Colloquium: Nonequilibrium effects in superconductors with a spin-splitting field
F. Sebastian Bergeret, Mikhail Silaev, Pauli Virtanen, Tero T. Heikkila
CColloquium: Nonequilibrium effects in superconductors with a spin-splittingfield
F. Sebastian Bergeret,
1, 2, ∗ Mikhail Silaev, Pauli Virtanen, and Tero T. Heikkil¨a † Centro de Fisica de Materiales (CFM-MPC),Centro Mixto CSIC-UPV/EHU,Manuel de Lardizabal 4,E-20018 San Sebastian,Spain Donostia International Physics Center (DIPC),Manuel de Lardizabal 5,E-20018 San Sebastian,Spain University of Jyvaskyla,Department of Physics and Nanoscience Center,P.O. Box 35 (YFL),FI-40014 University of Jyv¨askyl¨a,Finland NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa,Italy
We review the recent progress in understanding the properties of spin-split superconduc-tors under non-equilibrium conditions. Recent experiments and theories demonstrate arich variety of transport phenomena occurring in devices based on such materials thatsuggest direct applications in thermoelectricity, low-dissipative spintronics, radiationdetection and sensing. We discuss different experimental situations and present a the-oretical framework based on quantum kinetic equations. Within this framework weprovide an accurate description of the non-equilibrium distribution of charge, spin andenergy, which are the relevant non-equilibrium modes, in different hybrid structures.We also review experiments on spin-split superconductors and show how transport mea-surements reveal the properties of the non-equilibrium modes and their mutual coupling.We discuss in detail spin injection and diffusion and very large thermoelectric effects inspin-split superconductors.
CONTENTS
I. Introduction 1II. Superconductor with an exchange field 4A. Brief overview of the quasiclassical theory of diffusivesuperconductors 6III. Nonequilibrium modes in spin-split superconductors 8A. Description of nonequilibrium modes insuperconductors with spin splitting 9B. Accumulations in terms of the non-equilibrium modes 10IV. Spin injection and diffusion in superconductors 11A. Detection of spin and charge imbalance: Non-localtransport measurements 11B. Non-local conductance measurements in spin-splitsuperconductors 12C. Spin Hanle effect 13D. Spin imbalance by ac excitation 14V. Thermoelectric effects 14A. Charge and heat currents at a spin-polarized interfaceto a spin-split superconductor 15B. Linear response: heat engine based on asuperconductor/ferromagnet structure 15 ∗ sebastian [email protected] † Tero.T.Heikkila@jyu.fi C. Spin Seebeck effect 18D. Thermophase in a S(FI)S contact 18VI. Summary and Outlook 18Acknowledgments 20References 20
I. INTRODUCTION
Ferromagnetism and spin-singlet superconductivityare antagonist orders and hardly coexist in bulk mate-rials. However, hybrid nanostructures allow for the pos-sibility of combining the two phenomena via mutual prox-imity effects. The combination leads to the emergence ofnovel features not present in either system alone. We canmake a distinction among those characteristics affectingthe spectral properties of the materials, showing up whenthe probed systems are in equilibrium, and those relatedto nonequilibrium phenomena. The emphasis of our textis in the latter phenomena, especially related to steady-state currents or voltages applied across the structures.Both superconductors and ferromagnets are examplesof electron systems with spontaneously broken symme-tries, and thereby characterized by order parameters. a r X i v : . [ c ond - m a t . s up r- c on ] J a n The order parameter for a conventional spin singlet su-perconductor is the amplitude of (Cooper) pairing be-tween electrons in states with opposite spins and mo-menta (Bardeen et al. , 1957). The presence of thiscomplex pairing amplitude F leads to two characteristicfeatures of conventional superconductivity (de Gennes,1999; Tinkham, 1996): An equilibrium supercurrent thatis proportional to the gradient of the phase of F and thatcan be excited without voltage, and to the quasiparti-cle spectrum exhibiting an energy gap proportional tothe absolute value of F . The resulting density of states(DOS, Eq. (1) for h eff = 0) is strongly energy dependentand results into a non-linear nonequilibrium response ofsuperconductors.The main defining feature of ferromagnets is the bro-ken spin-rotation symmetry into the direction of magne-tization, and the associated exchange energy h that splitsthe spin up and down spectra. This also leads to a strongspin dependence (spin polarization) of the observables re-lated to ferromagnets.There are two mechanisms that prevent most of theferromagnetic materials from becoming superconducting.One of them is the orbital effect due to the intrinsicmagnetic field in ferromagnets. When this field exceedsa certain critical value, superconductivity is suppressed(Ginzburg, 1957). The second mechanism is the para-magnetic effect Chandrasekhar (1962); Clogston (1962);and Saint-James et al. (1969). This is due to the intrinsicexchange field of the ferromagnet that shows up as a split-ting of the energy levels of spin-up and spin-down elec-trons and hence prevents the formation of Cooper pairs.We focus here on the regime where this spin-splitting fieldis present, but not yet too large to kill superconductivity.In superconductors the spin-splitting field can be gen-erated either due to the Zeeman effect in magnetic field oras a result of the exchange interaction between the elec-trons forming Cooper pairs and those which determinethe magnetic order. Such fields can lead to drastic mod-ifications of the ground state of a spin-singlet supercon-ductor. The best-known example is the formation of thespatially inhomogeneous superconducting state predictedby Fulde and Ferrell (1964) and Larkin and Ovchinnikov(1965) and dubbed as FFLO. Although extensively stud-ied in the literature, the FFLO phase only takes place in anarrow parameter window and therefore its experimentalrealization is challenging.Other more robust phenomena related to the spin-splitting fields in superconductors have their origin inthe quasiparticle spectrum modification. In the centralpanel of Fig. 1 we show the resulting spin-split den-sity of states. This was first explored experimentallyby Meservey et al. (1975, 1970) through the spin-valveeffect in the superconductor/ferromagnet (Al/Ni) tun-nel junctions (Fig. 1a). In these experiments the mag-netic field was applied in the plane of a thin supercon-ducting film, such that the paramagnetic effect domi- nates. The spin-split DOS was utilized to determine thespin polarization of an adjacent ferromagnet (Meservey et al. , 1980; Meservey and Tedrow, 1994; Paraskevopou-los et al. , 1977; Tedrow and Meservey, 1971, 1973). Thebasis of this spin-valve effect is the spin-resolved tunnel-ing into the superconductor with spin splitting, shownin Fig. 1a. This schematic picture illustrates how byproperly tuning the voltage across the junction, the elec-tronic transport is dominated by only one of the spinspecies. That results in peculiar asymmetric differentialconductance curves dI/dV ( V ) (cid:54) = dI/dV ( − V ) observedin experiments and revealing the spin polarization. Thisidea has been used more recently to probe the spatiallyresolved spin polarization of different magnetic materialsby means of scanning tunneling microscopy with spin-split superconducting tips (Eltschka et al. , 2014, 2015).Similar effects can also arise in thin superconducting filmsby the magnetic proximity effect from an adjacent ferro-magnetic material (Tedrow et al. , 1986). In such a case,the spin splitting of the density of states can be observedfor small magnetic fields or even at zero field, as discussedin Sec. II.The combination of spin-splitting fields with strongspin-orbit interaction in superconducting nanowires hasalso raised considerable interest as a platform for realiz-ing topological phases and Majorana fermions, with pos-sible applications in topological quantum computation(Aasen et al. , 2016). Although these effects are beyondthe focus of this review, the physics discussed below mayhelp in understanding transport properties of the devicesstudied in that context.Due to the different nature of their broken symme-try, combining superconductors (S) and ferromagnets(FM) in hybrid structures leads to a multitude of ef-fects where magnetism affects superconductivity and viceversa. Some of these effects show up already in equi-librium properties, especially studied in the context ofproximity effects in superconducting/metallic ferromag-net hybrids and reviewed for example by Buzdin (2005)and Bergeret et al. (2005). The latter usually focus onthe unusual behavior of Cooper pairs leaking from a su-perconductor into a metallic ferromagnet generating, forexample, oscillating pair wave functions analogous to theFFLO state (Buzdin et al. , 1982; Demler et al. , 1997)and long-range spin triplet correlations (Bergeret et al. ,2001b) induced by the coupling between the intrinsic ex-change field of the ferromagnet and the leaked supercon-ducting condensate (Bergeret et al. , 2001b). These effectsmanifest themselves in measurable equilibrium effects,such as the density of states and critical temperatureoscillations in S/FM bilayers (Jiang et al. , 1996; Kontos et al. , 2001), triplet spin valve effects in the critical tem-perature of FM/S/FM structures (Singh et al. , 2015),and unusual Josephson effects in SC/FM/SC junctions(Ryazanov et al. , 2001; Singh et al. , 2016). Inversely,a magnetic proximity effect can arise when the triplet FIG. 1 Central panel: quasiparticle spectrum and density of states in a superconductor with spin splitting, N is the normalmetal DOS. (a-d) Schematic pictures of various nonequilibrium phenomena occurring at normal metal/insulator/superconductor(NM/I/S) and ferromagnetic/insulator/superconductor (FM/I/S) interfaces discussed in this review. For clarity we show thelimit of half-metallic FM with N ↑ = 0. (a) Spin-resolved tunneling from a ferromagnetic metal to a spin-split superconductorthat leads to the spin valve effect, i.e. , the charge current in the parallel magnetic configuration is different from that in theanti-parallel one. (c) Creation of spin and charge accumulation in the voltage biased FM/S junction. (b,d) Schematic pictureof thermally excited currents in NM/S and FM/S junctions with a spin-split superconductor. (b) Spin Seebeck effect in NM/Sjunction: A pure spin current is generated by the temperature bias between a spin-split superconductor at temperature T SC and a normal metal at temperature T NM > T SC . (d) Thermoelectric effect in a FM/I/S junction: Here the spin current ispartially converted to the charge current due to the spin-dependent density of states in the ferromagnet. pairs, created in the FM region, leak back into the su-perconductor in a FM/S metallic bilayer, generating anon-vanishing magnetic moment in the SC within a co-herence length ξ s from the SM/FM interface (Bergeret et al. , 2004).In contrast to these equilibrium proximity effects, herewe focus on nonequilibrium properties of a superconduct-ing material with a built-in spin-splitting field. The in-terest in studying such systems has been intensified re-cently due to the technological advances which allow fora controllable generation of spin splitting in thin super-conducting films either by applying an external in-planemagnetic field (H¨ubler et al. , 2012; Quay et al. , 2013)or by an adjacent ferromagnetic insulator. Structureswith insulating FMs avoid the proximity effect suppress-ing superconductivity. Such nonequilibrium propertiesare studied by applying currents or voltages across thestructures. The focus of our Colloquium is on steady-state nonequilibrium effects with time independent driv-ing fields, but we also mention works studying alternatingcurrent (ac) responses.Often the nonequilibrium effects can survive to muchhigher distances than ξ s , as their decay scales are deter-mined via the various inelastic and spin-flip scatteringlengths. Moreover, they can be studied at a weak tun-neling contact to ferromagnets, making the analysis insome cases more straightforward than in proximity ex-periments. Nonequilibrium properties are related to thedeviation of the electron distribution function from its equilibrium form, which leads to a nonequilibrium distri-bution (imbalance) of charge, energy or spin degrees offreedom. We refer to these different types of deviationsfrom equilibrium as nonequilibrium modes. Specifically,we explore the coupling between these modes in super-conductors with a spin-splitting field, and discuss unusu-ally strong thermoelectric response and long-range spinsignals.The above mentioned ability to characterize the spinpolarized Fermi surface of metallic magnets with thehelp of spin-split superconductors has a direct connec-tion with spintronics, and in particular with the searchfor spin valves with larger efficiencies than in the struc-tures exhibiting large magnetoresistance (Baibich et al. ,1988; Binasch et al. , 1989; Moodera et al. , 1995). Indeed,a superconductor with a spin-splitting field has also anintrinsic energy dependent spin polarization around theFermi level. This allows for studying different spintroniceffects in a setting of a controllable non-linearity arisingfrom the superconducting gap. Some of these effects areschematically shown in Fig. 1. This review explains thosephenomena in detail. The term ”mode” here refers to the changes of the electron dis-tribution function with respect to its equilibrium form. It shouldbe distinguished from collective modes such as the Carlson andGoldman (1973) or the amplitude mode (Higgs, 1964) that af-fect the response of superconductors at temperatures close to thecritical temperature or at high frequencies.
In normal metals and superconductors a spin accumu-lation, or spin imbalance, can be created by injection of acharge current from a ferromagnetic electrode (Gu et al. ,2002; Jedema et al. , 2001; Johnson, 1994; Johnson andSilsbee, 1985; Poli et al. , 2008; Shin et al. , 2005; van Son et al. , 1987; Takahashi and Maekawa, 2003). This stateis characterized by the excess population in one of thespin subbands, determined by the balance between spininjection and relaxation or spin diffusion rates. In normalmetals the nonequilibrium spin imbalance decays due tospin-flip scattering at typical distances of several hun-dreds of nanometers. In the superconducting state, atlow temperatures k B T (cid:28) ∆ the injection of any amountof carriers just above the energy gap shifts the chemi-cal potential of quasiparticles rather strongly due to thelarge amount of quasiparticles at the gap edge [Fig. 1(c)].This leads to a strong spin signal in SF junctions (Poli et al. , 2008; Takahashi and Maekawa, 2003).The spin relaxation length in normal metals dependsonly weakly on the temperature T . In the superconduct-ing state, however, the scattering length is drasticallymodified with T . According to the first theory and ex-periments on spin injection in superconductors, the spinrelaxation length was found to be reduced compared tothe normal state (Morten et al. , 2004; Poli et al. , 2008).However, subsequent experiments showed, contrary toexpectations, an increase of the spin decay length (H¨ubler et al. , 2012; Quay et al. , 2013). It is now understoodthat these findings can only be explained by taking intoaccount the spin-splitting field inside the superconduc-tor (Bobkova and Bobkov, 2015, 2016; Krishtop et al. ,2015; Silaev et al. , 2015a). Due to this field, as shownin Sec. III.B, it is necessary to take into account fourtypes of nonequilibrium modes describing spin, charge,energy, and spin-energy imbalances. These modes pro-vide the natural generalization of the charge and energyimbalances introduced by Schmid and Sch¨on (1975). InSec. IV we show how the spin-splitting field couples pair-wise these modes: charge to spin energy and spin to en-ergy. Such a coupling leads to striking effects. For ex-ample, the coupling between the spin and energy modesleads to the long-range spin-accumulation observed inthe experiments by H¨ubler et al. (2012) and Quay et al. (2013). As we show in Sec. IV this long-range effect is re-lated to the fact that the energy mode can only relax viainelastic processes which at low temperatures are rare.The coupling between different modes shows up alsoin tunnel contacts with spin-split superconductors. Be-cause the spin-splitting field shifts the spin-resolved DOSaway from the chemical potential of the superconductor,the system exhibits a strong spin-dependent electron-holeasymmetry. The spin-averaged density of states is stillelectron-hole symmetric, and therefore does not violatefundamental symmetries of the (quasiclassical) supercon-ducting state. This spin-resolved electron-hole asymme-try leads to a large spin Seebeck effect shown schemati- cally in Fig. 1b and discussed in Sec. V.C. A tempera-ture difference across a tunneling interface between a nor-mal metal and a spin-split superconductor drives a purespin current between the electrodes, without transport ofcharge. If one of the electrodes is small so that the spininjection rate is large or comparable to the rate for spinrelaxation, a spin accumulation forms in this electrode.However, it was noticed in several recent works(Kalenkov et al. , 2012; Machon et al. , 2013, 2014; Ozaeta et al. , 2014) that in certain situations the relevant ob-servables are not spin-averaged, resulting in an effectiveelectron-hole asymmetry showing up also in the chargecurrent. The spin components are weighted differentlyin a setup consisting of the spin-filter junction connectedto the spin-split superconductor (Ozaeta et al. , 2014),shown schematically in Fig. 1d. As a result of this ef-fective electron-hole symmetry breaking, the system ex-hibits a very large thermoelectric effect. This is discussedin Sec. V.The main body of the review is organized as follows.In Sec. II we describe spin-split superconductors and givean overview of the quasiclassical theory that can be usedfor describing both their equilibrium and nonequilibriumproperties. In Sec. III we describe the nonequilibriummodes in superconducting systems driven out of equilib-rium in terms of the quasiclassical formalism. SectionIV focuses on the spin injection and diffusion in super-conducting systems, and reviews experiments performedto detect spin and charge imbalance in superconductorswith and without spin-splitting. In Sec. V we describethe giant thermoelectric response of a system exhibit-ing spin-polarized tunneling into a superconductor witha spin-splitting field. Finally, we present our conclusionsand an outlook on possible future developments in thefield in Sec. VI. A longer version of this review, along withcomprehensive technical detail, can be found at (Bergeret et al. , 2017). II. SUPERCONDUCTOR WITH AN EXCHANGE FIELD
The main focus of this colloquium is on superconduc-tors with a spin-split density of states (DOS). As dis-cussed in the introduction such a splitting can originateeither by an external magnetic field (Meservey et al. ,1970) or by the exchange field induced by an adjacentferromagnetic insulator (Tedrow et al. , 1986). The splitDOS was observed in spectroscopy experiments (Hao et al. , 1990; Meservey et al. , 1980, 1970; Paraskevopou-los et al. , 1977; Tedrow and Meservey, 1971; Xiong et al. ,2011).Formally, the normalized DOS of a spin-split Bardeen-Cooper-Schrieffer (BCS) superconductor is expressed asthe sum of the DOS of each spin species, N = N ↑ + N ↓ , N = 12 Re ε + h eff (cid:113) ( ε + h eff ) − ∆ + 12 Re ε − h eff (cid:113) ( ε − h eff ) − ∆ , (1)where ± h eff is the effective spin-splitting field. Equation(1) is a simplified description because it does not take intoaccount the effect of magnetic impurities or spin-orbitcoupling (SOC) (Meservey and Tedrow, 1994) discussedbelow. Often inelastic processes are described by ε (cid:55)→ ε + i Γ, where Γ is the Dynes et al. (1984) parameter.In the case when the exchange field is induced by anadjacent ferromagnetic insulator (FI) there is no needof applying an external magnetic field (Hao et al. , 1990;Moodera et al. , 2007; Senapati et al. , 2011; Tedrow et al. ,1986; Wolf et al. , 2014b; Xiong et al. , 2011). Microscop-ically, the spin splitting originates from the exchange in-teraction between conduction electrons and the magneticmoments of the FI localized at the S/FI interface (Izyu-mov et al. , 2002; Khusainov, 1996; Tokuyasu et al. , 1988).The ferromagnetic ordering in the FI is due to a directexchange coupling between the localized magnetic mo-ments. In usual FIs the direct coupling is strong enoughthat one can assume that the magnetic configuration ofthe FI is only weakly affected by the superconductingstate (Bergeret et al. , 2000; Buzdin and Bulaevskii, 1988).The modification of the DOS is non-local and survivesover distances away from the FI/S interface of the or-der of the coherence length ξ s (Bergeret et al. , 2004;Tokuyasu et al. , 1988). If the thickness d of the S filmis much smaller than ξ s , the spin splitting can be as-sumed as homogeneous across the film. Thus the densityof states can be approximated by Eq. (1) with an effec-tive exchange field h eff ≈ J ex (cid:104) S r (cid:105) a/d (de Gennes, 1966a;Khusainov, 1996; Tokuyasu et al. , 1988), where a is thecharacteristic distance between the localized spins, J ex isthe exchange coupling between conduction electrons andlocalized moments, and (cid:104) S r (cid:105) is the average of the latter.In Fig. 2 we show an example of the measured differen-tial conductance of an EuS/Al/Al O /Al junction. TheAl layer adjacent to the EuS has a spin-split density ofstates that shows up as the splitting peaks (bright stripesin the figure) in dI/dV . Even at zero applied magneticfield the splitting is nonzero. The magnetization reversalof EuS at H c ≈ − . et al. , 2017). As afirst approach the DOS inferred from Fig. 2 can be welldescribed by the expression (1).The advantage of using a FI instead of an externalmagnetic field is that one avoids the depairing effects andall complications caused by the need to apply magneticfields in superconducting devices. Moreover, because theelectrons of the superconductor cannot propagate intothe FI, superconducting properties are only modified bythe induced spin-splitting field at the S/FI interface, andnot by the leakage of Cooper pairs into the FI as would FIG. 2 Color plot of measured differential conductance, dI/dV of a EuS/Al/Al O /Al junction as a function of the ap-plied voltage and external magnetic field. H co denotes the co-ercive field of the EuS layer when the magnetization switches.Figure adapted from the work by Strambini et al. (2017) happen in the case of metallic ferromagnets. In addi-tion, FIs can also be used as spin-filter barriers (Moodera et al. , 2007), in some cases with a very high spin-filteringefficiency, and therefore they play a crucial role for dif-ferent applications as discussed below.In Table I we show a list of FI/S combinations andthe reported induced exchange splittings and spin-filterefficiencies (barrier spin polarizations).The paramagnetic effect, that leads to the spin-splitting, is modified by spin relaxation and orbital de-pairing. In their absence the superconductivity survivesthe spin-splitting field up to the Chandrasekhar-Clogstonlimit (Chandrasekhar, 1962; Clogston, 1962) h = ∆ / √ is the order parameter at zero-field and zero-temperature. At this field the system experiences a first-order phase transition into the normal state when theorder parameter changes abruptly from ∆ to zero. Thispicture changes qualitatively due the presence of mag-netic impurities and spin-orbit scattering. Even at T = 0and h = 0 the spin-flip processes induced by magnetic im-purities result in the pair breaking effect closing the en-ergy gap (Abrikosov and Gor’kov, 1960a) at τ sf ∆ = 3 / τ sf . For valuesof τ sf larger than the critical one the phase transitionswitches from the first to the second order at (Bruno andSchwartz, 1973) τ sf ∆ = 0 .
461 and the gapless state ap-pears at a certain value of h ( τ sf ) (see Fig. 3a) . TABLE I Magnetic properties of different ferromagnetic insulator-superconductor junctions used in experiments. Middlecolumn shows the spin-filter efficiency characterized by the polarization P = ( G ↑ − G ↓ ) / ( G ↑ + G ↓ ) of the FI barrier (red) withnormal-state conductance G σ for spin σ . The exchange splittings measured in the superconductor (blue) are listed in the rightcolumn. The data is extracted from (Tedrow et al. , 1986); (Moodera et al. , 1988); (Hao et al. , 1990); (Moodera et al. ,1993); (Senapati et al. , 2011); (Pal and Blamire, 2015). Note that µ B · µ eV ∼ = 670 mK.Material Combination Barrier polarization Exchange Splitting (applied field)EuO/Al/AlO /Al no spin-filter barrier 1 T (0.1 T)-1.73 T(0.4 T)Au/ EuS/Al > /Ag no spin-filter barrier 4 T (0.6 T)NbN/ GdN/NbN Figures. (Dated: May 5, 2017)
PACS numbers: −3 −2 −1 0 1 2 3−4−202 −3 −2 −1 0 1 2 3−6−4−202 (a) N ↑ ε/ ∆ ε/ ∆ (b) N ↑ FIG. 1: (Color online) Hanle signal at the distance L D = 2 λ sn . (a) β = − .
9, temperatures
T/T c =0 . , . , . , . , . , . , . , . β = 0 .
5, temperatures
T/T c = 0 . , . , . , . , . , . , . , . FIG. 3 Calculated density of states of a thin superconductingfilm at T →
0. We only show the DOS for one of the spinspecies, N ↑ . Shown by dashed red lines is the DOS in theabsence of relaxation τ sn = 1 / ( τ − sf + τ − so ) = ∞ and zeroexchange field h = 0 which corresponds to a gap ∆ . Othercurves are plotted for h = 0 . and different spin relaxationrates. (a) Spin-flip relaxation β = ( τ so − τ sf ) / ( τ so + τ sf ) =1, curves from top to bottom correspond to an increasing( τ sn ∆ ) − , varying equidistantly from 0 by 0 .
04 steps. (b)Spin-orbit relaxation β = −
1, curves from top to bottomcorrespond to an increasing ( τ sn ∆ ) − , varying equidistantlyfrom 0 by steps of 3 .
4. For clarity the curves are shifted alongthe vertical axis.
Contrary to the spin-flip processes, the spin-orbit scat-tering alone does not have any effect on the supercon-ducting state. However, in combination with h (cid:54) = 0it tends to smear out the spin-splitted DOS singulari-ties provided the spin-orbit relaxation time, τ so , is notvery short (see Fig. 3b). At short relaxation times τ so (cid:28) Γ / ∆ , where Γ is the depairing parameter (Dynes et al. , 1984) the effect of spin splitting is eliminated andthe usual BCS density of states is recovered (see Fig. 3b).Therefore in this case the critical spin-splitting field isstrongly increased above the Chandrasekhar-Clogstonlimit(Bruno and Schwartz, 1973).Besides broadening of the DOS singularities, the spin-orbit and spin-flip relaxation processes have an impor-tant effect on the paramagnetic spin susceptibility of the superconductor as it becomes non-vanishing even in thezero-temperature limit (Abrikosov and Gor’kov, 1960b;Bruno and Schwartz, 1973; Yosida, 1958). The static spinsusceptibility characterizes the paramagnetic response ofthe superconductor to an external magnetic field. In ausual normal metal the Zeeman field produces the samemagnetization as a spin-dependent chemical potentialshift δµ of the same magnitude when the distributionfunctions in different spin subbands are given by f ↑ ( E ) = f ( E + δµ ) and f ↓ ( E ) = f ( E − δµ ). This is differentin superconductors where the paramagnetic susceptibil-ity is determined by both the spin-polarized quasiparti-cles and the emergent spin-triplet superconducting cor-relations (Abrikosov and Gor’kov, 1960b, 1962). On theother hand, the non-equilibrium spin modes as system-atically described in Sec. III are determined only by thequasiparticle contribution.In the next sections we review the transport propertiesof diffusive hybrid structures with spin-split supercon-ductors by contrasting existing theories and experiments.For this sake, in the next section we briefly introducethe quasiclassical Green’s function formalism for super-conductors in the presence of spin-dependent fields andspin-polarised interfaces. It is in our opinion the mostsuitable formalism for the description of diffusive hybridstructures. A. Brief overview of the quasiclassical theory of diffusivesuperconductors
Quasiclassical Keldysh Green’s function technique isa useful and well-established way to describe transportand nonequilibrium properties of good metals, where therelevant physical length scales affecting different observ-ables are long compared to the Fermi wave length λ F ,and where in particular disorder plays a major role. Sev-eral reviews explain this technique for various applica-tions (Belzig et al. , 1999; Bergeret et al. , 2005). Herewe just outline the main features relevant for spin-splitsuperconductors. Briefly, the Keldysh Green’s functions(GFs), ˇ G ( r , r (cid:48) ; t , t (cid:48) ) are two-point correlation functionswhich depend on two coordinates and two times. Herethe “check” ˇ G denotes GFs that live in a structure formedby the direct product of Keldysh, spin and Nambu spaces.The equation of motion for ˇ G can be written as a kinetic-like equation for the Wigner transformed GF, ˇ G ( R , p ),where R and p are the center of mass coordinate and p the momentum after Fourier transformation with respectto the relative component. A significant simplificationcan be done in the case of metals by noticing that theGreen’s functions are peaked at the Fermi level. This al-lows for an integration of the equations over the quasipar-ticle energy, related to the magnitude of p . This proce-dure leads to the quasiclassical GFs, ˇ g ( R , n ), which onlydepend on the direction of the momentum at the Fermilevel and on two times in the case of non-stationary prob-lems, or only on a single energy ε in the stationary case.These functions obey the Eilenberger (1968) equation.One of the advantages of using the quasiclassical GFs isthat in the normal state, the spectral part is trivial, i.e. ,the retarded and advanced GFs are energy independent.All transport information of the normal metal is encodedin the quasiclassical Wigner distribution function ˆ f ( R , n )and quasiclassical equation for it resembles the classicalBoltzmann equation (Langenberg and Larkin, 1986).In contrast, the superconducting case distinguishes it-self by a non-trivial spectrum, and therefore requires tak-ing into account the full Keldysh structure of the GFs, i.e. ˇ g = (cid:32) ˆ g R ˆ g K g A (cid:33) . (2)This GF satisfies the normalization condi-tion(Eilenberger, 1968) ˇ g = ˇ1 . (3)In the diffusive limit the elastic mean free path l dueto scattering at non-magnetic impurities is much smallerthan any other length involved in the problem except λ F .Within this limit the Eilenberger equation can be reducedto a diffusive-like equation, in the same way as the Boltz-mann equation is simplified in the diffusive limit. Thisquasiclassical diffusion equation for superconductors isthe Usadel (1970) equation (we set (cid:126) = k B = 1) D ∇ · (ˇ g ∇ ˇ g ) + [ iετ − i h · σ τ − ˇ∆ − ˇΣ , ˇ g ] = 0 . (4)Here D is the diffusion coefficient, ˇ g ( r , ε ) is the isotropic(momentum independent) quasiclassical GF, h the spin-splitting field either generated by an external field or bythe magnetic proximity effect in a FI/S junction, andˇ∆ = ∆ e iϕτ τ depends on the superconducting order pa-rameter ∆ that has to be determined self-consistently.Here τ i and σ i are Pauli spin matrices in Nambu and spin space, respectively. The self-energy ˇΣ in Eq. (4) describesdifferent scattering processes, such as elastic spin-flip orspin-orbit scattering, ˇΣ el and inelastic electron-phononand electron-electron scattering, ˇΣ in .Equation (4) is central in the description of diffusivesuperconducting structures. Whereas the spectral prop-erties can be obtained by solving the retarded (R) andadvanced (A) components of this equation, nonequilib-rium properties are described by the kinetic equation ob-tained by taking the Keldysh (K) component of Eq. (4).This can be compactly written as ∇ k j akb = H ab + R ab + I ab coll , (5)where we introduce the spectral current tensor j akb , j akb = 18 Tr τ b σ a (ˇ g ∇ k ˇ g ) K . (6)The different current density components (charge, spin,energy, spin-energy) can be obtained from Eq (6). Forexample, the charge current density reads J k = σ N e (cid:90) ∞−∞ dε j k , (7)Here σ N = e ν F D and ν F are the normal-state conduc-tivity and density of states at the Fermi level respectively.In Eq. (5) the term H ab = Tr τ b σ a [ − i h · σ τ , ˆ g K ] / R ab = Tr τ b σ a [ ˆ∆ , ˆ g K ] / I ab coll = Tr τ b σ a [ ˇΣ , ˇ g ] K / Elastic self-energy terms.
We consider elastic contri-butions to ˇΣ el due to scattering at impurities with spin-orbit coupling (relaxation time τ so ) and the spin flipsat magnetic impurities ( τ sf ) (Maki, 1966). Within theBorn approximation, they read ˇΣ so = σ · ˇ g σ / (8 τ so ),ˇΣ sf = σ · τ ˇ gτ σ / (8 τ sf ). In the normal state they con-tribute to the energy-independent total spin-relaxationtime τ − = τ − + τ − . In contrast, in the supercon-ducting case the spin-relaxation time and length acquireenergy dependence, which is different for the spin-orbitand spin-flip scattering (Maki, 1966; Morten et al. , 2004,2005). Therefore it is convenient to describe the relativestrength of these two scattering mechanisms in terms ofthe parameter β = ( τ so − τ sf ) / ( τ so + τ sf ). In diffusive su-perconducting thin films one can also describe the depair-ing effect of an in-plane magnetic field with a self-energyterm ˇΣ orb = τ ˇ gτ /τ orb characterized by the orbital de-pairing time τ orb (Anthore et al. , 2003; de Gennes, 1999).This term also contributes to charge imbalance relaxation(Nielsen et al. , 1982; Schmid and Sch¨on, 1975).The parameters τ − and β are material specific. Forexample, in Al films, the reported values from a set ofspin injection experiments are τ sn ≈
100 ps (Jedema et al. , 2002; Poli et al. , 2008) and β ≈ . τ sn in Nb is only 0.2ps, and is strongly dominated by spin-orbit scattering(Wakamura et al. , 2014). They affect both the spectrumof a bulk superconductor (see Fig. 3) and the spin relax-ation as described in Sec. IV.A. Inelastic self-energies.
The relevant inelastic processesentering the self-energy in Eq. (4), are the particle–phonon and particle–particle collisions. These processesdo not conserve the energies of colliding quasiparticles,but conserve the total spin.The coupling between quasiparticles and phonons lim-its some of the effects discussed in the following sections.Due to the energy dependence of the phonon density ofstates, this coupling decreases rapidly towards low tem-peratures, and eventually phonons decouple from elec-trons, and the main heat relaxation occurs via other pro-cesses such as quasiparticle diffusion. Superconductivitymodifies the electron-phonon heat conduction (Eliash-berg, 1972; Kaplan et al. , 1976; Kopnin, 2001), as alsothe electronic spectrum is energy dependent, and is af-fected by the spin splitting (Grimaldi and Fulde, 1997;Virtanen et al. , 2016).Particle-particle collisions in superconductors and su-perfluids are discussed by Eliashberg (1972); Kopnin(2001); and Serene and Rainer (1983), although mainlywithin contact interaction models disregarding screeningeffects (Feigel’man et al. , 2000; Kamenev and Levchenko,2009; Narozhny et al. , 1999). The collision inte-grals can have spin structure also in the normal state(Chtchelkatchev and Burmistrov, 2008; Dimitrova andKravtsov, 2008).The far-from-equilibrium results discussed in Sec. IVdisregard the particle-particle collisions, as the simplertheory already describes effects not very far from themeasured ones. On the other hand, Sec. V mostly con-centrates on the quasiequilibrium limit, where also spinaccumulation is lost due to a strong spin relaxation.
Hybrid interfaces . In subsequent sections we apply thekinetic equation, Eq. (5), in different situations. For thedescription of transport in hybrid structures, we needin addition a description of interfaces between differentmaterials in the form of boundary conditions. Such inter-faces usually are described by sharp changes of the po-tential and material parameters over atomic distances,and thus cannot be included directly in the quasiclas-sical equations which describe properties over distancesmuch larger than λ F . The description of hybrid inter-faces requires then the derivation of suitable boundaryconditions, first done in the quasiclassical approach byZaitsev (1984).Boundary conditions for the Usadel equation traceback to the work of Kupriyanov and Lukichev (1988).These boundary conditions are applicable for non- magnetic N-N, S-S and S-N interfaces with low trans-missivity (Lambert et al. , 1997). Later Nazarov (1999)generalized these boundary conditions for an arbitraryinterface transparency.Tokuyasu et al. (1988) derived the boundary condi-tion for an interface between a superconductor and a fer-romagnetic insulator and introduced the concept of thespin-mixing angle, which describes the spin-dependentphase shifts acquired by the electrons after being scat-tered at the FI/S interface. Later Cottet et al. (2009)and Zhao et al. (2004) extended these boundary condi-tions to magnetic metallic structures, such as F-S or S-F-S systems, though with low polarization. Boundary con-ditions for large polarization and low transmission havebeen presented by Bergeret et al. (2012) and Machon et al. (2013). General boundary conditions for arbitraryspin polarization and transmission have been extensivelydiscussed by Eschrig et al. (2015).Here we mainly deal with low transmissive barriers be-tween a mesoscopic superconductor and normal and mag-netic leads and use the description presented by Bergeret et al. (2012). In this description, the component of thespectral current density multiplied by σ N perpendicularto the interface is continuous across it, and given by σ N j a ⊥ ,b = − eR (cid:3) Tr τ b σ a (cid:104) ˆΓˇ g ˆΓ † , ˇ g (cid:105) K , (8)where R (cid:3) is the spin-averaged barrier resistance per unitarea, and the spin-dependent transmission is character-ized by the tunneling matrix ˆΓ = tτ + uσ , assumingpolarization in the z -direction. The normalized trans-parencies satisfy t + u = 1 and are determined fromthe interface polarization | P | ≤ ut = P . TheGreen’s function ˇ g in the r.h.s of Eq. (8) corresponds tothe opposite side of the junction. III. NONEQUILIBRIUM MODES IN SPIN-SPLITSUPERCONDUCTORS
The out-of-equilibrium state in superconducting sys-tems is characterized by the presence of nonequilibriummodes associated with the different electronic degrees offreedom. For example, injection of an electric currentfrom a normal electrode into a superconductor generatesa charge imbalance mode (Clarke, 1972; H¨ubler et al. ,2010; Tinkham, 1972; Tinkham and Clarke, 1972; Yagi,2006) that diffuses into the S region. This nonequilibriummode reflects an imbalance of the quasiparticle popu-lation between the electron-like and hole-like spectrumbranches. The charge imbalance measurements madein the 1970s were to our knowledge the first to studysuch nonequilibrium modes in non-local multiterminalsettings. This technique was later adapted to spintronics,to study the nonequilibrium spin accumulation inducedby spin-polarized electrodes (Johnson and Silsbee, 1985).
FIG. 4 Schematic illustration of the quasiparticle distribution function components in a superconductor with spin splitting2 µ B B . The occupied states are represented by filled circles. (a) Equilibrium distribution, (b) charge imbalance f T , (c) spinimbalance f T , (d) spin energy imbalance f L , and (e) energy imbalance f L . The dashed and dotted arrows show elasticprocesses which lead to the formation — and the reverse processes to the relaxation — of a particular nonequilibrium mode.In (c,d) the dashed black lines show particle-hole branch transitions while the dotted blue lines correspond to the spin-flipprocesses. Schematically, nonequilibrium modes can be repre-sented in terms of the electron/hole branches in the spec-trum of the superconductor (Tinkham, 1996), as illus-trated in Fig. 4. For example the charge mode can beunderstood as the imbalance between the electron andhole branches (Fig. 4b). In the absence of spin-dependentfields there is one more nonequilibrium mode: the energyimbalance mode (Fig. 4e) . It describes the excess en-ergy stemming from an equal change in the quasiparticlepopulations of the electron-like and hole-like branches.This energy mode affects charge transport properties in-directly via the self-consistency equation for ∆. Thismechanism explains, for example, the enhancement ofthe superconducting transition temperature in the pres-ence of a microwave field (Ivlev et al. , 1973; Klapwijk et al. , 1977).In this section we generalize the description in termsof nonequilibrium modes to account for superconductorswith spin-split density of states. The spin splitting (en-ergy difference 2 h = 2 µ B B between the black and reddispersion curves in Fig. 4 for spin up/down quasipar-ticles) gives rise to four distinct quasiparticle branches,electron/hole and spin up/down. These four nonequilib-rium modes and their coupling are at the basis of themain effects discussed in this review. A. Description of nonequilibrium modes in superconductorswith spin splitting
At this point we combine the pictorial description ofthe nonequilibrium modes (Fig. 4) with the quasiclassicalformalism introduced in Sec. II.A and in particular, theUsadel equation. For a description of non-equilibriumproperties we need to consider the Keldysh componentˆ g K of the quasiclassical GF [Eq. (2)]. For clarity we firstconsider a unique spin polarization direction parallel tothe z -axis. From the normalization condition, Eq. (3),ˆ g K can be expressed in terms of the retarded and ad-vanced components and the generalized matrix distribu-tion function ˆ f (Langenberg and Larkin, 1986)ˆ g K = ˆ g R ˆ f − ˆ f ˆ g A . (9)In the case of only one spin polarization axis, the 4 × f can be written as the sumof four components ˆ f = f L ˆ1 + f T τ + ( f T σ + f L σ τ ) . (10) Here we assume a unique spin polarization direction. In the mostgeneral case the distribution function has all spin componentsˆ f = f L ˆ1 + f T τ + (cid:80) j ( f Tj σ j + f Lj σ j τ ). L -labeled functions describe longi-tudinal modes, the (spin) energy degrees of freedom, andare antisymmetric in energy with respect to the Fermilevel, ε = 0. The T -labeled functions describe transversemodes and are symmetric in energy. In equilibrium, thedistribution function is proportional to the unit matrixin Nambu and spin space, and given byˆ f eq ( ε ) = (1 − n F )ˆ1 = tanh( ε/ T )ˆ1 . (11)We can now turn to the pictorial description of Fig. 4and associate each component of ˇ f in Eq. (10) with anonequilibrium mode as discussed next.As shown in Figs. 4(b)–(e), two of these modes haveelectron-hole branch imbalance, f T and f L , while f T and f L are particle-hole symmetric. The filled circles inFig. 4 represent the occupied states. As a reference, panel(a) corresponds to the equilibrium distribution functionˆ f = f L ˆ1 = tanh( ε/ T )ˆ1. In order to excite the nonequi-librium modes, f T , f T and f L , one only needs to movethe populated states (filled circles) between the differ-ent spectral branches in an elastic process, i.e., betweenequal-energy states (marked by horizontal dashed ar-rows). These modes can also relax back to equilibriumdue to elastic scattering processes. The relaxation mech-anisms depend on intrinsic material properties, degreeand type of disorder, and also on the superconductingspectrum, and are discussed in more detail below.The last nonequilibrium mode, the deviation of f L from f L , is characterized by a change in the total quasi-particle number and energy content, corresponding to anincrease or decrease of the effective temperature. It canbe excited by increasing the number of occupied statesto higher energies, and its relaxation requires inelasticprocesses.In the absence of spin splitting, the charge imbalanceis determined by f T , and the energy imbalance by f L .The spin splitting changes the system properties, mixingthe coupling between spin-dependent modes and physicalobservables [see Eqs. (16) and (17) below]. Qualitatively,the outcome can be seen by counting the number of oc-cupied states on the different branches in Fig. 4. Forexample, the charge imbalance µ is determined by thedifference between the number of occupied states in theelectron and hole branches. Both f T and f L componentscontribute to it, as seen in Figs. 4b and d.On the other hand, a nonzero spin accumulation µ z can be induced by exciting the modes f T or f L [Figs. 4(c),(e)]. These two contributions to the total spinaccumulation have important differences: The mode f T contributes to spin imbalance also in the absence of spinsplitting. Spin imbalance in this mode can be induced forexample by a spin-polarized injection from a ferromag-netic electrode, in both the normal and the supercon-ducting state. The relaxation of the spin accumulation created in this way is determined by elastic scatteringprocesses. The second mechanism of inducing spin ac-cumulation is by exciting the longitudinal mode f L , inthe presence of spin splitting [Fig. 4(e)]. Since energy-conserving transitions do not result in the relaxation ofthe f L mode, this component of the spin imbalance is notsuppressed by elastic scattering. In other words, its re-laxation can be only provided by inelastic processes, e.g.,electron-phonon and electron-electron scattering. Thisresult, obtained here on a phenomenological level, is cru-cial in understanding the long-range spin signal observedin superconductors, for example by H¨ubler et al. (2012)and discussed in the next sections. B. Accumulations in terms of the non-equilibrium modes
Quantitatively, we define the charge and spin accu-mulations based on the Keldysh component of the GF,Eq. (9), µ ( r , t ) = − (cid:90) ∞−∞ dε
16 Tr ˆ g K ( ε, r , t ) (12) µ sa ( r , t ) = (cid:90) ∞−∞ dε
16 Tr τ σ a [ˆ g K eq ( (cid:15), r , t ) − ˆ g K ( ε, r , t )] , (13)whereas the local energy and spin-energy accumulationsare given by q ( r , t ) = (cid:90) ∞−∞ dε ε Tr τ [ˆ g K eq ( ε, r , t ) − ˆ g K ( ε, r , t )] (14) q sa ( r , t ) = (cid:90) ∞−∞ dε ε Tr σ a [ˆ g K eq ( ε, r , t ) − ˆ g K ( ε, r , t )] . (15)Above, a = 1 , , µ S of the superconducting conden-sate (see below).In terms of the distribution functions, the charge andspin accumulations read (we assume magnetization in z -direction) µ = − (cid:90) ∞−∞ dε ( N + f T + N − f L ) (16) µ z = − (cid:90) ∞−∞ dε [ N + f T + N − ( f L − f eq )] , (17)where N + = N ↑ + N ↓ is the total density of states (DOS), N − = N ↑ − N ↓ is the DOS difference between the spinsubbands, and f eq ( ε ) = tanh( ε/ T ) is the equilibriumdistribution function. Similarly for (14,15) we get q = 12 (cid:90) ∞−∞ dεε [ N + ( f L − f eq ) + N − f T ] (18) q sa = 12 (cid:90) ∞−∞ dεε [ N − ( f L − f eq ) + N + f T ] . (19)1All these quantities, Eqs. (12-15) are directly related toexperimental observables. The charge imbalance µ char-acterizes the potential of the quasiparticles in the super-conductor (Artemenko and Volkov, 1979). In nonequilib-rium situations, µ can differ from the condensate poten-tial µ S . In the problems discussed in this Colloquium, ∆can be chosen time-independent and µ S = 0. The chargedensity depends on µ via ρ = − ν F e φ − eν F µ where φ isthe electrostatic scalar potential (Kopnin, 2001). In met-als, local charge neutrality is maintained on length scaleslarge compared to the Thomas–Fermi screening length,so that − eφ = µ and charge imbalance is associated withstatic electric fields.In the quasiclassical formulation used here, electro-chemical potential differences appear explicitly in energyshifts in the boundary conditions for the distributionfunctions (Belzig et al. , 1999). The Fermi distributionat potential V corresponds to f eq ,L ( T ) ( E ) = 12 [tanh (cid:0) E + eV T (cid:1) + ( − ) tanh (cid:0) E − eV T (cid:1) ] . (20)For superconductor at equilibrium, V = 0 in this descrip-tion. However, V = φ (cid:54) = 0 can describe voltage-biasednormal (∆ = 0) reservoirs.Spin accumulation is a standard observable in spin-tronics (Jedema et al. , 2002; Johnson and Silsbee, 1985).The local energy accumulation is typically measured viaelectron thermometry (Giazotto et al. , 2006). The spin-energy accumulation was measured recently in normal-state nanopillar spin valves (Dejene et al. , 2013). To ourknowledge this quantity has not been directly studiedexperimentally in superconducting systems.In the normal state the spectrum is trivial, g R ( A ) = ± τ σ . Thus, according to Eq. (9), the Keldysh compo-nent is simply proportional to the distribution function.In other words, the different modes decouple in Eqs. (12-15). Moreover, in the normal state it is unnecessary toseparate between transverse and longitudinal modes, andrather consider the spin-dependent full distribution func-tion f j ( E ) = [1 − f Lj ( E ) − f T j ( E )] /
2. Solutions of thekinetic equation in the normal state are discussed for ex-ample by Brataas et al. (2006).In the superconducting case the situation is more com-plex. First, the spectrum is strongly energy dependentaround the Fermi level and the spectral GFs have a non-trivial structure in spin space. Components proportionalto the unit matrix in spin space describes the BCS singletGFs, whereas terms proportional to the Pauli matrices σ j , j = 1 , ,
3, describe the triplet state (Bergeret et al. ,2001b, 2005). Second, due to this energy dependenceand non-trivial spin structure, the spectral functions en-ter (12-15) and lead to a coupling between the differentnon-equilibrium modes that in turns couple all electronicdegrees of freedom, as discussed next.
FIG. 5 Scanning electron microscopy image of the lateralstructure used by H¨ubler et al. (2012). From H¨ubler et al. (2012).
IV. SPIN INJECTION AND DIFFUSION INSUPERCONDUCTORS
Non-equilibrium modes can be experimentally studiedby means of non-local transport measurements. In thissection we review experiments on charge and spin injec-tion in superconductors, and apply the kinetic equationapproach described in the previous sections to describedifferent experimental situations.
A. Detection of spin and charge imbalance: Non-localtransport measurements
Studies of the nonequilibrium modes started with thepioneering experiment of Clarke (1972), who realized away of detecting the charge imbalance in a supercon-ductor. The main idea of this experiment is to injecta current from a normal metal (injector) into a super-conductor. This current creates a charge imbalance thatcorresponds to a shift of the chemical potential of thequasiparticles with respect to the one of the condensate.This shift of the chemical potential can be detected bya second electrode (detector) that probes the voltage be-tween the superconductor and the detector.More recent experiments used the same non-local mea-surement to explore the charge, energy and spin modesin mesoscopic superconducting lateral structures (Beck-mann et al. , 2004; H¨ubler et al. , 2010; Poli et al. , 2008;Quay et al. , 2013; Wolf et al. , 2014a, 2013). A scanningelectron microscopy image of such a lateral structure isshown in Fig. 5. A detailed overview of the experimentson charge and energy imbalance can be found in the re-cent topical reviews by Beckmann (2016) and Quay andAprili (2017).Whereas the charge and energy modes were known fora long time, it was first in the 1990s that theorists pre-dicted that electronic charge and spin degrees of free-dom can be separated in a superconductor (Kivelson andRokhsar, 1990; Zhao and Hershfield, 1995). First experi-ments on F-S-F layered structures, however, did not showany evidence of such a spin-charge separation (Gu et al. ,2002; Johnson, 1994) and the different relaxation timesfor spin and charge accumulation in superconductors re-2mained an open question.First clear insight into the separation of the spin andcharge modes was obtained in experiments using lateralnanostructures with ferromagnetic injectors and detec-tors (Beckmann et al. , 2004; Cadden-Zimansky et al. ,2007; H¨ubler et al. , 2012; Kolenda et al. , 2016; Poli et al. , 2008; Quay et al. , 2013; Shin et al. , 2005; Wolf et al. , 2013, 2014b; Yang et al. , 2010). First theoreti-cal works on spin injection into mesoscopic superconduc-tors (Morten et al. , 2004, 2005) showed that the spin-relaxation length in the superconducting state stronglydepends on the energy of the injected quasiparticles andon the spin relaxation mechanism. In particular, for adominating spin-orbit scattering, superconductivity sup-presses the spin relaxation rate τ − s , which can be qual-itatively understood as the decrease in the cross sectionof the quasiparticle momentum scattering at the ener-gies near the gap edge ε ∼ ∆. The suppression of τ − s is however compensated by the decrease in the quasipar-ticle group velocity v g ∼ v F (cid:112) − | ∆ | /ε so that thespin relaxation length λ so ∼ v g τ s remains almost un-changed in the superconducting state. On the contrary,if the spin-flip mechanism dominates, the spin relaxationis not related to the momentum scattering because theinteraction with magnetic impurities does not depend onthe propagation direction and the quasiparticle spin doesnot depend on energy. This results in an increase of τ − s which is equivalent to a decrease of the spin-relaxationlength in the superconducting state. Although theseworks provided an explanation to some experiments, twoimportant features observed in subsequent works couldnot be explained in terms of that theory: First, the spinaccumulation was detected at distances from the injectormuch larger than the spin-relaxation length measured inthe normal state (H¨ubler et al. , 2012; Quay et al. , 2013;Wolf et al. , 2013). Second, an unexpected spin accumu-lation was observed even if the current was injected froma non-magnetic electrode (Wolf et al. , 2013). In orderto explain these two observations one needs to take intoaccount the spin splitting in the superconductor. B. Non-local conductance measurements in spin-splitsuperconductors
Specifically, one of the setups studied by H¨ubler et al. (2012), was a lateral non-local spin valve (see Fig. 5)where the experimentalists determined the non-local dif-ferential conductance g nl = dI det dV inj . (21)Typical experimental curves are shown in Fig. 6a,adapted from H¨ubler et al. (2012) and Fig. 6b shows theresults calculated from the kinetic equations.If the detector is a ferromagnet with magnetizationcollinear with the spin accumulation in the wire, the current at the detector for V det = 0 is obtained fromEqs. (7,8), I det = ( µ + P det µ z ) /R det , (22)where R det = R (cid:3) /A is the detector interface resistancein the normal state, A is the cross-sectional area of thedetector, µ is the charge imbalance and µ z the spin im-balance defined in Eqs. (12,13). According to the explicitexpressions (16,17), the full description of the non-localcurrent requires all four non-equilibrium modes.Particularly interesting is the contribution from thesecond term in the r.h.s. of Eq. (17). It is nonzerowhen the spin splitting described by N − is nonzero andit provides a qualitative explanation of the experimentsby H¨ubler et al. (2012); Quay et al. (2013); and Wolf et al. (2013): The spin imbalance µ z , being related tothe energy nonequilibrium mode f L , once excited canonly relax via inelastic processes, especially mediated bythe electron-phonon interaction. At low temperaturesthe corresponding decay length can be much larger thanthe spin decay length in normal metals. This explainsthe long-range non-local signal observed in the experi-ments. The observed long-range spin accumulation canthus be understood to result from the spin accumulationgenerated by the effective heating of the superconduct-ing wire caused by the injection of nonequilibrium quasi-particles with energies larger than the superconductinggap (Bobkova and Bobkov, 2015, 2016; Krishtop et al. ,2015; Silaev et al. , 2015a,b; Virtanen et al. , 2016). Such aheating can originate, for example, by an injected currenteven from the non-ferromagnetic electrode. The heatingis not sensitive to the sign of the bias voltage at the injec-tor and hence the generated spin imbalance must be aneven function of the voltage, µ z ( V inj ) = µ z ( − V inj ). Thisleads to an antisymmetric shape of the non-local spinsignal in g nl with respect to V inj , in agreement with theexperimental observation (Wolf et al. , 2014a). All thesefeatures occur only if the superconductor has a spin-splitdensity of states induced either by an external magneticfield or by the proximity to a ferromagnetic insulator.A quantitative description of these effects can be pro-vided by solving the kinetic equations for superconduc-tors with spin-split subbands (Silaev et al. , 2015a). Inthis case the diffusion couples non-equilibrium modespairwise. In particular, the kinetic equations (5) takethe form ∇ · j e j s j c j se = S T R T R L R L R T + S L f L f T f T f L , (23)where the spectral energy j e , spin j s , charge j c and spin3 Figure 2(a) shows the local differential conductance ofone contact as a function of the injection bias voltage V inj for different applied magnetic fields B at T ¼
50 mK . Forsmall B pronounced gap features at V (cid:2) (cid:3) (cid:1) V areobserved as well as a negligible subgap conductance. Uponincreasing the magnetic field, the gap features broaden dueto orbital pair breaking, and for B > : the Zeemansplitting is seen. We describe our data with the standardmodel of high-field tunneling [8] to obtain the spin-dependent density of states n (cid:3) ð E Þ , where (cid:3) ¼ (cid:3) standsfor spin up and down, respectively. From n (cid:3) ð E Þ , we cal-culate the current for each spin I (cid:3) ¼ G N e Z ð (cid:1) (cid:3)P Þ n (cid:3) ð E Þ½ f ð E Þ (cid:1) f ð E þ eV Þ(cid:4) dE (1)where G N ¼ G þ G " is the normal-state junction conduc-tance, and f is the Fermi function. The total chargecurrent is I ¼ I " þ I , and the spin current is proportionalto I s ¼ I " (cid:1) I . Fits of this model to the measured con-ductance spectra yield G N , the pair-breaking parameter (cid:1) ,and the spin-orbit scattering strength b so . Details of the fitprocedure have been given previously [9,10]. The spinpolarization P ¼ : (cid:3) : obtained from these fits isthe same as obtained from the spin-valve experiments. Therelatively small P is typical of ultrathin alumina tunnelbarriers [11]. Figure 2(b) shows a contour plot of thecomplete dataset of the local conductance as a functionof bias and magnetic field. The gap observed at B ¼
100 mT is slightly larger than at zero applied field. Weattribute this to the presence of stray fields of the ferro-magnetic contacts. At higher fields, in the wedge-shapedregions indicated by the lines, a single spin band dominatesconductance.Next, we focus on the nonlocal differential conductance.To eliminate the effect of small variations of the junctionconductances, we plot the normalized nonlocal conduc-tance g nl =G inj G det throughout this Letter. In Fig. 3(a) g nl =G inj G det is displayed as a function of the applied biasvoltage V inj for different magnetic fields B and a contact distance d (cid:2) (cid:1) m . The data were measured simulta-neously with the local conductance of Fig. 2(a), in theconfiguration shown in Fig. 1. For comparison, we showdata obtained from the NISIN reference sample in Fig. 3(b).At B ¼ , there is no conductance below the gap, and abovethe gap, both the FISIF and NISIN samples show a nearlylinear increase due to charge imbalance [9]. With increas-ing magnetic field, the charge imbalance signal decreases,as clearly seen for the NISIN sample. The FISIF sampleshows a qualitatively different behavior: ( i ) in the biasrange corresponding to the Zeeman splitting, a positivepeak arises for V inj < , and a negative peak for V inj > ;( ii ) for higher bias j V inj j * (cid:1) V , an additional asym-metry evolves on top of the charge imbalance signal.While the observation ( i ) is systematic for all nine samples,( ii ) was observed only in a few samples, whereas othersamples showed the symmetric charge imbalance signalalso seen in the NISIN sample at high bias. In the followingwe therefore only concentrate on the asymmetric peakfeatures. Upon increasing the field, the peak heightsincrease gradually to their extremal values at B (cid:5) : (cid:1) :
75 T , before the peaks start to decline, broaden and moveinwards, simultaneously. The positive peak (at negativebias) is slightly larger than the negative peak (at positivebias). Above the critical field B c (cid:2) :
15 T the asymmetricfeatures disappear and one finds a small bias-independentsignal (not shown).
FIG. 2 (color online). (a) Local differential conductance g loc ¼ dI inj =dV inj of one junction as a function of injector bias V inj for different applied magnetic fields B . (b) The same dataplotted on a color scale. The lines indicate the regions where asingle spin band dominates conductance. FIG. 3 (color online). Normalized nonlocal differential con-ductance g nl =G inj G det as a function of injector bias V inj fordifferent applied magnetic fields B for one pair of contacts (a),nonlocal conductance of a pair of contacts of the NISIN refer-ence sample (b), the data from panel (a) plotted on a color scale(c), and calculated differential spin current (d). PRL week ending16 NOVEMBER 2012 -2 -1 0 1 2 V inj / Δ -8-6-4-202468 g n l / g h =0; P inj =0h =0.2 Δ ; P inj =0h =0.2 Δ ; P inj =0.5 (b) FIG. 2: (Color online) Temperature dependencies of the figure of merit ZT ( T ) = α / ( G th G − α ) calculated according to theexpressions Eq.(39,40,41) in the Review draft in the presence of spin-orbital scattering β = −
1. Curves from top to bottomcorrespond to the spin relaxation times 1 / ( τ sn T c ) = 0;0 . . . . . FIG. 6 (a) Nonlocal conductance measured as a functionof the injecting voltage, g nl ( V inj ) adopted from H¨ubler et al. (2012). (b) The same quantity calculated using the ki-netic theory for α orb = 1 . β = 0 .
5, ( τ sn T c ) − = 0 . T = 0 . T c , effective inelastic relaxation length L = 20 λ sn , L det = 5 λ sn . Black solid and red dash-dotted curves cor-respond to the injection from non-ferromagnetic ( P inj = 0)and ferromagnetic ( P inj = 0 .
5) electrodes, respectively at thespin-splitting h = 0 . . Blue dashed line corresponds to h = 0. The conductance is normalized to g = R ξ / ( R inj R det ),where R ξ = ξ/ ( A s σ N ) is the normal-state resistance of thewire with length ξ and cross section A s . energy j se currents derived from the general Eq.(6) are j e j s j c j se = D L D T D T D L D T D L D L D T ∇ f L ∇ f T ∇ f T ∇ f L . (24)Here D L/T/T /L are kinetic coefficients related to thespectral GFs (Silaev et al. , 2015a), S T /L are parts ofcollision integrals describing spin relaxation, and R T/L the coupling between the quasiparticles and the super-conducting condensate.On the one hand, the charge is coupled to the spin-energy mode [lower right block of Eq. (24)]. The relax-ation of both of these modes, right hand side of Eq. (23),is nonvanishing for all energies, below and above the gapdue to the magnetic pair breaking effects (Nielsen et al. ,1982; Schmid and Sch¨on, 1975). On the other hand, thespin-splitting field couples the spin and energy modes, f L and f T respectively [upper left block of Eq. (24)].As explained above, the energy mode can only decay viainelastic scattering which at low temperature can be dis-regarded compared to the spin relaxation.Solutions of Eqs. (23,24) along with Eqs. (16,22) repro-duce the main features of the measured non-local conduc-tance presented in Fig. 6a. Depending on the magnitudesof the spin-splitting field h and the injector polarization P inj , we can identify three distinct parameter regimes af-fecting the symmetry of g nl . (i) When h = P inj = 0 (bluedashed curve in Fig. 6b), the only contribution to thedetector current comes from charge imbalance and g nl isa symmetric function of the injection voltage. In the ab-sence of spin splitting and depairing effects, R T = 0 for ε > ∆, and hence charge imbalance decays only via in-elastic scattering neglected here. This explains the mono-tonic increase of g nl in Fig. 6b at large voltages. (ii) For P inj = 0 but in the presence of an applied field leading to h (cid:54) = 0 (black solid curve), charge relaxation is stronglyenhanced due to the orbital depairing. The main long-range contribution comes from µ z produced by the heat-ing effect described above. The resulting g nl is an anti-symmetric function of V inj . (iii) When both h (cid:54) = 0 and P inj (cid:54) = 0 (red dash-dotted curve), an additional symmet-ric long-range contribution in g nl results due to a thermo-electric effect at the injector. Note that in the case h = 0, P inj (cid:54) = 0, there would be another symmetric contributionto g nl due to the regular spin injection also present inthe normal state. However, this is a short-range mode(decays via spin relaxation), and therefore does not showup beyond the spin relaxation length.In the experiments by H¨ubler et al. (2012); Quay et al. (2013); and Wolf et al. (2014a) the spin-splitting fieldwas caused by an external magnetic field. Therefore oneneeds to take into account the orbital depairing effectof the magnetic field in addition to the Zeeman effect.The relative strength of the orbital depairing and thespin-splitting field is described by the dimensionless pa-rameter α orb = ( hτ orb ) − . In Fig. 6 we choose the value α orb = 1 .
33, which should correspond to the experimentby H¨ubler et al. (2012).In the presence of a supercurrent, all coefficients of thematrix in Eq. (24) are nonzero (Aikebaier et al. , 2017).As a result, for example the spin and charge modes aredirectly coupled by diffusion.
C. Spin Hanle effect
In the previous sections we assume that all magneti-zations and the applied field are collinear. If one liftsthis assumption, the applied field leads to a precessionof the injected spin around the field direction. This isthe spin Hanle effect that in the normal state has beenextensively studied in the literature and observed in sev-eral experiments (Jedema et al. , 2001, 2003; Johnson andSilsbee, 1985; Villamor et al. , 2015; Yang et al. , 2008).The Hanle precession can be measured via the non-localconductance in a setup such as the one shown in Fig. 5.The non-local measured signal oscillates and decays as afunction of the amplitude of the applied field.Formally the Hanle effect is described by the first termon the r.h.s of Eq. (5). Indeed, one can derive theBloch-Torrey transport equation (Torrey, 1956) for themagnetic moment m ( ε, x ) = Tr( τ σ g K ) / et al. , 2015b). It reads ∂ m ∂t + ∇ · j s = γ m × h s − m /τ S . (25)Here γ = − j s is the spin current density tensor. In the normal state4the spin relaxation τ S and Zeeman field h s are energyindependent. This explains why the nonlocal resistancevs. field curve does not depend either on temperature oron the type of spin relaxation (magnetic or spin-orbit im-purities). In contrast, they are predicted to be stronglyenergy dependent in the superconducting state, and theprecession and decay of the nonlocal signal disappear at T →
0, whereas the shape of the curves at interme-diate temperatures depends on the type of spin relax-ation (Silaev et al. , 2015b). Experimental evidence ofthe Hanle effect in the superconducting state has notbeen reported so far.
D. Spin imbalance by ac excitation
The quasiparticle f T,j mode — or equivalently, thequasiparticle magnetic moment m ( ε, x ) above — can beexcited by an external ac magnetic field, which via theZeeman coupling generates a conduction electron spinresonance (Aoi and Swihart, 1970; Maki, 1973; Nemes et al. , 2000; Vier and Schultz, 1983; Yafet et al. , 1984).This was recently studied experimentally in spin-splitthin Al films by Quay et al. (2015). As the f T,j mode canrelax rapidly via elastic spin-flip scattering, the linewidthseen in such experiments is generally τ − S (cid:39) τ − sn in-stead of the time scale of the long-ranged non-local spinsignal. Spin-flip scattering also provides a channel viawhich electromagnetic fields can generate spin imbalancethrough the orbital coupling (van Bentum and Wyder,1986; Virtanen et al. , 2016). For high enough driving am-plitude, the imbalance modifies the self-consistent ∆( T )relation, which develops additional features in the spin-split case (Eliashberg, 1970; Virtanen et al. , 2016). Ef-fects related to spin-splitting and relaxation can more-over be probed with tunnel junctions at low frequen-cies (Quay et al. , 2016) or via photoassisted tunneling(Marchegiani et al. , 2016). V. THERMOELECTRIC EFFECTS
Thermoelectric effects relate temperature differencesto charge currents, and electrical potentials to heat cur-rents. Thermoelectric effects are typically described viathe linear response relation between charge and heat cur-rents I , ˙ Q and bias voltage and temperature difference V and ∆ T across a contact: (cid:32) I ˙ Q (cid:33) = (cid:32) G αα G th T (cid:33) (cid:32) V − ∆ T /T (cid:33) . (26)Here G is the conductance and G th the heat conductanceof the contact. α is the thermoelectric coefficient.With a non-zero α , electrical energy may be convertedto heat or cooling, or reciprocally a temperature differ-ence may be converted to electrical power. The efficiencyof this conversion is typically described by thethermoelectric figure of merit, ZT = α G th GT − α = S GT ˜ G th , (27)where S = α/ ( GT ) is the thermopower (Seebeck coef-ficient) and ˜ G th = G th − α / ( GT ) is the thermal con-ductance at a vanishing current. In particular, the max-imum efficiency of a thermoelectric heat engine is (Sny-der and Ursell, 2003) max η = η Carnot √ ZT − √ ZT +1 with η Carnot = ∆
T /T . Maximum efficiencies of the deviceare obtained when ZT → ∞ . At or above room tem-perature, the record-high figures of merit are obtainedin certain strongly doped semiconductor structures (Kim et al. , 2015; Zhao et al. , 2016). A typical record value forthose cases is ZT (cid:38) . . . et al. (1974) showed that α = α N G (∆ /T ) , G ( x ) = 32 π (cid:90) ∞ x y dy cosh ( y/ , (28)where the latter form comes from the reduction of thequasiparticle density in the superconducting state, and α N is the value of the thermoelectric coefficient in the In the case of thermoelectric effects, it is customary to talk aboutheat currents ˙ Q instead of energy currents ˙ U , and we adapt thisconvention here. These are related by (Ashcroft and Mermin,1976) ˙ Q = ˙ U − µI/e , where µ is a reference energy compared tothe Fermi level. At linear response we can set µ = 0 in whichcase ˙ Q = ˙ U . On the other hand, when considering heat balanceat non-vanishing voltages as in Sec. V.A, the two are not thesame and rather the heat current ˙ Q should be used. α N depends onthe exact electronic spectrum. For example, for a sim-ple quadratic dispersion α N = π G T k B T eE F , where E F isthe Fermi energy. At temperatures T (cid:28) ∆ /k B , α isthus expected to be a product of two small coefficients, α N ∝ k B T /E F , and G (∆ /T ). This is very small and noteasy to measure quantitatively.However, superconductors do contain some ingredientsfor strong thermoelectric effects, because the latter typi-cally require strongly energy dependent density of statesof the charge carriers. This is provided by the BCS den-sity of states. Hence, if one can break the electron-holesymmetry of the transport process via some mechanism,superconductors can become very strong thermoelectrics.This is precisely what happens in spin-split superconduc-tors, as an exchange field breaks the symmetry in eachspin sector, but so that the overall spin-summed energyspectrum remains electron-hole symmetric. Transportthrough a spin filter to a spin-split superconductor thencan provide large thermoelectric effects because the twospins are weighed differently (Machon et al. , 2013, 2014;Ozaeta et al. , 2014). We discuss these effects in this sec-tion. A. Charge and heat currents at a spin-polarized interfaceto a spin-split superconductor
Consider a tunnel contact from a non-superconductingreservoir R to a superconductor S in a spin-splitting field.Let us assume that the tunnel contact is magnetic, sothat the conductance through it is spin-polarized. De-noting the spin-dependent conductances in the normalstate by G ↑ , G ↓ we can parameterize them by the to-tal conductance G T = G ↑ + G ↓ and the spin polariza-tion P = ( G ↑ − G ↓ ) /G T . The total tunneling quasipar-ticle charge and heat currents are now expressed as asum over spin-dependent contributions, but otherwise ofthe standard form (Giaever and Megerle, 1961; Giazotto et al. , 2006). Denoting the spin-dependent reduced den-sity of states via N + = N ↑ + N ↓ and N − = N ↑ − N ↓ the spin-averaged tunnel currents can be obtained fromthe Keldysh component of Eq. (8) after taking the cor-responding traces: I = G T e (cid:90) ∞−∞ dε ( N + + P N − ) ( f R − f S ) (29)˙ Q i = G T e (cid:90) ∞−∞ dε ( ε − µ i )( N + + P N − )( f R − f S ) . (30)Here f R/S = n F ( E − µ R/S ; T R/S ), n F ( E ; T ) = { exp[ E/ ( k B T )] + 1 } − are the (Fermi) functions of thereservoirs biased at potentials µ R/S and temperatures T R/S . The reduced density of states in the superconduc-tor for spin σ is N σ ( ε ). The heat current ˙ Q iσ is calcu-lated separately for i = R, S , using the potential µ R/S , eV= " -1.5 -1 -0.5 0 0.5 1 1.5 e _ Q = ( G T " ) -0.3-0.25-0.2-0.15-0.1-0.0500.05 h = 0 h = 0 : " h = 0 : " h = 0 : " h = 0 : " h = 0 : " h = 0 : " FIG. 7 Cooling power from reservoir R vs. voltage for differ-ent values of the exchange field h , assuming a unit polarization P = 1 at the temperature k B T = 0 .
3∆ close to that yieldingoptimal cooling for P = h = 0. The exchange fields are givenin units of ∆. Changing the sign of P or h inverts the voltagedependence with respect to V = 0. because the two heat currents differ by the Joule power I ( µ R − µ S ) /e . In the analysis below, we disregard the spinrelaxation effects on the density of states, because thisassumption allows for some analytically treatable limitsand because it is a fair approximation in the case of oftenused Al samples.The heat current from R is a non-monotonous func-tion of voltage even in the absence of spin polariza-tion or temperature difference. In particular, for volt-age V = ( µ R − µ S ) ≈ ∆ /e , it is positive, i.e., reservoir R cools down (Leivo et al. , 1996; Nahum et al. , 1994; Pekola et al. , 2004). This heat current is quadratic in the volt-age, and therefore it does not result from the usual Peltiereffect [Eq. (26) for ˙ Q ] where the cooling power is linearin voltage.Interestingly, in the presence of spin polarization P and with a non-zero spin-splitting field h in the super-conductor, the cooling power is nonzero even in the linearresponse regime, i.e. low voltages (Ozaeta et al. , 2014).As an example we show in Fig. 7 the cooling power fromreservoir R as a function of voltage for various values of h ,assuming the ideal case of unit spin polarization P = 1.Contrary to the spin-independent case, the N-FI-S ele-ment can also be used to refrigerate the superconductor.Electron refrigeration using magnetic elements have beenstudied by Rouco et al. (2018). B. Linear response: heat engine based on asuperconductor/ferromagnet structure
As can be seen in Fig. 7, the simultaneous presence ofthe non-vanishing spin polarization P and a spin-splitting6field h lead to a heat current that has a linear componentin the voltage V . This component is nothing else thanthe Peltier effect. In the limit k B T (cid:28) ∆ − h the linear-response coefficients evaluate to (Ozaeta et al. , 2014) G ≈ G T (cid:112) π ˜∆ cosh(˜ h ) e − ˜∆ , (31) G th ≈ k B G T ∆ e (cid:114) π e − ˜∆ (cid:104) e ˜ h ( ˜∆ − ˜ h ) + e − ˜ h ( ˜∆ + ˜ h ) (cid:105) , (32) α ≈ G T Pe (cid:112) π ˜∆ e − ˜∆ (cid:104) ∆ sinh(˜ h ) − h cosh(˜ h ) (cid:105) , (33)where ˜∆ = ∆ / ( k B T ) and ˜ h = h/ ( k B T ). These yield thethermopower S = αGT ≈ P ∆ eT [tanh(˜ h ) − h/ ∆] . (34)At low temperatures the thermopower is maximized for h = k B T arcosh[∆ / ( k B T )], where it is S max ≈ k B e P (cid:34) ∆ k B T − arcosh (cid:32)(cid:114) ∆ k B T (cid:33)(cid:35) . (35)It can hence become much larger than the “natural scale” k B /e , and even diverge towards low temperatures. How-ever, such a divergence comes together with the vanishingof the conductance, Eq. (31), and therefore is in practiceeither cut off by circuit effects, where the impedance tothe voltmeter becomes lower than the contact impedance,due to spin relaxation neglected above, or alternativelyby additional contributions beyond the BCS model. Thelatter ones are described in more detail by Ozaeta et al. (2014). Nevertheless, with proper circuit design oneshould be able to measure a thermopower much exceed-ing k B /e in this setup.The above theoretical predictions in the linear responseregime were confirmed experimentally by Kolenda et al. (2017, 2016). In particular, they prepared a sample con-taining a crossing of three types of metals, a normal-metallic Cu, ferromagnetic Fe, and superconducting Al.The measured configuration is sketched in Fig. 8a. Theelectrons in the ferromagnetic wire were heated with theheater current I heat , producing a temperature differencebetween the ferromagnet and the superconductor. Thecontact between the ferromagnet and the normal metal isohmic and therefore the temperature difference betweenthem is negligibly small. Then the thermoelectric cur-rent was measured as a function of the magnetic field B applied parallel to the ferromagnetic wire. The agree-ment between the experimental results and the above de-scribed tunneling theory was excellent (see Fig. 9). Thetemperature difference between the ferromagnet and thesuperconductor was a fitting parameter, whereas the po-larization P was fitted from the conductance spectrum.In the experiment it was fitted to the value P = 0 .
08, a
FIG. 8 a) Schematic setup for measuring the thermoelectri-cally induced current, used by Kolenda et al. (2016). S, F,and N stand for a superconductor, ferromagnet and a normalmetal, whereas FI is a ferromagnetic insulator. b) Setup usedfor a direct measurement of the Seebeck effect. c) Heat en-gine realized in a lateral setup with “n-doped” and “p-doped”junctions using a FNF trilayer with antiparallel magnetiza-tion directions. To disregard spin accumulation, the islandhas to be large compared to the spin relaxation length. d)Heat engine with a spin-split superconducting island. Theferromagnets can also be replaced by a normal metal if theinterfaces to the superconductor contain a ferromagnetic in-sulator. In (c) and (d), the heating power P heat is partiallyconverted to “useful” work P work dissipated on the load. modest value attributed to the thin oxide barrier betweenthe Fe and the Al layers. In principle larger values of P can be obtained by increasing the thickness of the ox-ide barrier (M¨unzenberg and Moodera, 2004), but this ofcourse would reduce the amplitude of the thermoelectriccurrent.In the experiment, the thermoelectric current was mea-sured rather than the voltage. In that case the impedanceof the sample dominated that of the measurement lines.This is why the measurement yielded the exponentiallylow thermoelectric current, which nevertheless was size-able. The measurement configuration in Fig. 8b wouldhave directly measured the generated voltage drop ( i.e. ,Seebeck effect) instead of the current. This voltage re-sults from the ratio of two exponentially small functions,the thermoelectric coefficient α and the conductance G ,and itself is not small. Such a measurement would thentell about spurious effects, for example due to spin re-laxation, or due to the presence of fluctuations or statesinside the gap. These effects would limit the divergingSeebeck coefficient at low temperatures (Ozaeta et al. ,2014). Better still, replacing the normal metal with an-other superconductor with an inverse spin-splitting field,would have resulted to twice as large signal (correspond-ing to a series of p- and n-doped thermoelectric elements),7 FIG. 9 Thermoelectric current as a function of the appliedmagnetic field, measured in (Kolenda et al. , 2016). The cir-cles show the measurement values, the solid lines show acomparison to Eq. (29). The three solid lines correspond toslightly different temperature differences; for further details,see (Kolenda et al. , 2016). From Kolenda et al. (2016). but would not be possible to create as such with a mag-netic field. The solution would be furthermore to re-place the ferromagnetic wire by an FNF heterostructure[Fig. 8c, where the ferromagnets have antiparallel mag-netizations, for example due to different coercive fields,and the normal metal N would serve as a spacer betweenthem]. To reach high figures of merit, the ferromagneticmetals should also be replaced by ferromagnetic insula-tors, which can reach very high values of spin polarization(see Table I), with P exceeding 0.9.The island setup in Figs. 8(c) and (d) also realizes athermally isolated structure, in contrast to those in (a)and (b). This allows realizing a heat engine, where thevoltage measurement is replaced by the “device” to bepowered with the engine, with resistance that should bematched to the thermoelectric element. If only the elec-trons of the ferromagnetic island are heated, the mainspurious heat conduction mechanism is due to electron-phonon coupling. In that case it is advantageous to usethe structure (d), because the electron-phonon heat con-ductance is weaker in a superconductor (Heikkil¨a et al. ,2017; Kaplan et al. , 1976) than in a normal metal (Well-stood et al. , 1994). For example, Fig. 10 shows a pre-diction for the resulting temperature dependence of thethermoelectric figure of merit ZT in structure (d), in-cluding this spurious heat conduction. In an optimizedstructure, very large ZT could thus be expected. In thepicture, g = 5 k B √ πe ΣΩ∆ / (2 G T ) is a dimensionlessquantity characterizing the relative strength of electron- k B T= " ( ! P ) Z T = P P = 0 : ; ! = 10 ! " P = 0 : ; ! = 10 ! " P = 0 : ; ! = 10 ! " P = 0 : ; ! = 10 ! " P = 0 : ; ! = 10 ! " P = 0 : ; ! = 10 ! " FIG. 10 Figure of merit in a N-FI-S-FI-N heat engine as afunction of temperature for polarizations P of the junction.The figure has been calculated with h = 0 .
5∆ and g = 1000,without calculating ∆ self-consistently. The solid lines cor-respond to Γ = 10 − ∆ and the dashed lines to Γ = 10 − ∆.The figure of merit at low temperatures reaches very close to P / (1 − P ) unless P is very close to unity, but the exacttemperature scale where this happens depends on the valueof polarization. At the lowest temperatures ZT is limitedby another spurious heat conduction process, due to nonzerodensity of states inside the gap, described here by the DynesΓ parameter. phonon coupling (characterized by Σ (Giazotto et al. ,2006)) to the tunnel coupling of the thermoelectric el-ement in an island with volume Ω. For example, forΩ = 0 . µ m , Σ = 10 W µ m − K − and 1 /G T = 30kΩ, g = 1000.Note that it is really the presence of the spuriouselectron-phonon heat conduction that limits the highestavailable values of ZT . Often such spurious mechanismsare disregarded from the theoretical analysis, for examplein the case of quantum dots (Hwang et al. , 2016).Even if the true figure of merit of the type of heat en-gine discussed above can be made high, these systemscannot obviously be used to replace room-temperaturethermoelectric devices to be applied for example in en-ergy harvesting. However, there are other applicationswhere the large figure of merit may turn out to be essen-tial. For example, this type of thermoelectric heat enginecan be used for thermal radiation sensing at low temper-atures (Giazotto et al. , 2006; Heikkil¨a et al. , 2017). An-other possible use of the thermoelectric effects would bein non-invasive low-temperature thermometry (Giazotto et al. , 2015b), where the temperature (difference) profilescould be read from the thermopower, without having toapply currents. In a scanning mode this would hence bea low-temperature version of the method used by Menges et al. (2016).8Note that the above discussion disregards the effect ofspin-orbit or spin-flip scattering on the superconductingstate. It limits ZT especially in heavy-metal supercon-ductors. The associated effects were considered by Berg-eret et al. (2017) and Rezaei et al. (2017). C. Spin Seebeck effect
Besides the large thermoelectric effect, the contact be-tween spin-split superconductors with other conductingmaterials can exhibit a large (longitudinal) spin See-beck effect, where a temperature difference drives spincurrents to/from the spin-split superconductor (Ozaeta et al. , 2014). In this case the charge, heat, spin and spinheat currents are described by the full (Jacquod et al. ,2012; Machon et al. , 2013; Onsager, 1931) Onsager linear-response matrix I ˙ QI s ˙ Q s = G α P G ˜ αα G th T ˜ α P G th TP G ˜ α G α ˜ α P G th T α G th T V − ∆ T /TV s / − ∆ T s / T , (36)where for k B T (cid:28) ∆ − h the coefficients G , G th and α aregiven in Eqs. (31-33), and ˜ α = α/P . Here V s and ∆ T s refer to spin-dependent biases (Bergeret et al. , 2017).The spin currents induced in the case of two spin-splitsuperconductors, and the additional effects of Josephsoncoupling, magnetization texture and spin-orbit effects arediscussed by Bathen and Linder (2017) and Linder andBathen (2016). When either of the two materials real-izes an island, the spin current can be converted into aspin accumulation µ z that is determined from the bal-ance between thermally induced spin currents and spinrelaxation within the island. The above discussion onheat engines assumes a structure size much longer thanthe spin-relaxation length, and hence disregards this spinaccumulation. The effect of the thermally induced spinaccumulation on the superconducting gap was consideredby Bobkova and Bobkov (2017), who predicted the asso-ciated changes in the critical temperature.This spin Seebeck effect should be contrastedto the analogous phenomenon discussed in non-superconducting materials (Uchida et al. , 2014). There,a major contribution to the spin Seebeck signal is dueto the thermally induced spin pumping (Hoffman et al. ,2013). D. Thermophase in a S(FI)S contact
The large thermoelectric effect described above allowsfor a large thermally induced phase gradient. This wastheoretically investigated by Giazotto et al. (2015a). The total current in this case consists of the sum of a ther-moelectric current I th and the supercurrent, I = I th + I c sin( ϕ ) , (37)where I th is obtained from (29) and I c is the critical cur-rent for the junction with a phase difference ϕ of theorder parameters across the contact. The critical currentis proportional to √ − P (Bergeret et al. , 2012) anddepends on the spin-splitting field in S (Bergeret and Gi-azotto, 2014)In an electrically open configuration, the two currentsmust cancel, and instead a thermophase ϕ th developsacross the junction. This is obtained fromsin( ϕ th ) = − I th I c . (38)The thermophase can be detected using a bimetallic loopwith two contacts, characterized by critical currents I c , and thermophases ϕ th1 , . For non-zero exchange fieldand spin polarization P , the resulting thermophases canbe much larger than in ordinary bulk superconductors.Hence the temperature dependence of the inductancesplay a more minor role than in the case of superconduc-tors without spin splitting (Shelly et al. , 2016; Van Har-lingen et al. , 1980). For junctions with non-equal ther-mophases and for negligible loop inductance (in practice,2 eLI c , / (cid:126) (cid:28)
1) in the absence of an external flux thecirculating current is I circ = I c I c I c + I c (cid:2) sin( ϕ th1 ) − sin( ϕ th2 ) (cid:3) . (39)In the case of symmetric junctions both thermophasesare the same and the circulating current in the absenceof an external flux vanishes. However, as discussed byGiazotto et al. (2015a), the thermoelectric current affectsthe response of the circulating current to the externalflux, allowing for their measurement also in that case.Equation (38) requires that both sides of the equationhave an absolute value of at most unity, i.e., | I th | < I c .For a very large thermoelectric current, its cancellationwith a supercurrent is no longer possible, and instead avoltage across the contact forms. In this case the directcurrent response of the junction is more similar to thecase discussed above in the linear response limit for a N-FI-S junction. This regime was investigated in detail byLinder and Bathen (2016). Moreover, the nonvanishingdc voltage across the superconducting junction leads toJosephson oscillations at the frequency 2 eV /h , where h is the Planck constant. Hence, the device can be usedas a temperature (difference) to frequency converter asdiscussed in more detail by Giazotto et al. (2015b). VI. SUMMARY AND OUTLOOK
This review focuses on transport and thermal prop-erties of superconducting hybrid structures with a spin-9split density of states. Such a splitting can be achievedeither by an external magnetic field, or, more interest-ingly, by placing a ferromagnetic insulator (FI) adjacentto a superconducting layer (S) (Sec. II). We discuss sev-eral experimental situations with the help of a theoret-ical framework (see Sec. II.A and III.B) based on thequasiclassical formalism, with which one can account forboth thermodynamical and nonequilibrium properties ofsuch hybrid structures. In order to account for effectsbeyond quasiclassics, as for example strong spin polar-ization, we combine the quasiclassical equations with ef-fective boundary conditions.Out-of equilibrium superconductivity by itself leads toa decoupling between the charge and energy degrees offreedom of the electronic transport. In this review weshow that the combination between superconductivityand magnetism requires on one hand a description of ad-ditional nonequilibrium modes, spin and spin energy, andon the other hand to couples them all. This leads to noveland intriguing phenomena discussed in this review withdirect impact in latest research activities and proposedfuture technologies based on superconductors and spindependent-fields (Eschrig, 2011, 2015; Linder and Robin-son, 2015). By using the theoretical formalism presentedin this review one can predict and explain phenomenasuch as the spin injection and relaxation (Sec. IV) in su-perconductors with an intrinsic exchange field along withtheir consequences in the transport properties. We alsodiscuss a number of striking thermoelectric effects in su-perconductors with a spin-splitting field (Sec. V).The best scenario for the phenomena and applicationsdiscussed here, and in particular for the thermoelectriceffects, are FI-S systems where the spin splitting canbe achieved without the need of an applied magneticfield. Hence it becomes important to look for ideal FI-Smaterial combinations. So far europium chalcogenides(EuO, EuS and EuSe) together with Aluminum filmshave shown large splittings and hence these are the bestcombination. In addition, thin films of EuO or EuS canbe used as almost perfect spin filters (see Table I) andhence they are good candidates for realizing the near-optimal heat engines proposed in Sec. V. One of the mainchallenges from this perspective is to find FI-S combi-nations with large superconducting critical temperatureand simultaneously a large spin splitting. Superconduc-tors like Nb or Pb on the one hand increase T C withrespect to Al-based structures, but on the other the spin-orbit coupling may spoil the sharp splitting as discussedin Sec. II. Recent experiments on GdN-NbN suggest largesplittings (Pal and Blamire, 2015) but further research inthis direction is needed.In Sec. IV.D we briefly discuss the dynamics of spin-split superconductors in rf fields. Historically, magneticresonance effects in superconductors are well studied, butfewer experiments have probed spin-split thin films.Besides the effects discussed in this review, several the- oretical studies made striking predictions in mesoscopicsystems with spin-split superconductors, such as the cre-ation of highly polarized spin currents (Giazotto andBergeret, 2013b; Giazotto and Taddei, 2008; Huertas-Hernando et al. , 2002), large supercurrents in FI-S-I-S-FIjunctions (Bergeret et al. , 2001a), junctions with switch-able current-phase relations (Strambini et al. , 2015), andan almost ideal heat valve based on S-FI elements (Gia-zotto and Bergeret, 2013a).Although many of the transport phenomena in spin-split superconductors are now well-understood, we fore-see a number of exciting avenues for future research.One further perspective of the present work is the ex-tension of the Keldysh quasiclassical formalism used inthis review to include magneto-electric effects associatedwith the spin-orbit coupling (SOC). For a linear in mo-mentum SOC the generalization of this can be done byintroducing an effective SU(2) gauge potential. The qua-siclassical equations in this case have been derived byBergeret and Tokatly (2013, 2014, 2016). Effects such asthe spin-Hall and spin-galvanic effect in superconductorshave been studied in the equilibrium case (Konschelle et al. , 2015). Extending these results to a nonequilib-rium situation, and also to time-dependent fields, wouldbe an interesting further development and would allowfor a detailed study of the well-controlled non-linearitiesassociated to these effects in superconductors. First stepsin this direction have been taken in (Espedal et al. , 2017).Recent discoveries of skyrmionic states in chiral mag-nets (Nagaosa and Tokura, 2012) have attracted a lotof attention due to the effects resulting from the inter-play of magnetism and SOC (Soumyanarayanan et al. ,2016) which can induce chiral Dzyaloshinskii-Moriya in-teractions between magnetic moments. Currently it isvery interesting to study these effects in the presenceof the additional component — superconductivity, whenthe exchange interaction is mediated by the Cooper pairs(de Gennes, 1966b). One can expect that in such systemssuperconductivity can induce a non-trivial magnetic or-dering and dynamics. These effects can show up in var-ious systems including ferromagnet/superconductor bi-layers, surface magnetic adatoms and bulk magnetic im-purities inducing the localized Yu-Shiba-Rusinov statesmodified by the SOC (Pershoguba et al. , 2015).Superconducting structures with strong spin-orbit cou-pling and exchange fields are also of high interest inview of engineering a platform for realization of topo-logical phases and Majorana bound states (Alicea, 2012;Beenakker, 2013; Hasan and Kane, 2010; Qi and Zhang,2011). Understanding and controlling the behavior andrelaxation of nonequilibrium quasiparticles in these sys-tems is also of importance, not least because of their in-fluence on the prospects of solid-state topological quan-tum computation (Nayak et al. , 2008).This review focuses exclusively on the nonequilibriumproperties of superconductors in proximity to magnets.0We expect the inclusion of the magnetization dynam-ics and its coupling to the electronic degrees of free-dom via the reciprocal effects of spin transfer torqueand spin pumping (Tserkovnyak et al. , 2005) in the far-from equilibrium regime to lead to completely new typeof physics, as the two types of order parameters affecteach other. The coupling of supercurrent on magnetiza-tion dynamics and texture has been studied during thepast decade (Houzet, 2008; Richard et al. , 2012; Waintaland Brouwer, 2002), but the work where both systemsare out of equilibrium has been mainly concentrated onJosephson junctions (Hikino et al. , 2011; Holmqvist et al. ,2014, 2011; Kulagina and Linder, 2014; Mai et al. , 2011)and much less attention has been paid to quasiparticleeffects (Linder et al. , 2012; Skadsem et al. , 2011; Trif andTserkovnyak, 2013).Besides the rich physics offered by spin-split supercon-ductors, they have been long used as tools to charac-terize equilibrium properties of magnets, especially theirspin polarization. In this review (see end of Sec. V.B)we outline two further possibilities related to their largethermoelectric response: accurate radiation sensing andnon-invasive scanning thermometry. We believe there arealso many other avenues to be uncovered, opened by thepossibility for realizing a controlled combination of mag-netism and superconductivity. ACKNOWLEDGMENTS
We thank Faluke Aikebaier, Marco Aprili, DetlefBeckmann, Wolfgang Belzig, Irina Bobkova, Alexan-der Bobkov, Matthias Eschrig, Yuri Galperin, FrancescoGiazotto, Vitaly Golovach, Kalle Kansanen, AlexanderMel’nikov, Jagadeesh Moodera, Risto Ojaj¨arvi, AsierOzaeta, Charis Quay, Jason Robinson, Mikel Rouco,and Elia Strambini for useful discussions. This workwas supported by the Academy of Finland Centerof Excellence (Project No. 284594), Research Fel-low (Project No. 297439) and Key Funding (ProjectNo. 305256) programs, the European Research Council(Grant No. 240362-Heattronics), the Spanish Ministe-rio de Econom´ıa y Competitividad (MINECO) (ProjectsNo. FIS2014-55987-P and FIS2017-82804-P), the Euro-pean Research Council under the European Union’s Sev-enth Framework Program (FP7/2007- 2013)/ERC Grantagreement No. 615187-COMANCHE.
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