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Thermospin effects in superconducting heterostructures I. V. Bobkova1,2 and A. M. Bobkov1 1Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia 2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia (Dated: January 24, 2017) Recently the thermally created pure spin currents were predicted for Zeeman-split superconductor/normal metal heterostructures. Here it is shown that this ”thermospin” current can lead to an 7 accumulationofapurespinimbalanceinasystem. Thethermallyinducedspinimbalancecanreach 1 thevalueofZeemansplittingofthesuperconductingDOSandstronglyinfluencessuperconductivity 0 in the heterostructure. Depending on the temperature difference between the superconductor and 2 the normal reservoir it can enhance the critical temperature of the superconductor or additionally n suppress the zero-temperature superconducting state. The last possibility gives rise to an unusual a superconductingstate, which starts to develop at finitetemperature. J 3 2 The coupling of superconductivity and magnetism in with respect to their analogues in normal systems. The hybridstructuresisinfocusofthe researchincondensed thermally induced pure spin currents were predicted at ] matter physics now. It leads to many interesting fun- S/N interfaces with Zeeman-split superconductors29, at n o damental effects. Among them are Cooper pairs with interfaces between two Zeeman-split superconductors33 c finite center of mass momentum, triplet odd frequency and also in more complicated systems, composed of two - Cooper pairs1–13, longer spin lifetimes and spin charge Josephsonjunctioninterlayers33andJosephsonjunctions r p separation14–25, fully spin-polarized electric currents26. with spin-textured materials34, where the additional su- u The most part of the mentioned effects is based on cou- perconducting phase control of the spin current can be s pling charge and spin degrees of freedom and, therefore, achieved. . t is of interest of superconducting spintronics27. a Inthisworkweshowthattheexistenceofsucha”ther- m Recently it has been also shown that coupling of heat mospin”effectisanaturalsourceofathermallyactivated - transportandchargedegreesoffreedominsuperconduc- pure spin imbalance in superconductors. We explore the d tor/ferromagnet (S/F) hybrid structures also results in maximal value of this imbalance and its influence on the n quite interesting effects. The so-called giant thermoelec- superconductivityofS/Nbilayer. WepredictthatinS/N o c tric effect was predicted in S/F heterostructures with heterostructures with a temperature difference between [ applied in-plane magnetic field28,29 and in superconduc- N and S parts the spin imbalance can result in the en- tors with magnetic impurities30. The theoretical predic- hancement of its critical temperature. More interesting 1 v tionofthegiantthermoelectriceffectinS/Fheterostruc- case is when it leads to the enhancement or even ap- 2 tures was experimentally realized in Refs. 31 and 32. It pearance of superconductivity upon heating the sample. 7 was also predicted that use of doubly Zeeman-split su- Thiseffectcanbeaclearsignatureofthe presenceofthe 3 perconducting bilayers enhances the thermoelectric ef- thermallyinducedspinimbalanceinthesuperconductor. 6 fectsignificantly33. Furtherthethermoelectriceffectwas There are other ways to create spin imbalance in hy- 0 proposed in superconducting hybrids with spin-textured brid structures. One of them is the injection of an elec- . 1 materials without need to apply an external magnetic tric currentfromastrongferromagnet37. Itis inevitably 0 field34. accompanied by electric currents or voltages, that is, it 7 leads to coupling charge, spin and heat degrees of free- 1 It was reported that in addition to thermally cre- : ated electric currents in Zeeman-split superconducting dom simultaneously. Another way is the photo-assisted v spin-imbalance24. But in this case the spin imbalance systems there should exist also thermally created spin i X currents29,33. In contrast to thermoelectric currents, generation mechanism is strongly different. There is no anexternalsourceofspinsinthesystem. Thespinimbal- r these spin currents do not require a spin polarization a anceis acquiredhere notdue to spatialmovementofthe of the barrier between the materials of the hybrid and, spin but due to internal redistribution of the quasipar- therefore,canexisteveninsuperconductor/normalmetal ticles between the Zeeman-split energy subbands. This hybridsprovidedthatthere is a spin-dependentparticle- spin imbalance is absent in the absence of the spin-flip holeasymmetryonatleastonesideoftheinterface. Itis processes. worthto note that this spin Seebek effect (it can be also called by ”thermospin” effect) is also known for normal At first we describe qualitatively the mechanism of (nonsuperconducting)systems35,36, butthe greatadvan- thermally induced pure spin currentgeneratedatthe in- tageofusingsuperconductorsisthatthe spin-dependent terface between the normal metal and the Zeeman-split particle-hole asymmetry here is very large due to the superconductor. Letusconsiderthecorrespondinginter- presence of the superconducting gap at the Fermi level. face, as depicted in the insert to Fig. 1(b). The external Due to this reason the thermally induced electric and magnetic field is applied parallel to the superconduct- spin currents in superconducting structures are ”giant” ing film in order to avoid large orbital deparing of su- 2 perconductivity. In this situation the main effect of the 0.6 h=0.4∆ (a) field on the superconductivity is the Zeeman splitting of P✐PH 0.4 P the spin subbands. The gappedsuperconducting DOS is ∆ N spin-split and the DOS for spin-up and spin-down spin / 0.2 S N subbands are shifted by the value of the Zeeman field µ 0.0 h = µ H in the opposite directions from the the Fermi level, aBs it is shown in panels (b) and (c) of Fig. 1. At -0.2 0.3 TS=0.4∆ finitetemperaturethespin-upDOSabovethegapispar- 0.01 0.1 0.2 tiallyoccupiedbyquasiparticlesandthe spin-downDOS -0.4 T∆N 0.0 0.2 0.4 0.6 0.8 1.0 is partially empty below the gap. If the superconductor ttuemrepTera,tuitreleTaSdsdtiffoetrhsefraopmpetahreanncoermofalthmeeqtaulastiepmaprteircale- (b) µTNS,↑<→<−TSh N S (c) µTSN,↑>→>∆T−Sh N currents between each of the subbands and the normal N↑ N↓ N↑ N↓ metal. Thisspin-upandspin-downquasiparticlecurrents ✲ flow in the opposite directions and the electric currents, ✛ ✛ carried by these flows cancel each other exactly due to the overall particle-hole symmetry of the superconducting DOS. Therefore, there is no thermoelectric effect in ✲ thissituation38,butthereisthermallyinducedpurespin current. It can be calculated as follows29: ∞ G I = dε(N N )(f f ), (1) FIG.1. (a)Spinimbalance,accumulatedinthenormalmetal s ↑ ↓ N S 2e−Z∞ − − aenstatefumnpcetrioantuorefsToNf.thDeiffsuerpeenrtcocnudruvecstincgorrreessperovnodirtToSd.iffIenr-- sert: sketch of the corresponding S/N system. (b) and (c) where G is the conductance of the S/N interface, N is ↑,↓ Schematic illustration of the thermospin effect for the case thenormalizedLDOSforthecorrespondingspinsubband when the reservoir is normal. The DOS in the N layer (left) inthesuperconductorandf arethequasiparticledis- N,S andintheSlayer(right)areshownasfunctionsofε(vertical tribution functions for the normal metal and the super- axis). The filled parts are occupied by electrons. The pink conductor, respectively. fN,S are assumed to be of the linesarethedistributionfunctioninS.Thearrowscorrespond Fermi distribution form, corresponding to different tem- tothedirectionsoftheelectronflowbetweenthespin-upsub- peraturesTN,S. Thesuperconductingspin-resolvedDOS band and theN layer. takes the form N = ε h/ (ε h)2 ∆2 if one ne- ↑,↓ | ± | ± − glectsthedepairingeffectofthemagneticfieldandother p distribution function takes the form f = f(ε µ,T), deparing factors, such as magnetic and spin-orbital im- ↑(↓) ∓ whereµistheresultingspinimbalance. Itautomatically purities,forsimplicity. Itisseenthatthespin-dependent leads to zero charge and spin flows across the interface, particle-holeasymmetryisverylargeherefortheenergies but the heat flow is nonzero due to the finite tempera- oftheorderof∆ascomparedtothenonsuperconducting case35. ture difference at the interface. Consequently, we should also take into account the heat balance equation, which Spin imbalance. If the thermally injected spin is not describes the removal of this heat into the phonon sub- removed from the superconductor or the normal metal, system. The corresponding equation determines the ef- and the thickness of this layer is less than the spin re- fective temperature of the electron gas in the layer (see laxation length, the spin imbalance is accumulated in it Appendix B for the detailed discussion) but for strong due to the thermospineffect. The particularvalue ofthe enough energy relaxation this temperature is very close spinimbalancedependsontheparametersofthesystem. to the phonon temperature in the layer Te Tph =T. Further we calculate this thermally induced spin imbal- ≈ In order to accumulate the spin imbalance in one of ance under the assumption τ τ τ , where τ is ε G sf ε ≪ ≪ thelayers(normalorsuperconducting)weshouldassume the characteristic energy relaxationtime in the layer, τ G that its size is restricted in order to obey the inequality isthecharacteristictime,whichaquasiparticlespendsin τ τ , and the other layer should be bulk and serve itandτ isthespinrelaxationtime. Underthesecondi- G sf sf ≪ as a spin reservoir. Let us first analyze the case of spin tionsquasiparticlesinthelayercanbedescribedbyadef- accumulation in the normal layer because of its simplic- inite temperature (due to the fact that the energy relax- ity. Under our assumptions µ can be found from the ationisthefastestprocess)andthedifferentspinspecies N conditionofzerototalspincurrentthroughtheinterface: have their own chemical potentials due to the weakness of the spin-flip processes. In this case the thermally in- ∞ ε σµ ε duced spin flow across the interface is compensated by dε σN tanh − N tanh =0, (2) σ the counterflow due to the different chemical potentials " 2TN − 2TS# Z σ forspin-upandspin-downquasiparticlesandthespinre- −∞ X laxation in the layer can be disregarded. Therefore, the whereσ = , inthesubscriptsand intheexpressions. ↑ ↓ ± 3 The spin imbalance µ , accumulated due to the ther- N 1.0 (a) mospin effect in the normal layer, is plotted in Fig. 1(a) as a function of TN for different temperatures of the su- 0.8 perconductor T . There are three important features in S 0 0.6 the behavior of the spin imbalance: (i) it is obvious that ∆ µN =0atTN =TS -thereis nothermospineffectatall; ∆/ 0.4 (ii) µ ∆ h at T T and (iii) µ h for the N → − N ≪ S N →− 0.2 opposite limit T T . These characteristic features S N ≪ determine the maximal value of the thermally induced 0.1 0.2 0.3 0.4 0.5 0.6 spin imbalance in Zeeman-split S/N heterostructures. 0.6 The opposite case of spin accumulation in the super- h=0.4D0 conductor µ is presented in Figs. 2(b) and 3(b). This 0.4 S 0.5 S N 0.6 case is more complicated because of the fact that the 0.2 0.7 0 superconducting order parameter ∆ is very sensitive to ∆ 0.79 0.0 the value of the spin imbalance and the particular tem- / perature of the superconductor. At each point of the µS -0.2 curves µS and ∆ are determined self-consistently from -0.4 (b) Eq.(2)with the substitutions (ε∓µN)/TN →ε/TN and -0.6 ε/TS (ε µS)/TS andtheself-consistencyequationfor 0.1 0.2 0.3 0.4 0.5 0.6 → ∓ the order parameter ∆ [Eq. (3)]. Here the limiting val- TS/∆0 ues of the spin imbalance are the same: µ ∆ h at S → − T T and µ h for the opposite limit T T , buSt≪nowN∆ is caSlcu→la−ted for the given h, TS, TNNan≪d µSS. aFsIGa.f2u.n(cat)ioSnupoferTcoSn.duHcetrieng∆o0rd≡er∆pa(rTaSme=teTrNfor=TN0,=h0=.1∆0)0. It is also obvious that µ 0 at ∆ 0 independent of S Different colors correspond to different magnetic fields. The → → TN and TS. dashed linesare resultsof theorderparametercalculation at Physically it is clear why the thermally induced spin µS = 0 (the dashed-dotted parts of the curves represent the imbalance has just these limiting values. If one neglects absolutelyunstablebranchesofthesolutions). Thesolidlines the spin relaxation, µ is determined by Eq.(2), which in are the results taking into account the real value of the spin fact means that the spin flow is zero for each of the spin imbalance. The dotted parts of these curves correspond to subbands separately. If T T , in the spin-up sub- absolutely unstable branches of the solution. (b) The appro- S N band quasiparticles are accu≪mulated at ε > ∆ h [see priate spin imbalance µS as a function of TS. − Fig. 1(c)]. Under the condition of fast thermalization processes the presence of large number of quasiparticles atε ∆ h andlowtemperatureofthe superconductor We begin by considering the case of small T , plot- ∼ − N just mean that the chemical potential µ↑ ∆ h for ted in Fig. 2. In order to clearly see the main physical ∼ − this spin subband. For the opposite case TN ≪ TS [see effects of µS on superconductivity, here we neglect the Fig. 1(b)] the spin balance for the spin-up subband re- depairing due to the orbital effect of the magnetic field. quiresthatthe spinflowfromStoNatε>∆ hequals The resulting order parameter as a function of the su- − to the opposite spin flow from N to S at ε< ∆ h. It perconductor temperature T is represented in Fig. 2(a) − − S is possible if the number of quasiparticles at ε > ∆ h for different magnetic fields. The order parameter with- − and ε< ∆ h are equal, that is µ↑ = h. out taking into account the thermospin accumulation is − − − Superconducting order parameter. Letusnowconsider plotted by dashed lines. The Zeeman depairing of su- how the thermally induced pure spin imbalance µS, ac- perconductivityis clearlyseenfromthese dashedcurves. cumulated in the superconductor, influences the super- Solidlines representthe orderparameterin the presence conductivity in S/N heterostructure similar to the one of the thermospin accumulation. Physically its main ef- sketched in Fig. 2. We assume that the superconductor fect is the compensation of the Zeeman depairing43. As is in dirty limit. The calculations are performed in the a result ∆ survives at larger temperatures as compared framework of the Usadel equations for the quasiclassi- to dashed lines. cal Green’s functions1–3,39, see Appendix A for details. This is the recoveringof superconductivity by the cre- The superconducting order parameter∆ depends on the ation of spin imbalance. The critical temperature in- anomalous Green’s function F and chemical potential σ creasesandcansufficientlyexceeditsvaluewithoutther- µ and is calculated from the self-consistency equation: S mospin effect and ∆(T ) ∆(T ) . The dotted S S H=0 → | parts of the curves represent absolutely unstable solu- ΩD λ ε σµ tions. In the regions of existing of these unstable solu- S ∆= dε Re[Fσ]tanh − , (3) tionsnormal(∆=0)andsuperconductingstatesaresta- 4 2T Xσ −ΩZD (cid:16) (cid:20) S (cid:21)(cid:17) ble simultaneously40–42. Therefore, the first-order transition from the superconducting to normal state should where λ is the dimensionless coupling constant. take place at high enough magnetic fields. However, the 4 problem of finding the exact value of the transition field absence of the thermospin effect the solution of the self- is beyond the scope of the present work because we as- consistencyequationbecomesmulti-valued. Theeffectof sumethespin-sliptimetobethelargesttimescale. Nev- appearing(orat leastenhancement) ofthe orderparam- ertheless, the most part of the curves, represented in eteruponheatingthesuperconductorcanbeviewedasa Fig. 2(a) is below the zero-temperature Pauli limiting hallmarkofspinimbalanceinit. Ifthecontactwithahot field h=∆ /√241. normal reservoir only leads to the heating of the super- 0 Taking into account the realistic values of the orbital conductor,the effect is not possible because the electron deparing due to magnetic field, we obtain qualitatively gas in the superconductor cannot have the temperature thesameresult,butitislesspronouncedduetothepres- higher than TN. enceofanotherdepairingfactorinthesystem. Thermally Now we discuss the applicability of our predictions inducedspinimbalanceonlycompensatestheZeemande- to real experimental systems. Our calculations are per- paring, but does not interact with the orbital deparing. formed under the assumption τ τ τ because ε G sf It is interesting to note here that the the spin imbal- ≪ ≪ it is the most simple and clear case from the theoret- ance, which recovers superconductivity, can be induced ical point of view. More important thing is that the not only by the thermal difference, but also by the spin spin-flip processes are destructive for this type of spin injection from ferromagnets43,44. imbalance. Atthe sametime inrealexperimentalsetups different regimes are possible. While for thick enough 1.0 Al film very weak spin-flip rates (the spin relaxation (a) length up to the hundreds of microns, that is the con- 0.8 dition τ τ works very well) were reported37, the ε sf ∆0 0.6 opposite c≪ase τsf < τε is realized in thin Al films (the / thicknessis 10nm)15–17,19,25,45. τ iscontrolledbythe ∆ 0.4 ∼ G S/N interface transparency and can be varied in wide 0.2 range. Therefore, from the experimental data one can conclude that it is more preferable to use thick enough 0.1 0.2 0.3 0.4 0.5 0.6 films. The film thickness is, in fact, restricted by two 0.3 conditions. (i) It should be small compared to the spin h=0 relaxation length. (ii) The film should be not too thick, such that the Zeeman is still the essential deparing fac- 0.2 0.4 0 ∆ tor. Quantitatively, the relative influence of the orbital 0.5 / depairingwithrespecttotheZeemanoneiscontrolledby S 0.1 0.57 µ theparameterα/h=De2Hd2/(6µ ~c2)46,whichshould B (b) notbe considerablylargerthanunity. For the typicalAl 0 parameters and h ∆ it means that d.. 0 ∼ 0.1 0.2 0.3 0.4 0.5 0.6 In conclusion, we have shown that the giant ”ther- TS/∆0 mospin” effect inZeeman-splitS/N heterostructurescan leadtoanaccumulationofapurespinimbalanceinsome FIG.3. (a)SuperconductingorderparameterforTN =0.4∆0 parts of a system. The maximal value of this imbal- as a function of TS. The different types of lines mean the ance is of the order of Zeeman splitting of the super- same as in Fig. 2. Different colors correspond to different conducting DOS. This thermally induced pure spin ac- magnetic fields. (b) µS as a function of TS. The orbital deparing parameter De2∆0d2/(6c2~µ2B)=0.2. cumulation strongly influences the superconductivity in theheterostructure. Differentregimesarepossible: ifthe superconducting part is more hot than the normal one, Further we turn to considering the case of large T , N the spin imbalance weakens the Zeeman depairing and, plotted in Fig. 3. Here the realistic orbital deparing is therefore,recoverssuperconductivityinthesystem. Oth- taken into account from the very beginning. By com- erwise,ifthenormalpartismorehot,thantheimbalance paring Figs. 3(a) and (b) it is seen that in this case the strengthens the Zeeman depairing of superconductivity. suppressionofsuperconductivitybytheeffectiveZeeman Itcanresultintheappearanceofsuperconductivityupon field is only increased by the spin imbalance. The sup- heatingthesample. Itisobviousthatthespinimbalance pressionisstrongestforthelowestsuperconductingtem- also strongly influences the weak superconductivity re- peratures,wherethevalueofthespinimbalanceismaxi- gions,such as Josephsonjunction interlayers. Therefore, mal. Inthiscasesuperconductingorderparametershows the discussed here effect can be used to control the ap- very unusual behavior on the superconductor tempera- pearence and positions of 0 π transitions in the S/N/S tureTS. ItisabsentforsmallTS andarisesuponheating junctions via the manipula−ting the temperature differ- thesuperconductor. Thereasonisthatthespinaccumu- ence between the leads and the interlayer. lationbecomesweakerwhenT increasesandapproaches S T . It is worthnoting that the effect is only possible for Acknowledgments. WearegratefultoM.SilaevandT. N large enough magnetic fields (h > 0.5∆ ), when in the Heikkila¨ for the fruitful discussions. 0 5 Appendix A: Usadel Green’s functions in terms of equation: the θ-parametrization 2Re[coshθ ] ε σ For the problem under consideration it is enough to ϕσ tanh =Iσ,e−ph+Iσ,e−e , − τ − 2T find the retarded and advanced Green’s functions in the G (cid:18) N(cid:19) (B3) approximation of the impenetrable S/N interface. In where we introduce the characteristic time, which a terms of the so-called θ-parametrization47 the Usadel quasiparticle spends in the superconducting layer τ = G equation for the retarded Green’s function in the super- (σ d)/(GD). S conducting film takes the form: MultiplyingthekineticequationEq.(B3)bythequasi- (ε h)sinhθ +∆coshθ + particle energy ε and integrating it over energy, one can ↑,↓ ↑,↓ ± obtain the heat balance equation, which takes the form: e2 iD H2d2coshθ sinhθ =0, (A1) 6c2~ ↑,↓ ↑,↓ ∞ 2 ε The last term in Eq. (A1) accounts for the depairing εdεRe[coshθ ] ϕ tanh = σ σ of superconductivity in thin films by the orbitaleffect of −τG Z (cid:18) − 2TN(cid:19) −∞ the magnetic field46. The normalizedDOS can be found ∞ as εdεI , (B4) σ,e−ph N↑,↓ =Re[coshθ↑,↓], (A2) −Z∞ and the anomalous Green’s function, which enters the the contribution of the electron-electron relaxation term self-consistency equation, takes the form equals zero because this term conserves the total en- F↑,↓ =sinhθ↑,↓. (A3) ergy. Eq.(B4)determinesthe effective temperatureTSe,σ of the electron subsystem of the superconductor for a spinσ. However,dueto theoverallparticle-holesymme- Appendix B: heat balance equation try Re[coshθσ(ε)] = Re[coshθ−σ( ε)], Re[sinhθσ(ε)] = − Re[sinhθ ( ε)] and ϕ ( ε) = ϕ ( ε) (the last −σ σ −σ − − − − equality is only valid for the symmetric type of nonequi- If we neglect the elastic spin-flip processescompletely, librium) the electron-phonon integral23,24,48 that is assume τ , from the Keldysh part of the sf → ∞ Usadelequationweobtainthe followingequationforthe distribution function ϕ 23: ∞ σ I =8g dε′(ε′ ε)2sgn(ε′ ε) σ,e−ph ph Dκ ∂2ϕ I I =0 , (B1) − − × σ x σ− σ,e−ph− σ,e−e −Z∞ rwehneorremκaσliz=ati1on+o|fcotshheθσd|i2ffu−si|osninhcoθnσs|t2anactcbouynstuspfeorrcothne- Re[coshθσ(ε)]Re[coshθσ(ε′)]− h ductivity. The collision integrals Iσ,e−ph and Iσ,e−ph de- Re[sinhθσ(ε)]Re[sinhθσ(ε′)] scribe the electron-phonon and electron-electron relax- × ε′ ε i ation processes, respectively. Please note that in the coth − [ϕ (ε) ϕ (ε′)] ϕ (ε)ϕ (ε′)+1 (B5) σ σ σ σ problem under consideration we only consider the so- 2TS − − n o called ”symmetric” types of the quasiparticle nonequilibrium, when the electron and hole components of the also manifests the same symmetry I ( ε) = σ,e−ph distributionfunctionarethesame,thedistributionfunc- I (ε). In this case Eq. (B4) is valid at th−e same −σ,e−ph tion has no τ3 contribution. This is because the source −temperature Te = Te = Te for the both spin direc- S,↑ S,↓ S of the nonequilibrium here is the temperature gradient, tions. In addition, if the electron-phonon relaxation is which is just of this symmetric type. strong, that is τ τ , the electron temperature Te We assume that the film thickness in the x direction is very close toet−hpehp≪honGon temperature T . It is thiSs S issmallerthanthe characteristicrelaxationlengthofthe case that is considered in the main text of our paper. distribution function. Then ϕ↑,↓ does not depend on x. If the electron-phonon interaction is not so strong, that Integrating Eq. (B1) over the width d of the film along is τ . τ , all our results for the order parameter e−ph G the x-direction and taking into account the boundary are still valid, but Te (which is higher than T ) should conditions for the distribution function23 be substituted for TS in all the plots in Figs. 1S-3 of the S main text. The opposite limit τ τ is not of in- e−ph G 2G ε ≫ κ ∂ ϕ = Re[coshθ ] ϕ tanh , (B2) terest, because in this case the electron temperature in σ x σ σ σ (cid:12)(cid:12)x=0 σS (cid:18) − 2TN(cid:19) the superconductor is very close to the temperature of where σS(cid:12)is the normal state conductivity of the super- the normal reservoir TN and the thermally induced spin (cid:12) conducto(cid:12)r, one can obtain that ϕσ obeys the following imbalance is negligible. 6 1 M. Eschrig, Reports on Progress in Physics 78 104501 26 F. Giazotto and F. Taddei, Phys. Rev. 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