R compensatingelectriccurrentcondition. Thedropofthe FN (73) chemical potential across the F/N junction is shown in The thermal spin injection efficiency for the tunnel junc- Fig. 9 (b) and the heat current flowing from the hot to tion is determined by the spin-polarization properties of the cold end of the junction in Fig. 10 (b). The spin in- thecontactandtheconductivitymismatchissuedoesnot jection compensation should be useful for experimental ariseinthiscase. Asimilarresulthasalsobeenobtained investigation of the purely electronic contribution to the recently in Ref. 52. spin Seebeck effect.