Spin-Orbit-Induced Spin, Charge, and Energy Transport in Diffusive Superconductors
Camilla Espedal, Peter Lange, Severin Sadjina, A. G. Mal'shukov, Arne Brataas
SSpin Hall Effect and Spin Swapping in Diffusive Superconductors
Camilla Espedal , Peter Lange , Severin Sadjina, A. G. Mal’shukov , and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Institute of Spectroscopy, Russian Academy of Sciences, 142190, Troitsk, Moscow oblast, Russia
We consider the spin-orbit-induced spin Hall effect and spin swapping in diffusive superconduc-tors. By employing the non-equilibrium Keldysh Green’s function technique in the quasiclassicalapproximation, we derive coupled transport equations for the spectral spin and particle distribu-tions and for the energy density in the elastic scattering regime. We compute four contributions tothe spin Hall conductivity, namely, skew scattering, side-jump, anomalous velocity, and the Yafetcontribution. The reduced density of states in the superconductor causes a renormalization of thespin Hall angle. We demonstrate that all four of these contributions to the spin Hall conductivityare renormalized in the same way in the superconducting state. In its simplest manifestation, spinswapping transforms a primary spin current into a secondary spin current with swapped currentand polarization directions. We find that the spin-swapping coefficient is not explicitly but onlyimplicitly affected by superconducting correlations through the renormalized diffusion coefficients.We discuss experimental consequences for measurements of the (inverse) spin Hall effect and spinswapping in four-terminal geometries. In our geometry, below the superconducting transition tem-perature, the spin-swapping signal is increased an order of magnitude while changes in the (inverse)spin Hall signal are moderate.
I. INTRODUCTION
The coupling between a quasiparticle’s spin and its mo-mentum causes an initially unpolarized current in con-ductors to become spin dependent. The resulting spin-orbit-induced effects can be intrinsic or extrinsic. Intrin-sic effects are due to the manifestation of spin-orbit cou-pling in the quasiparticle band structure in combinationwith spin-conserving scattering events. Extrinsic effectsare due to spin-orbit scattering off impurities. We focuson extrinsic effects that give rise to spin relaxation, spinswapping, spin Hall and inverse spin Hall effects.The simplest manifestation of the spin-orbit interac-tion is spin relaxation . This causes a nonequilibriumspin polarization to decay with time or an injected spincurrent to decay with distance. Below the superconduct-ing transition temperature, measurements of the temper-ature dependence of the spin relaxation length can beused to determine the ratio between spin-orbit-inducedand magnetic-impurity-induced spin relaxation. Our fo-cus is on how the spin Hall effect and the spin swappingare affected by superconducting correlations.The correlation between the momentum and spin di-rections in the impurity scattering process can cause aninjected primary spin current to transform into a sec-ondary transverse spin current, even in the absence ofelectric (charge) currents. This effect is called spin swap-ping. In its simplest manifestation, the secondary currentflows along the polarization of the injected current andwith a polarization direction that is along the primarycurrent flow - the spin currents have been ‘swapped’.This effect was first studied theoretically for extrinsicspin-orbit coupling. More recently, an intrinsic (Rash-ba spin-orbit-induced) spin swapping effect was identi-fied in two-dimensional diffusive metals. Spin swappingdriven by electric fields in these systems has also beenconsidered . We will determine how the spin swapping differs in the superconducting state compared to the nor-mal state.The spin Hall effect has attracted considerableattention . There are two main contributions to theextrinsic spin Hall effect: skew scattering due to the spin-dependent quasiparticle scattering cross-section and theside-jump mechanism that arises from a spin-dependentdisplacement during the scattering events. Calculatingthe side-jump contribution to the spin Hall effect is asubtle issue because several terms contribute to this con-tribution. In the stationary regime and in the absence ofa magnetic field, we study three contributions in detail.The onset of superconductivity can renormalize thevarious spin transport effects and introduce new phenom-ena. The temperature dependence of the spin transportparameters below the critical temperature of the super-conductor can be used to identify and quantify the com-peting spin-orbit-induced effects . A giant enhancementof the spin signal of up to five orders of magnitude in thesuperconducting state was reported experimentally in anonlocal measurement setup. In niobium, there are mea-surements of a factor of four enhancement of the spinrelaxation time in the superconducting state comparedto the normal state .In the inelastic transport regime, a giant increase in thenonlocal spin and charge accumulation signal due to thespin Hall effect was computed at low temperatures.Moreover, these studies indicated that the magnitudes ofthe skew scattering and the side-jump contributions arerenormalized by different amounts below the supercon-ducting critical temperature in spin Hall devices. Recentnon-local measurements found an inverse spin Hall signalthat is 2000 times stronger in the superconducting statecompared to the normal state .Quasiparticle transport is elastic when the quasipar-ticle energy is conserved during the scattering events.In the opposite regime, quasiparticle interactions cause a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t transport to be inelastic, and the nonequilibrium distri-bution of the quasiparticles approaches equilibrium Fermidistributions that may be position, spin, and energy de-pendent. Spin transport in normal metals typically doesnot differ in the inelastic and elastic transport regimessince the temperature is considerably smaller than therelevant energy scale, that is, the Fermi energy. How-ever, in superconductors, the typical temperatures areon a considerably smaller energy scale, namely that ofthe superconducting gap, and (spin) transport in theelastic and inelastic transport regimes can significantlydiffer . Since inelastic scattering rates increase withtemperature, it is plausible that transport below the su-perconducting critical temperature is elastic .In this paper, we study the elastic transport of spin,particle, and energy in a diffusive superconductor. Forthis purpose, we use Keldysh nonequilibrium Green’sfunctions. We include scattering from impurities, takingthe spin-orbit coupling into account. We also comple-ment our results with known effects of magnetic impurityscattering. We compute the renormalization of the spinHall effect and spin swapping effect below the supercon-ducting critical temperature. In contrast to recent theo-retical works on inelastic scattering effects on trans-port in superconductors, we find the same renormaliza-tions of all spin Hall contributions in the elastic transportregime. Moreover, we extend these studies to arbitraryspin polarizations and provide a rigorous discussion onthe various contributions to the side-jump mechanism,including the anomalous velocity, the Yafet term, and anadditional expression in the self-energy. Thus far, therehave been no studies on the spin-swapping effect in su-perconductors. We demonstrate that the spin-swappingcoefficient is only implicitly renormalized by supercon-ducting correlations via the renormalized diffusion coef-ficients.We apply our transport formalism to study the (in-verse) spin Hall and spin-swapping effects in a four-terminal geometry. In this geometry, we demonstratethat the signal resulting from the spin swapping can be-come an order of magnitude larger in the superconduct-ing state compared to the normal state. On the otherhand, change in the signal resulting from the (inverse)spin Hall effect are only moderate.The remainder of this paper is organized as follows.In Sec. II, we first present the microscopic Hamiltonianand the resulting transport equations for spin, particle,and energy transport, including scattering from magneticand nonmagnetic impurities and from spin-orbit cou-pling. Sec. III presents the four-terminal geometry andthe calculation of the signals that result from the spinHall effect and the spin swapping mechanism. Subse-quently, Sec. IV provides an overview of the microscopicderivation of our results and a discussion of the side-jumpmechanism and its contributions to the spin Hall effect.Finally, we present our conclusions in Sec. V. The appen-dices contain more details of our calculations. II. TRANSPORT EQUATIONS
Let us first describe the microscopic model of the su-perconductor and then our primary results. The main re-sults are the relation between the currents and the quasi-particle distributions and the diffusion equations.We describe the system using a four-component basisvector in spin ⊗ particle-hole space. We use a ’hat’ tolabel vectors and matrices in this 4 × ψ † = ( ψ †↑ , ψ †↓ , ψ ↑ , ψ ↓ ) , (1)where ψ σ is the field annihilation operator for spin σ .The field operators are described by the BCS one-particle Hamiltonianˆ H ( ) = − (cid:126) m D i ( ) D i ( )+ eφ ( ) − µ + ˆ∆( )+ ˆ U tot , (2)where φ is the scalar potential, µ is the chemical po-tential, ˆ∆( ) describes superconducting correlations,and ˆ U tot describes both spin-conserving and spin-orbit-induced impurity scattering. We use an abbreviated no-tation for the coordinates that includes both spatial andtemporal coordinates, where = ( r , t ).The kinetic energy in Eq. (2) is expressed in terms ofthe co-variant derivative, D µ ( ) ˆ f ( ) = ∂ ˆ f ( ) ∂ µ + iˆ τ A µ ( ) ˆ f ( ) , (3)where ˆ τ = diag(1 , , − , −
1) is a generalization of thethird Pauli matrix and A µ ≡ ( φ/c, − A ) e/ (cid:126) contains thescalar potential φ and the electromagnetic vector poten-tial A . We also introduce its conjugate operatorˆ f ( ) D † µ ( ) = ∂ ˆ f ( ) ∂ µ − i ˆ f ( )ˆ τ A µ ( ) . (4)We use the standard four-vector notation, where Greekletters refers to both spatial and temporal coordinates,whereas Latin letters only take values referring to thespatial coordinates, i.e. µ = 0 , , , i = 1 , , ) = )0 0 − ∆( ) 00 ∆ ∗ ( ) 0 0 − ∆ ∗ ( ) 0 0 0 (5)which contains the s-wave superconducting scalar orderparameter ∆( ) = λ ( r ) h ψ ↓ ( ) ψ ↑ ( ) i , where λ is theinteraction strength and h . . . i denotes a quantum statis-tical average.The local potentialˆ U tot ( r ) = ˆ U ( r ) + ˆ U so ( r ) (6)includes elastic impurity scattering and spin-orbit cou-pling. We express elastic impurity scattering asˆ U ( r ) = X i u ( r − r i ) , (7)where u ( r − r i ) is the i th elastic scattering potential atposition r i . We consider extrinsic spin-orbit scatteringgoverned by the impurities. The extrinsic spin-orbit cou-pling is described byˆ U so ( r ) = X i ˆ u so ( r − r j ) (8)= i γ X j (cid:0) ˆ τ ˆ α × ∇ u ( r − r j ) (cid:1) i D i ( r ) , where ˆ α = diag( ¯ σ , ¯ σ ∗ ) is a generalized vector of 4 × × σ . γ is thespin-orbit interaction strength .The spin-orbit coupling at impurities can be under-stood as a consequence of the effects of intrinsic spin-orbitcoupling in the band structure. The latter renormalizesthe interaction strength γ from its vacuum value. Therenormalization can be interpreted as a shift in the phys-ical position operator, r → ˆ r eff = r + ˆ r so , (9)such that, to the first order in γ ,ˆ U (ˆ r eff ) = ˆ U ( r ) + ˆ U so ( r ) , where the spin- and velocity-dependent correction to theposition operator ˆ r so = − γ (ˆ τ ˆ α × p ) (10)is known as the Yafet term or the anomalous coordi-nate, where p is the momentum. The Yafet term in Eq.(10) also contributes to the spin Hall effect, as we willdiscuss in more detail below.With this in hand, we find that the equation of motionfor the four-component basis vector ˆ ψ is h i (cid:126) ˆ τ D ( ) − ˆ H ( ) i ˆ ψ ( ) , (11)and for its conjugate, ˆ ψ † ˆ ψ † ( ) h − iˆ τ D † ( ) − ˆ H ( ) i = 0 , (12)where the ’prime’ means that the covariant derivativesshould be replaced by its conjugated form. In otherwords, ˆ H is the same as ˆ H , except that we let D µ → D † µ .Starting from these equations of motion, we use theKeldysh Green’s function formalism to obtain expres-sions that describe the quasiparticle transport of spin,particle, and energy. We employ the quasiclassical ap-proximation, which is valid for length scales that areconsiderably larger than the Fermi wavelength, and thenthe diffusion approximation, which is applicable when thesystem is far greater than the mean free path. We nowpresent and discuss the main results. A rigorous deriva-tion for interested readers is included in Sec. IV. A. Current Expressions
In the elastic transport regime, energy is conservedand transport can be described at each energy (cid:15) relativeto the chemical potential in terms of spectral ( energy-dependent ) currents and distributions. Our main resultconsists of two parts: i) the relations between the quasi-particle spectral currents and the spectral distributionsand ii) the spectral diffusion equations. We discuss thespectral currents in this section and the spectral diffusionequations in the next section.The spectral currents are the spectral particle current j i ( (cid:15) ), the spectral spin current j ij ( (cid:15) ), the spectral en-ergy current j (cid:15)i ( (cid:15) ), and the spectral spin-energy current j (cid:15)ij ( (cid:15) ). Spectral currents with one subindex are particle( j i ) or energy ( j (cid:15)i ) and flow along the i direction. Quanti-ties with two subindices describe spin ( j ij ) or spin-energy( j (cid:15)ij ) current, where the first index ( i ) is the direction ofthe current flow and the second ( j ) denotes the spin po-larization direction.The corresponding spectral particle and energy distri-bution functions are h ( (cid:15) ) and h (cid:15) ( (cid:15) ), respectively. Sim-ilarly, the spectral spin and spin-energy distributionfunctions are h sj ( (cid:15) ) and h (cid:15)sj ( (cid:15) ), respectively, where thesubindex (here, j ) denotes the spin polarization direc-tion.From the spectral densities and spectral currents intro-duced in this section, the relevant physical quantities canbe extracted. For example, the electric current density isa sum of all the spectral particle currents, j tot i ( R ) = − eN Z d (cid:15) j i ( R , (cid:15) ) , (13a)and the electroneutrality dictates that the electrostaticpotential is eφ ( R ) = − Z d (cid:15) N S ( R , (cid:15) ) h ( R , (cid:15) ) , (13b)where N s is the renormalization of the density of statesdue to superconducting correlations, which are intro-duced and discussed further below. The expressions forthe spin, energy, and spin-energy properties are similar.To the first order in the spin-orbit coupling, we com-pute that there are three contributions to the spectralcurrent, namely, the conventional diffusion and super-current terms j (0) , the spin Hall effects j (sH) , and thespin-swapping effect j (sw) : j ( (cid:15) ) = j (0) ( (cid:15) ) + j (sH) ( (cid:15) ) + j (sw) ( (cid:15) ) . (14)We will now discuss these contributions to the spectralcurrent. To the zeroth order in the spin-orbit interactionstrength, the currents are well known (see, e.g. Ref. 35,): j (0) i = − D p ∇ i h + j sci h (cid:15) , (15a) j (0) ij = − D (cid:15) ∇ i h sj + j sci h (cid:15)sj , (15b) j (cid:15) (0) i = − (cid:2) D (cid:15) ∇ i h (cid:15) + j sci (1 − h ) (cid:3) , (15c) j (cid:15) (0) ij = − (cid:2) D p ∇ i h (cid:15)sj + j sci h sj (cid:3) . (15d)The diffusion coefficients D (cid:15) and D p are well known, andtheir energy dependencies are governed by the super-conducting correlations and the nonequilibrium state.One simple limit of these expressions is the normal state,where D (cid:15) = D p = D , where D is the diffusion con-stant. Another simple limit is the BCS approximation ofa dirty superconductor with no pair-breaking processes: D p /D = (cid:15) / ( (cid:15) − ∆ ) and D (cid:15) /D = 1 for energies abovethe gap, | (cid:15) | > | ∆ | . The current in Eq. (15) also includes asupercurrent j sc , which is proportional to the gradient ofthe superconducting phase. Microscopic expressions forthe generalized diffusion constants D p ( (cid:15) ) and D (cid:15) ( (cid:15) ) aswell as the supercurrent j sc in general out-of-equilibriumconditions are given in Sec. IV E.Let us now turn to the spin-orbit-induced correctionsto the conventional spectral current, which is one of ournew central results. To the first order in the spin-orbitcoupling strength γ , we compute contributions that cor-respond to the spin Hall and the inverse spin Hall effectsand to the spin-swapping effect. We find that the contri-butions to the spectral current due to the spin Hall andthe inverse spin Hall effects are j (sH) i = − χ sH ε ijk D ∇ j ( N S h sk ) , (16a) j (sH) ij = χ sH ε ijk D ∇ k ( N S ( h − , (16b) j (cid:15) (sH) i = − χ sH ε ijk D ∇ j ( N s h (cid:15)sk ) , (16c) j (cid:15) (sH) ij = χ sH ε ijk D ∇ k ( N S h (cid:15) ) , (16d)where N S ( (cid:15) ) is the ratio between the (energy-dependent)density of states in the superconducting state and thedensity of states in the normal state. The normal statespin Hall angle χ sH = χ (sk)sH + χ (sj)sH is given in terms ofthe skew scattering constant, χ (sk)sH = 4 η τ tr τ sk , (17a)and the side-jump constant, χ (sj)sH = 3 γmτ tr . (17b)The dimensionless quantity η = γp / p F is the Fermi momentum, τ tr is the transport relaxation time, and τ sk is the skewscattering time.We find that the spin Hall angles that arise from skewscattering and side jump are all renormalized by equalamounts below the superconducting critical temperature via the renormalized density of states parametrized by N S ( (cid:15) ). In contrast, Ref. 28 computes the spin Hall con-ductivity in a different transport regime, the inelastictransport regime, and predicts that the renormalizationof the spin Hall angle due to side jump and skew scatter-ing differs.We note that both the spin Hall and inverse spin Halleffects described by Eq. (16) are created by quasiparti-cles, while contributions from the condensate are absent.The origin of this is that the inverse spin Hall effect is in-duced by a nonequilibrium spin accumulation governedby the distribution function, whereas the phase of thecondensate wave-function remains intact. This is in con-trast to the equilibrium magnetoelectric effect producedby a static Zeeman field in a spin-orbit-coupled super-conductor discussed in Ref. 26. In that case, the conden-sate current emerges due to mixing of spin-singlet andspin-triplet Cooper pairs. Such a situation could occurout of equilibrium by taking into account that an effec-tive Zeeman field may be created by spin-polarized elec-trons due to the Coulomb exchange interaction of itin-erant electrons . Here, we assume that the Coulombinteraction is weak and disregard this effect.We also disregard the condensate supercurrent associ-ated with the conversion of quasiparticle current into thesupercurrent due to the inelastic relaxation of quasiparti-cles (the so-called charge imbalance relaxation). At lowtemperatures, such a relaxation occurs at large lengthscales. We assume that our system is small enough todisregard charge imbalance relaxation.When the scattering potential is isotropic, the trans-port relaxation time τ tr equals the elastic scatteringtime τ , χ (sk)sH = 4 πηN u /
3, and χ (sj)sH = 3 γm/τ , where u = u ( q = 0) is the Fourier transformed scatteringpotential at q = 0 and N is the density of states atthe Fermi level in the normal state. Our results in Eq.(16) are valid for general anisotropic scattering poten-tials, except that the skew scattering contribution (17a)is computed to the lowest order in small anisotropies; seeSec. IV. Several factors contribute to the side jump (17b),and we discuss these factors in more detail in Sec. IV andAppendix F.Let us study the superconductivity-induced renormal-ization of the spin Hall angle in the elastic transportregime in more detail. For this purpose, we consider aweakly perturbed superconductor in which the density ofstates is constant and equal to the BCS density of states.We can then express the spin Hall contribution to thespectral particle current, the inverse spectral spin Hallcurrent, from the zeroth-order spin current, j (sH) i = θ sH ε ijk j (0) jk , (18)where we have defined the spin Hall angle in the su-perconducting state as θ sH = χ sH N s ( D/D (cid:15) ). Whereas( D (cid:15) /D ) typically only weakly depends on the energy, N s strongly varies as a function of energy. In the super-conducting state, N S is greatly enhanced close to thesuperconducting gap, which causes a significant increaseof the spin Hall angle θ sH . The fact that there is a gi-ant enhancement in the spin Hall angle for quasiparticleswith energies around the gap is consistent with the mainfindings of Refs. 28 and 29. We provide microscopicexpressions for the density of states in the superconduc-tor with respect to its normal state value N s ( (cid:15) ) and thescattering times in Eq. (17) in Sec. IV.The spin-swapping effect couples only spins. To dis-play the spin swapping current in a compact manner, wedefine the operator [ a , b ] ( sw ) ij ≡ δ ij a c b c − a j b i , and weobtain the spectral currents j (sw) ij = χ sw D (cid:15) [ ∇ , h s ] ( sw ) ij + χ sw ∇ D (cid:15) ) , h s ] ( sw ) ij (19a)+ χ sH N s j sc, + R p , h (cid:15)s ] ( sw ) ij ,j (cid:15) (sw) ij = χ sw D p [ ∇ , h (cid:15)s ] ( sw ) ij + χ sw ∇ D p ) , h (cid:15)s ] ( sw ) ij (19b)+ χ sH N s j sc, + R p , h (cid:15)s ] ( sw ) ij . The normal state spin-swapping constant is χ sw = 4 η τ tr τ sw , (20)where τ sw is the spin-swapping scattering time. The spin-swapping constant reduces to χ sw = 4 η/ χ sw remains unchanged by superconducting corre-lations.The additional terms in Eq. (19) appear when there arespatial variations in the magnitude and phase of the su-perconducting order parameter. The term proportionalto j sc, can be viewed as super-spin-swapping current. Inaddition to this term, we have a more complicated termthat is related to the gradient of θ , which is related togradients in the spectral properties of the superconduc-tor.The expressions for the spectral currents, Eqs. (16)and (19), satisfy Onsager’s reciprocal relations. For ex-ample, the spin Hall effect and the inverse spin Hall effectare governed by the same susceptibility χ sH .The spin Hall effect and the spin swapping mechanismscan be detected in nonlocal geometries. In these setups,the detected signals will also depend on the counterflowof currents due to spin and particle distribution build-ups. We will subsequently compute these effects and theresulting effect of the superconducting correlations on theelectrochemical potentials that can be detected. B. Diffusion Equations
We now turn to the presentation of the spectral( energy-dependent ) diffusion equation. We find that thediffusion of particle, spin, and energy is described interms of energy-dependent diffusion equations: ∇ i j i = αh + α (cid:15) h (cid:15) , (21a) ∇ i j ij = (cid:16) α so τ so + α m τ m (cid:17) h j , (21b) ∇ i j (cid:15)i = 0 , (21c) ∇ i j (cid:15)ij = α (cid:15) h sj + αh (cid:15)sj + (cid:16) α (cid:15) so τ so + α (cid:15) m τ m (cid:17) h (cid:15)j , (21d)The terms proportional to α and α (cid:15) in Eqs. (21a)and (21d) are proportional to the superconducting gapand are responsible for converting quasiparticle currentsinto supercurrents. For completeness, we have also in-cluded the effects of the magnetic impurities, where weuse the results from Ref. 35. The spin relaxation termsin Eqs. (21b) and (21d) are given in terms of the spinrelaxation scattering times τ so and τ m due to spin-orbitcoupling and magnetic impurities, respectively. Eq. (21c)expresses that spectral energy is conserved in the elastictransport regime.Superconducting correlations lead to the introductionof the renormalization factors α , α so , α m , α (cid:15) so , and α (cid:15) m35 . These factors are energy dependent and are gov-erned by the superconducting state. Microscopic expres-sions for these renormalization factors (and the scatter-ing times) are presented in Sec. IV. To obtain insightsinto how the various effects occurring in Eq. (21) arerenormalized, let us consider a scenario in which the su-perconductor has properties that are close to that of abulk BCS superconductor (BCS limit). This is, for in-stance, realized in large superconductors that are weaklycoupled to reservoirs that inject spin and particle cur-rents. Quasiparticles can propagate for energies abovethe gap, | (cid:15) | > | ∆ | , when there is no conversion of quasi-particle currents to supercurrents and α = 0. At thesame time, the spin relaxation renormalization factorsbecome α (cid:15) so = α (cid:15) m = (cid:15) / ( (cid:15) − ∆ ) for spin energy, and α so = 1 and α m = ( (cid:15) + ∆ ) / ( (cid:15) − ∆ ) for spin density.This implies that the spin-orbit-induced spin relaxationrate is identical in the superconducting and normal stateswhereas the spin-energy relaxation rate is enhanced inthe superconducting state. In contrast, when magneticimpurities dominate the relaxations of spins, both thespin relaxation rate and the spin-energy relaxation ratesare enhanced for quasiparticles with energies above andclose to the superconducting gap. III. SPIN TRANSPORT IN NON-LOCALGEOMETRIES
We will now compute the signatures of the (inverse)spin Hall effect and the spin swapping in non-local geome-tries. We consider the setups in Fig. 1. The left normalmetal (N L ) functions as a spin injector into the supercon-ductor (S) via a tunnel junction. The additional normalmetals to the right (N R and N R ) act as detectors of thespin-particle-coupled properties of the superconductor.We assume that the transparency of the tunnel con-tacts is low such that there are no proximity effects be-tween the normal metals and the superconductor. Theequilibrium properties of the superconductor are then thesame as if it were detached from the rest of the circuit.Furthermore, we assume that the resistances of the tun-nel contacts used for detecting the inverse spin Hall andspin-swapping effects in the superconductors are consid-erably larger than the resistance of the tunnel contactused for spin injection. In this limit, we can first com-pute non-equilibrium the spin distribution in the normalmetal, which is not influenced by the rest of the circuit.In turn, this spin distribution leaks into the supercon-ductor. Finally, the electrochemical potential differencebetween the normal metals to the right (N R and N R )detects the inverse spin Hall effect and spin swappingwithout influencing the spin and particle distributions inthe superconductor. Our geometry differs from the setupin since the spin current into the superconductor is in-jected along its long axis.We first focus on the spin injection that originates fromthe left normal metal. Since the tunnel resistances arelarge, we can consider the properties of the left normalmetal independently of the rest of the circuit. The nor-mal metal is biased so in a way to in a way to main-tain the particle distribution in the normal metal closeto the tunnel contact in equilibrium. We consider thatthis is achieved with a electrochemical potential − µ L / µ L / I = Gµ L /e , wherethe conductance is G = e N L D L A L /L L in terms of thedensity of states N L , diffusion coefficient D L , cross sec-tion A L , and length L L of the left normal metal. Theparticle current flowing along the y -direction generatesvia the spin Hall effect a spin current flowing along the x -direction that is spin polarized along the z -direction.In turn, the spin current induces a spin distribution atthe edges of the left (L) normal metal. The standard cal-culation explained below shows that the spin distributionin the left normal metal close to the tunnel interface is h s ( L ) z ( (cid:15) ) = ζ L h eq ( (cid:15), µ L / . (22)where the dimensionless particle-spin conversion effi-ciency ζ L is independent of the energy and h eq ( (cid:15), µ L /
2) = 12 (cid:20) tanh µ L / (cid:15) k B T + tanh µ L / − (cid:15) k B T (cid:21) (23)arises from the distributions of the quasi-particles in thereservoirs at electrochemical potentials µ L / − µ L / ζ L as follows. To the zeroth order in the spin- µ L j (0) xz j sHy µ R xyzN L N R N R (a) µ L µ R j (0) xz j ( sw ) zx N L (b) FIG. 1. Non-local geometries for measuring the spin Halleffect (a) and the spin swapping (b). In both cases, a par-ticle current flowing in the left normal metal generates viathe spin Hall effect a spin current that flows into the super-conductor. Inside the superconductor, the inverse spin Halleffect converts the spin current into a particle current, andspin swapping swaps the spin current polarization and flowdirections. In (a), the electrochemical potentials in the nor-mal metals measure the inverse spin Hall effect. In (b), thespin-polarized contacts can be switched between a paralleland anti-parallel configuration to detect the spin-swappingeffect. orbit coupling, we use Eq. (15a) to find the relation be-tween the spatial variation of the spectral particle dis-tribution and the spectral particle current, j (0) y ( (cid:15) ) = − D∂ y h ( (cid:15) ) and solve the diffusion equation of Eq. (21a)to find the spatially varying spectral particle distribu-tion h ( (cid:15) ). To the first order in the spin-orbit coupling,the spatial variation of the spectral particle distributiongives rise to a spin current j xz ( (cid:15) ) and an associated spindistribution h sz ( (cid:15) ) in the normal metal. The spatial vari-ation of the spin distribution h sz ( (cid:15) ) is determined by thediffusion equation (21b) with the boundary conditionsthat the spectral spin current vanishes at the edges ofthe normal metal. The spectral spin current is from Eq.(16b) j xz ( (cid:15) ) = − D L ∂ x h sz ( (cid:15) ) − χ sH ,L D∂ y h ( (cid:15) ). Solving thediffusion equation (21b) with these boundary conditionsand assuming the normal metal is wider than its dif-fusion length, we find that the spin distribution of Eq.(22) with with the particle-spin efficiency parameter ζ L = − χ sH ,L λ L /L L , which is a dimensionless property of theleft normal metal spin injector where the spin-diffusionlength is λ L .Next, we will compute the resulting spin and particledistribution in the superconductor. Since we will focus onspin-distribution-induced spin-particle conversion effectsin the supercurrent, we only need to focus on how spinspropagate from the left normal metal into the supercon-ductor. At a low transmission interface, the spectral spincurrents through the interfaces are j xz = N S ( (cid:15) ) g T ( h s ( N ) z − h sz ) , (24)where the spin distribution at the normal metal side, h s ( N ) z , was computed in Eq. (22). The conductance of thetunnel junction when the superconductor is in the normalstate is G T = N g T . We solve the spin-diffusion equation(21b) in the superconductor with the boundary conditionof continuity of the spin current. We also expand the re-sult to the lowest order in the tunnel conductance andassume that the superconductor is considerably longerthan the spin-diffusion length (along the x -direction).We then find that the spatially dependent spin distri-bution is governed by the ratio between the tunnel con-ductance N S ( (cid:15) ) G T and the conductance of the supercon-ductor within the spin-flip length l ( (cid:15) ), N D (cid:15) ( (cid:15) ) /λ ( (cid:15) ): h sz ( x, (cid:15) ) = ζ L h eq ( (cid:15), µ L / g T N s ( (cid:15) ) λ ( (cid:15) ) D (cid:15) ( (cid:15) ) e − x/λ ( (cid:15) ) . (25)The energy-dependent spin-flip length is λ ( (cid:15) ) =[ D (cid:15) ( (cid:15) ) τ sf ( (cid:15) )] / , where the spin-flip relaxation rate is1 /τ sf = α so /τ so + α m /τ m .In the following, we will show how the spatially depen-dent spin distribution of Eq. (25) in the superconductorgives rise to the inverse spin Hall effect and spin swap-ping. A. Inverse Spin Hall Effect
In the inverse spin Hall geometry, the inverse spin Halleffect is measured via normal metals in tunnel contactwith the superconductor. From Eq. (16a), we see thatthe spin-Hall-induced spectral particle current density is j sH y ( (cid:15) ) = χ sH ,S DN S ( (cid:15) ) ∂ x h sz ( x, (cid:15) ) , (26)where we computed h sz ( x, (cid:15) ) in Eq. (25).We assume that the transverse width W S of the su-perconductor is smaller than the charge-imbalance re-laxation length. The spin-Hall-induced spectral particlecurrent density of Eq. (26) must then be compensated bythe zeroth-order spectral particle current density inducedby a transverse gradient of the spectral particle distribu-tion. Since transport is assumed to be elastic, we usethe boundary conditions that the total (zeroth-order andspin Hall contributions) spectral particle current densityshould vanish at the lateral edges. We also only take intoaccount the difference between the particle distributionsat y = W S / y = − W S / y = W S / h ( (cid:15) ) = η psp h eq ( (cid:15), µ L / D N s ( (cid:15) )2 D p ( (cid:15) ) D (cid:15) ( (cid:15) ) e − x/λ ( (cid:15) ) , (27)where the dimensionless particle-spin-particle conversionis governed by η psp = − ζ L χ sH ,S W S g T /D .The particle distribution of Eq. (27) can be detectedas a electrochemical potential in another normal metalin tunnel contact with the superconductor. The spec-tral particle current between the superconductor and thistunnel contact is j y = N S ( (cid:15) ) g T [ h − h eq ( (cid:15), µ R / , (28)The electrochemical potential µ R in the normal metalthat we detect is determined by the integral equationthat the total current into the right normal metal shouldvanish, R d(cid:15)j y ( (cid:15) ) = 0, and is therefore indepedent of thedetector tunnel conductance g T .In linear response, we expand h eq ( (cid:15), µ ) ≈ − [ ∂ (cid:15) f ( (cid:15) ) − ∂ (cid:15) f ( − (cid:15) )] µ , (29)where f ( (cid:15) ) is the Fermi-Dirac distribution function. Wethen find that the ratio between the electrochemical po-tentials in the superconducting state and the normalmetal is µ ( S ) µ ( N ) = R ∞ ∆ d(cid:15) e − x/λ ( (cid:15) ) ∂ (cid:15) f ( (cid:15) ) N S ( (cid:15) ) [ D/D (cid:15) ( (cid:15) )] e − x/λ N R ∞ ∆ d(cid:15)∂ (cid:15) f ( (cid:15) ) N S ( (cid:15) )[ D p ( (cid:15) ) D (cid:15) ( (cid:15) ) /D ] , (30)where λ N is the spin-diffusion length in the normal state.The electrochemical potential when the superconductoris in its normal state is µ ( N ) = µ R = η psp e − x/λ N µ L .Naturally, the electrochemical potential is proportionalto the spin Hall angle in the left normal metal and thespin Hall angle in the superconductor via the particle-spin-particle conversion coefficient η psp .We consider first the case when spin-flip is predomi-nately due to spin-orbit scattering, 1 /τ so (cid:29) /τ m . Re-markably, there is then an exact compensation of the fac-tors in the numerator and denominator of Eq. (30) so that V ( S ) = V ( N ) . This is because N S ( (cid:15) ) = D p ( (cid:15) ) D (cid:15) ( (cid:15) ) /D and λ ( (cid:15) ) = λ N in that limit. This ensures that the par-ticle imbalance of Eq. (27) attains its normal state valueeven when superconducting correlations are taken intoaccount.When spin-flip scattering due to magnetic impuritiesbecome stronger, there is a decay of the spin-Hall signalwhen the temperature is reduced below the supercon-ducting transition temperature. This is caused by thereduction of the the spin-diffusion length λ ( (cid:15) ) with re-spect to its normal state value λ N in this regime.We conclude that the inverse spin Hall signal is equalto or smaller than its value in the normal state below thesuperconducting transition temperature. B. Spin Swapping
To study spin swapping, we consider the geometryshown in Fig. 1b). In this scenario, spin swapping impliesthat the spin current flowing in the superconductor alongthe x -direction that is polarized along the z -direction willbe swapped into a secondary (and smaller) spin currentthat flows along the z -direction and is polarized alongthe x -direction. From Eq. (19a), we find that the spin-swapped-induced spectral spin current density is j ( sw ) zx ( (cid:15) ) = − χ sw ,S D (cid:15) ∂ x h sz ( x, (cid:15) ) . (31)Maintaining that the transverse width of the supercon-ductors is smaller than the charge imbalance length, theswapped spectral spin current of Eq. (31) must be coun-terbalanced by a spin current induced by a transversegradient of the spin distribution. Requiring a vanish-ing spectral spin-current at the edges ( z = − d S / z = d S /
2) then determined the transverse secondaryswapped spin distribution h ( sw ) x ( x, y, (cid:15) ) = η pss λ ( (cid:15) ) d S N S ( (cid:15) ) DD (cid:15) ( (cid:15) ) e − x/λ ( (cid:15) ) , , (32)where η pss = ζ L g T d S χ sw ,S /D is a dimensionslessparticle-spin-spin conversion factor.We can already here note that the swapped spin dis-tribution of Eq. (32) becomes larger for energies aroundthe gap than in the normal state. As we will demonstratebelow, this also leads to an enhanced spin swapping sig-nal. The detection of the spin swapping signal of Eq. (32)requires the use of spin-polarized contacts. Hence, we as-sume a setup such as the one shown in Fig. 1, where theright tunnel contacts consist of ferromagnets with a spinpolarization along the x -direction. We also assume thatthe magnetization of the tunnel contact can be made tobe parallel or anti-parallel. Furthermore, to detect thespatial variation of the swapped spin distribution alongthe z -direction, we consider a situation in which the tun-nel contacts are attached on top of the superconductor.In such an experiment, we can detect the swapped spincurrent.We detect the electrochemical potential in large probereservoirs where there is no spin distribution. The parti-cle distributions in the detectors attain their local equilib-rium values h eq ( (cid:15), µ R ) with respect to the detector elec-trochemical potential µ R . The spectral particle currentthrough the detector spin-polarized tunnel barrier is j ± z ( (cid:15) ) = N S ( (cid:15) ) g T d h ± P T d (cid:16) − h ( sw ) x (cid:17) + (cid:16) h eq ( (cid:15), µ ± ) − h ( S ) (cid:17)i , (33)where the sign ± indicates whether the tunnel polarizeris parallel or anti-parallel to the x -direction. h ( S ) is theparticle distribution in the superconductor that will notplay a role in our spin-swapping detection scheme. Re-quiring no total current to the reservoir, such that also R d(cid:15) ( j + z − j − z ) = 0, we find an expression for the elec-trochemical potential difference ∆ µ = µ + − µ − in linearresponse by using the expansion of Eq. (29):∆ µ = P T d η css R d(cid:15)N s ( (cid:15) ) h swx ( (cid:15) ) R d(cid:15)N s ( (cid:15) ) ∂ (cid:15) h Ld ( (cid:15) ) , (34)where we computed the transverse swapped spin distri-bution h swx in Eq. (32).It is instructive to consider the ratio between the electrochemical potential difference in the superconducting stateversus the normal state ∆ µ ( N ) ∆ µ ( S ) = R ∞ ∆ d(cid:15)N s ( (cid:15) ) [ D/D (cid:15) ( (cid:15) )] [ λ ( (cid:15) ) /λ N ] [ ∂ (cid:15) f ( (cid:15) )] e − x/λ ( (cid:15) ) R ∞ ∆ d(cid:15)N s ( (cid:15) ) ∂ (cid:15) f ( (cid:15) ) e − x/λ N . (35)We evaluate Eq. (35) numerically and the result is pre-sented in Fig. 2. As announced, below the superconduct-ing transition temperature, there is an enhancement ofthe spin-swapping signal. This can be understood fromEq. (32) and Eq. (35). The spin-swapping spin distri-bution is enhanced for energies around the gap in thesuperconducting state. This leads to the enhancementof the spin-swapping signal at temperature below the su-perconducting transition temperature. IV. MICROSCOPIC DERIVATIONA. Definition of the Green’s function
In this section, we will derive our results presented inSec. II, the diffusion equations (21), and the relations be-tween the currents and the distribution functions of Eqs.(15), (16), and (19). Our starting point is the microscopicHamiltonian of Eq. (2), and we use the nonequilibriumKeldysh Green’s function formalism.We define the kinetic Green’s function in terms of the T / T c μ R ( S ) / μ R ( N ) FIG. 2. The temperature dependence of the spin-swappingsignal. G K ij ( , ) = − i(ˆ τ ) ii (cid:10)(cid:2) ˆ ψ i ( ) , ˆ ψ † j ( ) (cid:3) − (cid:11) , (36a)where [ A, B ] ± = AB ± BA . ˆ G K is a 4 × ⊗ particle-hole space, and we denote such matri-ces using a ‘hat’ superscript. Similarly, we define theretarded Green’s function,ˆ G R ij ( , ) = − iΘ( t − t )(ˆ τ ) ii (cid:10)(cid:2) ˆ ψ i ( ) , ˆ ψ † j ( ) (cid:3) + (cid:11) , (36b)and the advanced Green’s function,ˆ G A ij ( , ) = iΘ( t − t )(ˆ τ ) ii (cid:10)(cid:2) ˆ ψ i ( ) , ˆ ψ † j ( ) (cid:3) + (cid:11) , (36c)where Θ( t ) is the Heaviside function. Next, we con-struct an 8 × G ( , ) inspin ⊗ particle-hole ⊗ Keldysh space, ˇ G = (cid:18) ˆ G R ˆ G K ˆ0 ˆ G A (cid:19) , (37)which obeys the right-handed equation of motion (cid:0) i (cid:126) c ˆ τ D ( ) − ˆ H ( ) (cid:1) ˇ G ( , ) = δ ( − ) (38)and its corresponding left-handed equation of motionˇ G ( , ) (cid:0) − i (cid:126) c ˆ τ D ( ) − ˆ H ( ) (cid:1) = δ ( − ) (39)in terms of the Hamiltonian (2). We denote 8 × × × B. Derivation of the co-variant Eilenbergerequations
The Eilenberger equation is widely used . Neverthe-less, we include a derivation of the Eilenberger equation for systems in which the extrinsic spin-orbit interactionis essential. Spin-orbit interactions require careful han-dling of the spin-orbit-induced self-energy that appearsin the Eilenberger equation. The Eilenberger equationis obtained by taking the difference between the left-and right-handed equations of motion, Eqs. 39 and 38.By taking the Wigner transform in the mixed representa-tion and keeping terms to first the order in (cid:126) , we obtainfor a stationary systemi (cid:126) v i ˜ ∇ i ˇ G + [ (cid:15) ˆ τ + ˆ∆ , ˇ G ] − − [ ˆΣ imp − ˆΣ m ⊗ , ˇ G ] − − ( ˆΣ so ⊗ ˇ G − ˇ G ⊗ ˆΣ so ) = 0 , (40)where ˜ ∇ µ X ≡ ∇ µ X + i A µ [ˆ τ , X ] − and ˆΣ so contains allthe self-energy contributions involving ˆ U so .In a stationary state, the quasiclassical Green’s func-tion is defined asˇ g ( R , p F , (cid:15) ) = i π Z d ξ p ˇ G c ( R , p , (cid:15) ) , (41)which in the mixed representation is a function of thecenter-of-mass coordinate R = ( r + r ) / (cid:15) related to the relative time coordinate t = t − t by a co-variant Wigner transform as defined in Ap-pendix A. The momentum p F is related to the relativeposition r = r − r by a Fourier transform and isfixed at the Fermi level, and the integration variable is ξ p = p / m . The quasiclassical Green’s function (41) isdetermined by the Eilenberger equation which in a sta-tionary state reads0 = i (cid:126) v F · ˜ ∇ ˇ g + (cid:2) ˆ τ (cid:15) + ˆ∆ , ˇ g (cid:3) − − (cid:2) ˇ σ, ˇ g (cid:3) − , (42)where v F = p F /m is the Fermi velocity, and we haveinserted the various self-energies that we will address inthe next section. Upon impurity averaging, ˆΣ so and ˆΣ so become identical and are included in the commutator inEq. (42). Eq. (42) does not determine the quasiclassi-cal Green’s function uniquely; therefore, we also need anormalization condition, ˇ g = 1 . (43)We have now derived the Wigner-transformed Eilen-berger equation in the presence of spin-orbit interactions. C. Self-energies
We consider a diffusive system and will therefore com-pute average quantities relevant at length scales longerthan the mean free path and independent of the impu-rity configuration. First, we include the effects of thelocal potential ˆ U tot within the self-consistent Born ap-proximation. To the second order in the local potential,the self-energy is shown in Fig. 3(a) and reads asˇΣ( , ) = (cid:10) ˆ U tot ( r ) ˇ G c ( , ) ˆ U tot ( r ) (cid:11) c , (44)0where h . . . i c denotes averaging over all possible impu-rity configurations, and we assume that h ˆ U tot i c = 0. Theself-energy ˇΣ is a functional of the impurity-averaged fullpropagator ˇ G c = h ˇ G i c . The terms in the local potentialgive rise to various self-energy contributions that can betreated independently. In the absence of spin-orbit cou-pling, the effects of elastic impurity scattering are calcu-lated fromˇΣ imp ( , ) = n Z d r i u ( r − r i ) ˇ G c ( , ) u ( r − r i ) , (45a)where n is the impurity concentration. Spin swappingand side-jump scattering arise from contributions thatare linear in the spin-orbit coupling strength:ˇΣ (1)so ( , ) = n Z d r i ˆ u so ( r − r i ) ˇ G c ( , ) u ( r − r i )+ n Z d r i u ( r − r i ) ˇ G c ( , )ˆ u so ( r − r i ) . (45b)To include spin-orbit-induced spin relaxation, we also (a) (b) FIG. 3. Self-energy diagrams. (a) Self-consistent Born ap-proximation. (b) Third-order contribution that determinesskew scattering. calculate the second-order contributions from the spin-orbit coupling to the self-energy:ˇΣ so ( , ) = n Z d r i ˆ u so ( r − r i ) ˇ G c ( , )ˆ u so ( r − r i ) . (45c)We will also include contributions from skew scatteringto the spin Hall and inverse spin Hall effects. However,skew scattering does not appear within the frameworkof the self-consistent Born approximation. To includeskew scattering, we also include contributions that arethird order in the potential u : see Fig. 3(b). To thefirst order in the spin-orbit coupling, the skew scatteringcontributions to the self-energy areˇΣ sk ( , ) = n Z d r i Z d ˆ u so ( r − r i ) ˇ G c ( , ) u ( r − r i ) ˇ G c ( , ) u ( r − r i )+ n Z d r i Z d u ( r − r i ) ˇ G c ( , )ˆ u so ( r − r i ) ˇ G c ( , ) u ( r − r i )+ n Z d r i Z d u ( r − r i ) ˇ G c ( , ) u ( r − r i ) ˇ G c ( , )ˆ u so ( r − r i ) . (45d)Moreover, to have a closed set of equations and acomplete quasiclassical theory, the self-energy ˇΣ[ ˇ G c ] isapproximated by the quasiclassical self-energy ˇ σ [ˇ g ] inEq. (42) which is then a functional of the quasiclassi-cal Green’s function ˇ g . Performing impurity average and employing the quasiclassical approximation yield simpli-fied expressions for the various self-energy contributionsof Eqs. (45) evaluated at position R , Fermi momentum p F and energy (cid:15) :1ˇ σ imp = − i2 D τ ( p − q ) ˇ g ( R , q , (cid:15) ) E F , (46a)ˇ σ (1)so = ˜ γp F D τ ( p − q ) (cid:2) ˆ τ ˆ α · (ˆ p × ˆ q ) , ˇ g ( R , q , (cid:15) ) (cid:3) − E F + i˜ γ D τ ( p − q ) (cid:2) ˆ τ ˆ α × (ˆ p − ˆ q ) , ˜ ∇ ˇ g ( R , q , (cid:15) ) (cid:3) + E F , (46b)ˇ σ so = − i γ D τ ( p − q ) ˆ τ ˆ α · ( p × q )ˇ g ( R , q , (cid:15) )ˆ τ ˆ α · ( p × q ) E F , (46c)ˇ σ sk = − i γ D τ sk ( p , q , q ) (cid:0) ˆ τ ˆ α · ( p × q )ˇ g ( R , q , (cid:15) )ˇ g ( R , q , (cid:15) ) − ˇ g ( R , q , (cid:15) )ˇ g ( R , q , (cid:15) )ˆ τ ˆ α · ( p × q ) (cid:1)E F (46d) − i γ D τ sk ( p , q , q ) ˇ g ( R , q , (cid:15) )ˆ τ ˆ α · ( q × q )ˇ g ( R , q , (cid:15) ) E F , where h . . . i F denotes an angular average with respect to q (and q ) at the Fermi surface. We changed the notationof the self-energy in Eq. (46) by switching from ˇΣ to ˇ σ toemphasize that we use the quasiclassical approximationof Eq. (45). The elastic scattering rate is1 τ ( p − q ) = 2 πnN (cid:12)(cid:12) u ( p − q ) (cid:12)(cid:12) , (47a)where N is the density of states at the Fermi level in thenormal state. The skew scattering rate is1 τ sk ( p , q , q ) = 2 π nN u ( p − q ) u ( q − q ) u ( q − p ) . (47b)The skew scattering rate (47b) is on the order of 1 / ( N u )smaller than the elastic scattering rate (47a). A de-tailed derivation of Eqs. (46) and (47) is presented inAppendix E. D. Diffusion Limit
Since elastic impurity scattering is assumed to bestrong (dirty limit), the quasiclassical Green’s functionbecomes almost isotropic and we can use an expansionin spherical harmonics up to the first order,ˇ g ( R , p F , (cid:15) ) ≈ ˇ g s ( R , (cid:15) ) + e p · ˇ g ( R , (cid:15) ) , (48)where ˇ g s and ˇ g are the isotropic and anisotropic Green’sfunctions, respectively, and e p = p F / | p F | . Expandingthe normalization condition (43) to the first order yieldsthe useful relationsˇ g = 1 , (cid:2) ˇ g s , ˇ g (cid:3) + = 0 . (49)We use this expansion (48) in the self-energy contribu-tions of Eq. (46) and retain only the dominant terms.As a conventional example, consider the elastic impurityscattering. Inserting the expansion of the Green’s func-tion in spherical harmonics of Eq. (48) into the quasi-classical elastic impurity scattering self-energy (46a) and performing the angular average providesˇ σ imp ( p F ) ≈ − i2 D τ ( p − q ) (ˇ g s + e q · ˇ g ) E F = − i2 τ ˇ g s − i2 (cid:16) τ − τ tr (cid:17) ( e p · ˇ g ) , (50a)where we dropped the center coordinate R and the en-ergy (cid:15) for brevity, and1 τ = D τ ( p − q ) E F is the average elastic scattering rate and1 τ tr = D τ ( p − q ) (1 − e p · e q ) E F is the inverse transport relaxation time.Similarly, we insert the Green’s function’s expansion(48) into the remaining self-energy contributions and per-form the angular average. We make use of the fact thatˇ g (cid:28) ˇ g s and only retain the dominant contributions. Tothe leading order in the spin-orbit coupling, we thenobtain the spin-swapping (“sw”), side-jump (“sj”), andspin-orbit-induced relaxation (“so”) contributions to theself-energy:ˇ σ sw ( p F ) = − η τ sw e p · (cid:2) ˆ τ ˆ α × , ˇ g (cid:3) + , (50b)ˇ σ sj ( p F ) = − i˜ γ τ tr e p · (cid:2) ˆ τ ˆ α × , ˜ ∇ ˇ g s (cid:3) − , (50c)ˇ σ so ( p F ) = − τ so (cid:0) ˆ τ ˆ α × e p (cid:1) ˇ g s (cid:0) ˆ τ ˆ α × e p (cid:1) , (50d)and, using Eq. (49), we find the skew-scattering contri-butionˇ σ sk ( p F ) = − i η τ sk e p · (cid:2) ˆ τ ˆ α × , ˇ g s ˇ g (cid:3) − , (50e)where [ a × , b ] ± = a × b ± b × a and, again, we omitthe arguments R and (cid:15) for brevity. When evaluating theself-energy to the first order in the spin-orbit coupling2(45b), a contribution to the side-jump mechanism (50c)is also obtained; see Appendices E 2 and F. The elasticscattering rate is1 τ = 2 πnN D(cid:12)(cid:12) u ( e p − e q ) (cid:12)(cid:12) E F , (51a)and the inverse transport relaxation time is1 τ tr = 2 πnN D(cid:12)(cid:12) u ( e p − e q ) (cid:12)(cid:12) (1 − e p · e q ) E F . (51b)Spin relaxation is determined by the spin-flip scatteringrate due to magnetic impurities,1 τ m = 83 πn m N S ( S + 1) D(cid:12)(cid:12) u m ( e p − e q ) (cid:12)(cid:12) E F , (51c)where S is the impurity spin quantum number, and thespin-flip scattering rate due to spin-orbit coupling,1 τ so = 83 π ˜ γ p nN D(cid:12)(cid:12) u ( e p − e q ) (cid:12)(cid:12) ( e p × e q ) E F . (51d)The spin-swapping scattering rate is1 τ sw = 98˜ γ p τ so , (51e)and the skew scattering rate is1 τ sk = 2 π nN u . (51f)The results in Eqs. (50) and (51) are valid irrespective ofthe possible anisotropy of the scattering potential, withthe exception that the expression for the self-energy dueto skew scattering (50e) is included only to the lowestorder in the anisotropy of the scattering potentials tokeep the result compact and simple. The isotropic Green’s function ˇ g s and the anisotropicGreen’s function ˇ g are obtained using the expansion inspherical harmonics (48) and splitting the Eilenbergerequation (42) into an even and an odd part with re-spect to e p . To the first order in the spin-orbit couplingstrength, the odd part is0 = i v F ˜ ∇ ˇ g s + i τ tr ˇ g s ˇ g + (cid:2) ˆ τ (cid:15), ˇ g (cid:3) − + (cid:2) ˆ∆ , ˇ g (cid:3) − − η τ sw ˇ g s h(cid:2) ˆ τ ˆ α , ˇ g s (cid:3) + × , ˇ g s ˇ g i − + i˜ γ τ tr ˇ g s h ˇ g s (cid:2) ˆ τ ˆ α , ˇ g s (cid:3) + × , ˜ ∇ ˇ g s i − − i η τ sk h(cid:2) ˆ τ ˆ α , ˇ g s (cid:3) + × , ˇ g s ˇ g i + , (52)where we used the normalization condition (49).The first line on the r.h.s. contains the contributions tothe zeroth order in the spin-orbit coupling strength. Theremaining terms are the contributions to the first order,which we rewrote in a way that simplifies our furthercalculations. The anisotropic Green’s function can be computed to the zeroth order from the first line. Theterms involving the energy (cid:15) and the superconductingorder parameter ∆ can be neglected compared to thedominating contribution arising from the elastic impurityself-energy (50a), and we obtain the following well-knownresult: ˇ g (0) = − l tr ˇ g s ˜ ∇ ˇ g s , (53)where l tr = v F τ tr is the impurity mean free path.By using ˇ g = ˇ g (0) + δ ˇ g in Eq. (52) and multiplying byi τ tr ˇ g s from the left, the first-order corrections δ ˇ g to theanisotropic Green’s function stemming from spin-orbitcoupling are additive, δ ˇ g = δ ˇ g (sw) + δ ˇ g (sj) + δ ˇ g (sk) , and are readily obtained by using Eq. (49). The spin-swapping self-energy (50b) contributes with δ ˇ g (sw) = i ηl tr τ tr τ sw h(cid:2) ˆ τ ˆ α , ˇ g s (cid:3) + × , ˜ ∇ ˇ g s i − , (54a)and the correction due to the self-energy contribution tothe side-jump mechanism (50c) is δ ˇ g (sj) = ˜ γ h ˇ g s (cid:2) ˇ g s ˆ τ ˆ α , ˇ g s (cid:3) + × , ˜ ∇ ˇ g s i − . (54b)Lastly, the correction from the skew scattering self-energy (50e) reads as δ ˇ g (sk) = − ηl tr τ tr τ sk ˇ g s h(cid:2) ˆ τ ˆ α , ˇ g s (cid:3) + × , ˜ ∇ ˇ g s i + . (54c)We will see that the side-jump mechanism (54b) andskew scattering (54c) both contribute to the same effect,namely, the spin Hall effect.Using Eq. (53) in the part of the Eilenberger equa-tion (42) that is even with respect to e p , we obtain theUsadel equation D ˜ ∇ · (cid:0) ˇ g s ˜ ∇ ˇ g s (cid:1) = − i (cid:2) ˆ τ (cid:15), ˇ g s (cid:3) − − i (cid:2) ˆ∆ , ˇ g s (cid:3) − + 18 τ so (cid:2) ˆ α ˆ τ ˇ g s ˆ τ ˆ α , ˇ g s (cid:3) − + 18 τ m (cid:2) ˆ α ˇ g s ˆ α , ˇ g s (cid:3) − , ≡ U iso (55)where we, at this stage, have included the well-knowneffect of magnetic impurities causing pair breaking andspin relaxation represented by the scattering lifetime τ m .We also introduce D = v F l tr / h , ˆ g Ks ( R , (cid:15) ) = ˆ g Rs ˆ h − ˆ h ˆ g As , (56)3where the advanced Green’s function can be expressed interms of the retarded Green’s function, ˆ g As = − (ˆ τ ˆ g Rs ˆ τ ) † .We assume that the distribution matrix is diagonal withrespect to particle-hole space and decompose it accordingto ˆ h = h (cid:15) + ˆ α j h (cid:15)sj + ˆ τ ( h + ˆ α j h sj ) , (57)where h (cid:15) and h are the energy and particle distribu-tion functions, respectively, and h (cid:15)j and h j are thespin-energy and spin distribution functions, respectively,where the subscript (j) denotes the spin polarization di-rection.In equilibrium, all the distribution functions except theenergy distribution, h (cid:15) , vanish . At equilibrium, theKeldysh function can then be expressed in terms of theretarded and advanced functions in a simple mannerˆ g Ks ( R , (cid:15) ) = h eq (cid:0) ˆ g Rs − ˆ g As (cid:1) , (58)where h eq = tanh( (cid:15)/ T ).In general, g Rs and f Rs depend on position and en-ergy and determine how the various transport mech-anisms renormalize below the superconducting criticaltemperature. They are solved by using the retardedpart of Eq. (55) together with the normalization con-dition ( g Rs ) − ( f Rs ) = 1. For energies far above thegap, the functions approach their high-temperature lim-its ( g Rs → f Rs →
0) while they diverge for en-ergies close to the superconducting-induced energy gap.The presence of magnetic impurities in the system sup-presses superconductivity and reduces the gap in the en-ergy spectrum.
E. Current and Densities
Let us now turn to derive expressions that describe thephysical particle and spin currents and equations thatdetermine the distribution matrix. We begin by defininga particle density, n ( P ) ( ), which counts the number ofelectrons, n ( P ) ( ) = 12 lim → Tr h ˆ n ( , ) i = X σ = ↑ , ↓ h ψ † σ ( ) ψ σ ( ) i . (59a)Analogously, we define a spin density, n ( S ) ( ); a parti-cle energy density; n ( P,E ) ( ) and a spin energy density, n ( S,E ) ( ) n ( S ) ( ) = 12 lim → Tr h α ˆ n ( , ) i , (59b) n ( P,E ) ( ) = 14 lim → Tr h (cid:0) i (cid:126) ˆ τ ∂ t − i (cid:126) ˆ τ ∂ t (cid:1) ˆ n ( , ) i , (59c) n ( S,E ) ( ) = 14 lim → Tr h (cid:0) i (cid:126) ˆ τ ∂ t − i (cid:126) ˆ τ ∂ t (cid:1) α ˆ n ( , ) i , (59d) with ˆ n ( , ) being defined byˆ n ( , ) = − i2 ˆ G K ( , )+ i2 ˆ τ (cid:16) ˆ G R ( , ) − ˆ G A ( , ) (cid:17) . (60)The trace is taken over spin ⊗ particle-hole space.From the densities (59), we calculate corresponding cur-rents using the equations of continuity. The current ex-pressions are Wigner transformed, and in terms of thequasiclassical Green’s functions, we define a current den-sity matrixˆ ( R , (cid:15) ) = v F (cid:16) ˆ g K ( R , (cid:15) ) − ˆ τ (cid:0) ˆ g R ( R , (cid:15) ) − ˆ g A ( R , (cid:15) ) (cid:1)(cid:17) − γp F l tr τ tr (cid:2) ˆ τ ˆ α × , ∇ ˆ g Ks ( R , (cid:15) ) (cid:3) − , (61)where the second term is the anomalous current con-tribution that arises from the anomalous velocity, whichis explained in detail in Sec. F 1.The particle current density and the spin current den-sity in the quasiclassical approximation are found by tak-ing the proper traces of the current density matrix in Eq.61 J i ( R ) = N Z d (cid:15) Tr (cid:2) ˆ τ ˆ i ( R , (cid:15) ) (cid:3) (62a)= N Z d (cid:15) j i ( R , (cid:15) )and J ij ( R ) = − N Z d (cid:15) Tr (cid:2) ˆ τ ˆ α j ˆ i ( R , (cid:15) ) (cid:3) (62b)= N Z d (cid:15) j ij ( R , (cid:15) ) ,J (cid:15)i ( R ) = N Z d (cid:15) Tr (cid:2) (cid:15) ˆ i ( R , (cid:15) ) (cid:3) (62c)= N Z d (cid:15) j (cid:15)i ( R , (cid:15) ) J (cid:15)ij ( R ) = N Z d (cid:15) (cid:15) Tr (cid:2) (cid:15) ˆ α j ˆ i ( R , (cid:15) ) (cid:3) (62d)= N Z d (cid:15) (cid:15)j (cid:15)ij ( R , (cid:15) ) , respectively. Here, j i is the particle current and j ij isthe spin current, as introduced in Sec. II. The expres-sions for the energy and the spin-energy current densitiesare derived similarly, but are defined compared to someequilibrium value . The second term on the r.h.s. ofEq. (61) results from the anomalous velocity correctionsof Eq. (F1) and contributes to the side-jump effect; seeAppendix F.We can now express the gradient of the current usingthe Usadel equation (55) in terms of the divergence ofthe matrix current ˆ . Using Eqs. (53), (55), and (61), wefind ˜ ∇ · ˆ = − (cid:16) ( U iso ) K − ˆ τ (cid:2) ( U iso ) R − ( U iso ) A (cid:3)(cid:17) (63)4where the contributions arise from the respective matrixblock in Eq. (55). From Eq. (63), the diffusion Eq. (21)can be derived in terms of the distribution functions inEq. (57). The currents in Subsec. II A are defined asindicated by Eq. (62) and are calculated using Eqs. (53)and (54) in Eq. (61).The renormalization factors are determined by thecomponents of the retarded/advanced Green’s function,where we have inserted the parametrization explained inAppendix C, as well as the self-consistency expression.For gap scattering and spin relaxation, they read as α = 2 Im[sinh θ ] Re[ e − i χ ∆] , (64a) α (cid:15) = 2 Re[sinh θ ] Im[ e − i χ ∆] (64b) α so = Re[cosh θ ] − Re[sinh θ ] , (64c) α (cid:15) so = Re[cosh θ ] + Im[sinh θ ] , (64d) α m = Re[cosh θ ] + Re[sinh θ ] , (64e) α (cid:15) m = Re[cosh θ ] − Im[sinh θ ] , (64f)The renormalized diffusion constants are D (cid:15) = D (cid:0) | cosh θ | − | sinh θ | (cid:1) , (65a) D p = D (cid:0) | cosh θ | + | sinh θ | (cid:1) , (65b)and N S = Re[cosh θ ] (66)is the density of states in the superconductor normalizedby the density of states in the normal state. We alsodefine the following currents related to the supercurrentsin the system v s = ∇ χ − e (cid:126) A i (67a) j sci = (cid:8) θ ) (cid:9) v si (67b) j sc, i = (cid:8) − | cosh θ | + | sinh θ | (cid:9) v si (67c) R pi = − (cid:0) sinh θ (cid:1) ∇ i (Re θ ) (67d) R (cid:15)i = − (cid:0) sinh θ (cid:1) ∇ i (Im θ ) (67e)This completes our derivations of the diffusion equa-tions and the associated relations between the currentsand the spatial variations of the densities. V. CONCLUSION
We have derived diffusion equations for the trans-port of spin, particle, and energy in the elastic trans-port regime, including scattering from magnetic and non-magnetic impurities and from spin-orbit coupling. We find that the spin Hall angle is renormalized by the re-duced density of states in the superconducting state.However, the spin-swapping coefficient does not explicitlydepend on the superconducting correlations but ratheris influenced by the superconducting state through therenormalized diffusion coefficients.In a two-dimensional geometry, we find a large en-hancement of the spin-swapping effects. This result im-plies that it should be possible to measure the influence ofsuperconductivity on these largely unexplored transportproperties.We thank Jacob Linder for stimulating discussions.
Appendix A: Fourier Transform
We define the Fourier transforms as x ( r , t ) = Z d q (2 π (cid:126) ) e − i qr / (cid:126) Z d (cid:15) π (cid:126) e i(cid:15)t/ (cid:126) x F ( q , (cid:15) ) , (A1a) x F ( q , (cid:15) ) = Z d r e i qr / (cid:126) Z d t e − i(cid:15)t/ (cid:126) x ( r , t ) , (A1b)where the subscript ”F” indicates that we are referringto the Fourier transform and not the Wigner transform. Appendix B: Wigner Transform
In this section, we will introduce the Wigner transformwhich we will use extensively. We follow the conventionsin Ref. [44] for an Abelian and spin-independent vectorfield.We can relate the Fourier transform and the Wignertransform by using a translation operator. For a functionthat depends on the space-times = X + z/ = X − z/
2, where X and z are the absolute and relativespace-times, we have x F ( p, X ) = Z dz e − ipz/ (cid:126) h e z ∂ X x ( X, X ) e − z ∂ X i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = X , (B1)where z ≡ ( t, r ), X ≡ ( T, R ), and p = ( (cid:15), p ). Theinner product is defined according to the mostly minusmetric, as outlined in Sec. II. To arrive at a co-varianttransform, we let ∂ X µ → D µ ( X ) and ∂ X µ → D µ ( X ),where the co-variant derivative is defined according toEq. (3), which results in x ( p, X ) = Z dz e − ipz/ (cid:126) × (cid:20) exp (cid:18) z µ D µ (cid:19) x ( X, X )exp (cid:18) − z µ D µ (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = X , (B2)5which is how we define the co-variant Wigner trans-form. Note that we write the co-variant Wigner trans-form without a subscript ” F ”. We define the connector ,ˆ U , ˆ U ( b, a ) ≡ exp (cid:20) iˆ τ ( b − a ) µ Z dsA µ ( a + ( b − a ) s ) (cid:21) , (B3)and to ease notation, we also define ˆ U = ˆ U ( , X ) andˆ U = ˆ U ( X, ). In terms of the connectors, the Wignertransform becomes x ( p, X ) = Z dz h e − ipz/ (cid:126) ˆ U x ( , ) ˆ U . i The inverse transform is x ( , ) = Z dp (2 π ) h e i pz/ (cid:126) ˆ U † x ( p, X ) ˆ U † i . (B4)Our Green’s functions are matrices in electron-holespace. This carries over to a matrix structure in theconnector. In electron-hole space, the Wigner transformof our Green function is defined asˆ G ( p, X ) = Z dz e − i pz/ (cid:126) h ˆ U ˆ G ( , ) ˆ U . i , (B5)In the following, we will expand the connectors in thegradient approximation. Appendix C: Parametrization
To simplify the calculations, we apply the θ -parametrization. The retarded Green’s function isˆ g R ( (cid:15) ) = (cid:18) ˆ1 cosh [ θ ( (cid:15) )] i σ sinh [ θ ( (cid:15) )] e iχ ( (cid:15) ) i σ sinh [ θ ( (cid:15) )] e − iχ ( (cid:15) ) − ˆ1 cosh [ θ ( (cid:15) )] (cid:19) . (C1)The advanced function can be found using the relationˆ g A ( R , (cid:15) ) = − (cid:0) τ ˆ g R ( R , (cid:15) ) τ (cid:1) † . Inspecting the elements ofthe retarded Green’s function in Eq. (36b) of the Green’sfunction, and the normalization condition, we obtain thefollowing symmetries for θ ( R , (cid:15) ) and χ ( R , (cid:15) ) χ ( (cid:15) ) = χ ∗ ( − (cid:15) ) , (C2a) θ ( (cid:15) ) = − θ ∗ ( − (cid:15) ) . (C2b)We insert these when we calculate the current. Appendix D: Self-consistency
The superconducting gap must be calculated self-consistently, and we can express the gap using theKeldysh Green’s function in the following way∆( ) = − N λ Z d(cid:15) Tr " (ˆ τ − iˆ τ ) α g Ks ( R , (cid:15) ) , (D1)where λ is the strength of the pairing potential. Usingour ansatz for the distribution function in Eq. 56, we canexpress the gap in terms of distribution functions and theparametrization parameters∆( ) = N λ Z d(cid:15) e i χ h − Re(sinh θ ) h (cid:15) + i Im(sinh θ ) h i . (D2)We use this relation in our expressions. Since we havechosen χ to be a real number, the phase of the orderparameter is also real. Appendix E: The Self-Energy
Here, we will calculate the contributions to the qua-siclassical self-energy (46) and outline how the self-energy contributions to the Eilenberger equation (42)and the Usadel equations (52) and (55) are obtained.We include effects from elastic impurity scattering, mag-netic impurities and spin-orbit coupling within the self-consistent Born approximation; see the Feynman dia-grams in Fig. 3(a). Skew scattering only appears beyondthe self-consistent Born approximation to at least thethird order in the potential u , . We take this lowestorder contribution to the skew scattering into account;see Fig. 3(b). The self-energies caused by elastic scat-tering, magnetic impurities, and the contribution fromspin-orbit scattering to spin relaxation are well known,but we also include their brief derivation here for com-pleteness and consistency in the notation.
1. Elastic Impurity Scattering
Using the Fourier representation of the elastic impurityscattering potential u ( r − r i ), its contribution to the self-energy (45a) isˇΣ imp ( , ) = n Z d q (2 π (cid:126) ) (cid:12)(cid:12) u ( q ) (cid:12)(cid:12) e − i q · r / (cid:126) ˇ G c ( , ) , where r = r − r is the relative position. Next, weFourier transform the relative spatial and temporal coor-dinates using the Fourier transform of Eq. (A1):ˇΣ imp ( R , p , (cid:15) ) = n Z d q (2 π (cid:126) ) (cid:12)(cid:12) u ( p − q ) (cid:12)(cid:12) ˇ G c ( R , q , (cid:15) ) . (E1)6Within the quasiclassical framework we can approximate R d q (2 π (cid:126) ) ≈ N R d ξ q R d e q π , where N is the density ofstates and e q = q F / | q F | , such that the quasiclassical ap-proximation to the self-energy isˇ σ imp ( R , p F , (cid:15) ) = − i2 D τ ( p − q ) ˇ g ( R , q , (cid:15) ) E F , (E2)where h . . . i F = R d e q π . . . denotes an angular average overall momentum directions at the Fermi surface and theelastic scattering rate is1 τ ( p − q ) = 2 πnN (cid:12)(cid:12) u ( p − q ) (cid:12)(cid:12) . (E3)We changed the notation of the self-energy in Eq. (E2)to the symbol ˇ σ to reflect that it is the quasiclassicalapproximation of Eq. (E1). This is a standard result forthe elastic scattering contribution to the self-energy thatwe included for completeness.
2. First Order in Spin-Orbit Coupling
The contributions to the self-energies above are wellknown. Let us now consider the nontrivial effect of thespin-orbit interaction to the first order in the spin-orbitinteraction strength. Inserting the expressions for ˆ u so and u into Eq. (45b) yieldsˇΣ (1)so ( , )= − γn (cid:126) Z d q (2 π (cid:126) ) (cid:12)(cid:12) u ( q ) (cid:12)(cid:12) e − i q · r / (cid:126) ˆ τ ˆ α · ( ˜ ∇ r ˇ G c ( , ) × q ) − γn (cid:126) Z d q (2 π (cid:126) ) (cid:12)(cid:12) u ( q ) (cid:12)(cid:12) e − i q · r / (cid:126) ( ˇ G c ( , ) ˜ ∇ r × q ) · ˆ τ ˆ α , where ˜ ∇ is defined as˜ ∇ X = ( ∇ X ) − i e (cid:126) A [ˆ τ , X ] − . (E4a), X ˜ ∇ = ( X ← ∇ ) + i e (cid:126) A [ X, ˆ τ ] − . (E4b)In the quasiclassical approximation we obtain from thisˇ σ (1)so ( p F ) = ˜ γp F D τ ( p − q ) (cid:2) ˆ τ ˆ α · (ˆ p × ˆ q ) , ˇ g ( q ) (cid:3) − E F + i˜ γ D τ ( p − q ) (cid:2) ˆ τ ˆ α × (ˆ p − ˆ q ) , ˜ ∇ ˇ g ( q ) (cid:3) + E F , (E5)where we omitted R and (cid:15) for brevity. We also introducedthe dimensionless parameter ˜ γ = γp F / (cid:126) .The first term on the r.h.s. of Eq. E5 gives rise to thespin-swapping effect . The second contributes to theside-jump mechanism but is only present when consid-ering the next-to-leading order in the gradient approxi-mation. The side-jump mechanism is discussed in moredetail in Appendix F.
3. Second Order in Spin-Orbit Coupling
Similarly, we obtain from Eq. (45c) to the lowest orderin the quasiclassical approximation the self-energy to thesecond order in the spin-orbit coupling strength:ˇ σ so ( p F ) = − i˜ γ p F D τ ( p − q ) ˆ τ ˆ α · (ˆ p × ˆ q )ˇ g ( q )ˆ τ ˆ α · (ˆ p × ˆ q ) E F , (E6)where we again omitted R and (cid:15) . This self-energy con-tribution describes spin-orbit-induced spin relaxation.
4. Skew Scattering
We include skew scattering to the lowest order in the gradient approximation. Inserting Eqs. (7) and (8) into theskew-scattering contribution to the self-energy (45d) providesˇΣ sk ( , ) = − γn (cid:126) Z d q (2 π (cid:126) ) Z d q (2 π (cid:126) ) u ( q ) u ( − q − q ) u ( q ) Z d e − i q · r e − i q · ( r − r ) × " (ˆ τ ˆ α × q ) · (cid:16) D ( r ) ˇ G c ( , ) (cid:17) ˇ G c ( , ) + ˇ G c ( , )(ˆ τ ˆ α × q ) · (cid:16) D ( r ) ˇ G c ( , ) (cid:17) + ˇ G c ( , ) (cid:16) ˇ G c ( , ) D ( r ) (cid:17) · (ˆ τ ˆ α × ( q + q )) , (E7)where we performed a partial integration in the Dyson equation in the last term and D ( r ) acts to the left. Werewrite the Green’s functions in terms of their respective center-of-mass and relative coordinates and, for example,use ˇ G c ( , ) = ˇ G c (cid:16) r + r , r − r , t − t (cid:17) = ˇ G c (cid:16) R + r − r , r / − ( r − R ) , t − t (cid:17) . R = ( r + r ) / G c ( , ) ≈ ˇ G c (cid:0) R , r / − ( r − R ) , t − t (cid:1) , ˇ G c ( , ) ≈ ˇ G c (cid:0) R , r / r − R ) , t − t (cid:1) . After inserting the Wigner coordinates and Fourier transforming Eq. E7, we haveˇΣ sk ( R , p , (cid:15) ) = − γn (cid:126) Z d q (2 π (cid:126) ) Z d q (2 π (cid:126) ) u ( q ) u ( − q − q ) u ( q ) Z d r e i p · r / (cid:126) Z d r e − i q · r / (cid:126) e − i q · (cid:0) r − R + r / (cid:1) × " (ˆ τ ˆ α × q ) · (cid:16) D ( R + r /
2) ˇ G c ( R , R + r / − r , (cid:15) ) (cid:17) ˇ G c ( R , r − R + r / , (cid:15) )+ ˇ G c ( R , R + r / − r , (cid:15) )(ˆ τ ˆ α × q ) · (cid:16) D ( r ) ˇ G c ( R , r − R + r / , (cid:15) ) (cid:17) + ˇ G c ( R , R + r / − r , (cid:15) ) (cid:16) ˇ G c ( R , r − R + r / D † ( R − r / , (cid:15) ) (cid:17) · (ˆ τ ˆ α × ( q + q )) where we used that, in a stationary case, the convolution with respect to the time variables reduces to a simpleproduct Z d t e i (cid:15)t Z d t ˇ G c ( t − t ) ˇ G c ( t − t ) = ˇ G c ( (cid:15) ) ˇ G c ( (cid:15) ) . Next, we introduce new variables according to r = x + y , r − R = x − y , ∂ ( r , r ) ∂ ( x , y ) = − , and consequently, we obtainˇΣ sk ( R , p , (cid:15) ) = − γn (cid:126) Z d q (2 π (cid:126) ) Z d q (2 π (cid:126) ) u ( q ) u ( − q − q ) u ( q ) Z d x e i( p − q ) · x / (cid:126) Z d y e i( p − q − q ) · y / (cid:126) × " (ˆ τ ˆ α × q ) · (cid:16) ∂ x ˇ G c ( R , x , (cid:15) ) (cid:17) ˇ G c ( R , y , (cid:15) ) + ˇ G c ( R , x , (cid:15) )ˆ τ ( ˆ α × q ) · (cid:16) ( − ∂ x + ∂ y ) ˇ G c ( R , y , (cid:15) ) (cid:17) − ˇ G c ( R , x , (cid:15) ) (cid:16) ∂ y ˇ G c ( R , y , (cid:15) ) (cid:17) · (ˆ τ ˆ α × ( q + q )) , where we only retained the lowest-order terms in the quasiclassical approximation, which reduced the co-variantderivative to normal derivatives.Performing out the partial integration providesˇΣ sk ( p ) = i γn (cid:126) Z d q (2 π (cid:126) ) Z d q (2 π (cid:126) ) u ( p − q ) u ( q − q ) u ( q − p ) × (cid:16) ˆ τ ˆ α · ( p × q ) ˇ G c ( q ) ˇ G c ( q ) + ˇ G c ( q )ˆ τ ˆ α · ( q × q ) ˇ G c ( q ) − ˇ G c ( q ) ˇ G c ( q )ˆ τ ˆ α · ( p × q ) (cid:17) , (E8)where we omitted the arguments R and (cid:15) for brevity. Eq. (E8) is in agreement with recent results that are validfor a normal metal only. Our treatment is a generalization to include skew scattering in the superconducting state.In the quasiclassical approximation, the skew-scattering contribution to the self-energy isˇ σ sk ( p F ) = − i˜ γp F D τ sk ( p , q , q ) (cid:0) ˆ τ ˆ α · (ˆ p × ˆ q )ˇ g ( q )ˇ g ( q ) + ˇ g ( q )ˆ τ ˆ α · (ˆ q × ˆ q )ˇ g ( q ) − ˇ g ( q )ˇ g ( q )ˆ τ ˆ α · (ˆ p × ˆ q ) (cid:1)E F , where 1 τ sk ( p , q , q ) = 2 π nN u ( p − q ) u ( q − q ) u ( q − p )is the skew-scattering rate. Note that the skew-scattering rate 1 /τ sk is a factor on the order of 1 / ( N u ) smaller thanthe elastic scattering rate 1 /τ . Appendix F: The Side-Jump Mechanism
The derivation of the side-jump contribution to thespin Hall effect is a subtle issue because there are three contributions to this effect, and one or two8of these continue to be overlooked in many works: i)A contribution arises from the self-energy to the firstorder in the spin-orbit interaction of Eq. (50c). Thiscontribution only appears beyond the lowest-order gra-dient approximation and is therefore often disregarded.However, within the quasiclassical approximation, it isof the same order as the other spin-orbit-induced self-energy contributions of Eq. (50) and must be included.It enters in the first term of the matrix current (61) viathe correction to the anisotropic Green’s function due tothe side-jump self-energy (54b). ii) Additionally, there isan anomalous current contribution (F1a) from the spin-orbit-induced correction to the velocity operator. iii) Fi-nally, the spin-orbit coupling is expressed in an effectivemodel with a renormalized coupling strength that is typi-cally much larger than the vacuum value. In this effectivetheory, the position operator also acquires an additionalspin-dependent and velocity-dependent contribution, theso-called Yafet shift of the position (10). This leads toanother anomalous contribution to the velocity opera-tor (F1b) and to the matrix current (61).Here, we will discuss these anomalous current contri-butions to the side-jump mechanism and compare it tothe contribution from the side-jump self-energy obtainedpreviously.
1. Anomalous Contributions to the Matrix Current
The shift in the position operator (9) leads to a shiftin the velocity operator,ˆ v = ˙ˆ r eff = ˙ r + ˙ˆ r so . The velocity operator is calculated from the Heisenbergequation of motion in terms of the Hamiltonian of Eq. (2)and acquires two spin-dependent corrections as a conse-quence of spin-orbit coupling. The first emerges from˙ r = − i (cid:2) r , ˆ H (cid:3) − = v − em ˆ τ ~A (1) + ˆ v (1)so , whereˆ v (1)so = − i (cid:2) r , ˆ U so (cid:3) − = γ X i (cid:0) ˆ τ ˆ α × ∇ u ( r − r i ) (cid:1) . (F1a)The second correction ˆ v (2)so = ˙ˆ r so arises from the Yafetshift of the position operator (10) and is, to the first orderin the spin-orbit coupling strength,ˆ v (2)so = − i (cid:2) ˆ r so , ˆ U (cid:3) − = γ X i (cid:0) ˆ τ ˆ α × ∇ u ( r − r i ) (cid:1) . (F1b)Note that ˆ v (1)so and ˆ v (2)so are identical, giving rise to anoverall factor of 2. In total, the velocity operator is thusgiven by ˆ v ( r ) = − i m ∂ r + ˆ v so ( r ) . (F2) Note that the spin current density in this definition is notconserved in the presence of magnetic impurities or spin-orbit coupling. As discussed in Sec. IV, the velocity op-erator (F2) acquires the two spin-dependent correctionsof Eq. (F1) as a consequence of the spin-orbit coupling,giving rise to the overall anomalous velocity contributionˆ v so ( r ) = 2 γ X i (cid:0) ˆ τ ˆ α × ∇ u ( r − r i ) (cid:1) . (F3)This anomalous contribution to the matrix current de-fined in Eq. (61) reads asˆ so ( )= i2 N lim → (cid:0) ˆ v so ( r ) ˆ G K ( , ) + ˆ G K ( , )ˆ v so ( r ) (cid:1) . (F4)The challenge in evaluating this expression is comput-ing the impurity average. While the conventional veloc-ity operator is independent of the impurity configura-tion, the anomalous contribution explicitly depends onthe impurities and we need to evaluate h ˆ v so ˆ G K i c and h ˆ G K ˆ v so i c . This can be achieved by following the proce-dure in Ref. 51: from the Dyson equation, it follows that h ˆ U tot ˇ G i c = ˇΣ ˇ G c and h ˇ G ˆ U tot i c = ˇ G c ˇΣ, where ˇΣ is theself-energy. Consequently (cid:10) ˆ v so ( r ) ˆ G K ( , ) (cid:11) c = Z d (cid:16) ˇ Σ (l)sj ( , ) ˇ G c ( , ) (cid:17) K , (F5a) (cid:10) ˆ G K ( , )ˆ v so ( r ) (cid:11) c = Z d (cid:16) ˇ G c ( , ) ˇ Σ (r)sj ( , ) (cid:17) K , (F5b)where, within the self-consistent Born approximation tothe first order in the spin-orbit coupling, see Fig. 3(a),ˇ Σ (l)sj ( , )= 2 γn Z d r i (cid:0) ˆ τ ˆ α × ∇ u ( r − r i ) (cid:1) ˇ G c ( , ) u ( r − r i ) , (F5c)ˇ Σ (r)sj ( , )= 2 γn Z d r i u ( r − r i ) ˇ G c ( , ) (cid:0) ˆ τ ˆ α × ∇ u ( r − r i ) (cid:1) . (F5d)In the mixed representation, we haveˇ Σ (l)sj ( R , p , (cid:15) )= 2i γn Z d q (2 π ) | u ( p − q ) | ˆ τ ˆ α × ( p − q ) ˇ G c ( R , q , (cid:15) ) , (F6a)ˇ Σ (r)sj ( R , p , (cid:15) )= − γn Z d q (2 π ) | u ( p − q ) | ˇ G c ( R , q , (cid:15) )ˆ τ ˆ α × ( p − q ) , (F6b)9and in the quasiclassical approximation, we obtainˇ σ (l)sj ( p F ) = γp F τ tr (cid:0) ˆ τ ˆ α × e p (cid:1)(cid:0) ˇ g s − ( e p · ˇ g ) (cid:1) , (F7a)ˇ σ (r)sj ( p F ) = − γp F τ tr (cid:0) ˇ g s − ( e p · ˇ g ) (cid:1)(cid:0) ˆ τ ˆ α × e p (cid:1) , (F7b)where we used the expansion ˇ g ( R , q F , (cid:15) ) ≈ ˇ g s ( R , (cid:15) ) + e q · ˇ g ( R , (cid:15) ) and performed the angular average over q . We also omitted R and (cid:15) for brevity.Using Eq. (F5), the anomalous contribution to theimpurity-averaged matrix current (F4) in the Fourier rep-resentation is (cid:10) ˆ so (cid:11) c ( R ) = i4 πN Z d (cid:15) Z d p (2 π ) (cid:16) ˇ Σ (l)sj ( R , p , (cid:15) ) ˇ G c ( R , p , (cid:15) ) + ˇ G c ( R , p , (cid:15) ) ˇ Σ (r)sj ( R , p , (cid:15) ) (cid:17) K , (F8)to the lowest order in the gradient approximation. In the quasiclassical approximation, this becomes (cid:10) ˆ so (cid:11) c ( R ) = 14 Z d (cid:15) D(cid:0) ˇ σ (l)sj ( R , p , (cid:15) )ˇ g ( R , p , (cid:15) ) + ˇ g ( R , p , (cid:15) ) ˇ σ (r)sj ( R , p , (cid:15) ) (cid:1) K E F , (F9)where h . . . i F = R d e p π . . . denotes an angular average overall momentum directions at the Fermi surface. We cannow use the expansion of the Green’s functions in spher-ical harmonics, Eqs. (48) and (49); insert the results ofEq. (F7); and perform the angular average. Note that,in general, an additional term emerges when computingEq. (F7). However, this term vanishes when the angularaverage is performed on Eq. (F9) and is consequently ofno interest. With this, we finally obtain the anomalouscorrection to the matrix current in Eq. (61).
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