A Supplementary Material

# Manifestation of extrinsic spin Hall effect in superconducting structures: Non-dissipative magnetoelectric effects

## Abstract

We present a comprehensive quasiclassical approach for studying transport properties of superconducting diffusive hybrid structures in the presence of extrinsic spin-orbit coupling. We derive a generalized Usadel equation and boundary conditions that in the normal state reduce to the drift-diffusion theory governing the spin-Hall effect in inversion symmetric materials. These equations predict the non-dissipative spin-galvanic effect, that is the generation of supercurrents by a spin-splitting field, and its inverse – the creation of magnetic moment by a supercurrent. These effects can be seen as counterparts of the spin-Hall, anomalous Hall and their inverse effects in the superconducting state. Our theory opens numerous possibilities for using superconducting structures in magnetoelectronics.

The spin-orbit coupling (SOC) in normal systems is at the basis of striking magnetoelectric effects, such as the spin (SHE)(Sinova et al., 2015) and anomalous (AHE)(Nagaosa et al., 2010) Hall effects widely studied in normal systems (Zutic et al., 2004). What are the counterpart of these effects in the superconducting state is, in several aspects, still an open question.

According to its origin the SOC can be classified as intrinsic or extrinsic. Intrinsic SOC generates from the crystal potential associated with the electronic band structure, and in superconducting structures, in analogy with the normal state, might lead to non-dissipative magnetoelectric and spin-galvanic effects as shown in theoretical studies (Edelstein, 1995; Bergeret and Tokatly, 2015; Konschelle et al., 2015; Mal’shukov and Chu, 2008; Mal’shukov et al., 2010). In contrast, extrinsic SOC originates from a random potential due to impurities. Its influence on the thermodynamics of bulk superconductors was studied long ago by Abrikosov and Gorkov (AG) (Abrikosov and Gorkov, 1962), who explained non-vanishing magnetic susceptibility of superconductors at zero temperature. The AG model has been used later to describe the physics of superconductor-ferromagnet (S-F) structures with SOC. Within this model, the SOC acts only as a relaxation term for the spin in the normal and for triplet correlations in the superconducting state. The suppression of triplet correlations in S-F-S junctions is associated with the suppression of oscillatory behavior of the critical Josephson current (Demler et al., 1997; Bergeret et al., 2005).

It is well established in the theory of normal systems that SOC not only leads to spin relaxation, but also to the coupling between spin and charge currents, responsible for extrinsic SHE and AHE. One expects that this coupling translates to a singlet-triplet coupling in the superconducting state, by analogy to the case of non-centrosymmetric superconductors with intrinsic SOC(Bergeret and Tokatly, 2014). However, for superconductors with extrinsic SOC this coupling has never been considered, and there is no theoretical framework for its description.

In this letter we address this issue and derive from a microscopic model a diffusion equation for superconducting structures with extrinsic SOC. This equation, Eq.(5), generalizes the well known Usadel equation and contains not only the usual relaxation term due to the SOC, but also a coupling between spin and charge degrees of freedom that lead to the SHE and AHE in the normal case. By using the derived equations we demonstrate that the charge-spin coupling indeed translates in the superconducting state into singlet-triplet coupling. Moreover, our equations also show that the lack of a macroscopic inversion symmetry due, for example, to the presence of hybrid interfaces, leads to magnetoelectric effects. An example of these is a magnetic moment induced by a supercurrent. Inversely, SOC leads to the creation of a supercurrent when the system is polarized via the exchange field of a ferromagnet. In the latter case the magnitude of the induced supercurrent is, as the anomalous Hall voltage, proportional to , where is the SH-angle.

Basic equations for diffusive superconductors with extrinsic SOC.- We first explain how to derive the generalized Usadel equation and boundary conditions that allow for an accurate description of superconducting diffusive structures with extrinsic SOC(13). Following the standard derivation of the quasiclassical equations (see e.g. (Langenberg and Larkin, 1986)) the starting point is the kinetic equation for the Wigner transformed Keldysh matrix Green’s function ,

 τ3∂tˇG+∂t′ˇGτ3+pkm∂kˇG+i[hστ3+ˇΔ,ˇG]=I (1)

where is the spin-splitting field, is the anomalous self-energy (SE) describing superconducting correlations, and are Pauli matrices spanning the Nambu and spin spaces, respectively. The collision integral in Eq.(1) describes scattering at impurities,

 I=−i[ˇΣ,ˇG]+12{∇rˇΣ,∇pˇG}−12{∇pˇΣ,∇rˇG} (2)

where we performed the standard gradient expansion. We describe impurities by an operator , with being a random scalar potential, the SOC term, and the effective Compton wavelength. Within the Born approximation the SE in Eq.(2) is the Wigner transform of where the angular brackets denote averaging over impurities configuration. In we identify two types of terms: (i) those quadratic in the potentials, and , which lead to the relaxation of momentum and spin, respectively, and (ii) the mixed terms that account for the charge-spin coupling. The last terms are traditionally disregarded in the quasiclassical kinetic theory of superconductors (Alexander et al., 1985; Demler et al., 1997; Volkov et al., 2007). The importance of mixed terms has been recognized in the context of spin transport in normal conductors s (Raimondi and Schwab, 2010; Shen et al., 2014) where they are responsible for the extrinsic SHE and the spin current “swapping”. Our goal is to incorporate these magnetoelectric effects into the quasiclassical theory of diffusive superconductors, which requires reconsideration of the standard derivation procedure of the quasiclassical equations.

To consistently catch the charge-spin coupling one needs to include gradient terms in the collision integral Eq.(2). This brings momentum derivatives of the GF which do not allow for a straightforward integration over the particle energy to derive the Eilenberger equation for the quasiclassical GF that depends on the direction of the Fermi momentum. In order to overcome this difficulty we first obtain from Eq.(1) equations for the zeroth and first moments of the GF. At this level one can introduce the quasiclasical GF and consider directly the diffusive limit in which is approximated as , where is the isotropic part and is the leading anisotropic correction. The anisotropic part determines the “matrix current”

 ˇJk=vFgk−λ2pF4τϵkja{σa,[g,gk]}, (3)

where the second term is the “anomalous velocity” contribution due to SOC, and is the momentum scattering time. The physical charge and spin currents are obtained from the Keldysh component of the matrix current, and , respectively. In the diffusive limit one can solve the equation for the 1st moment and one finds the anisotropic component that allows to express the matrix current in terms of the isotropic part of GF

 ˇJk=−D(ˇg∂kˇg−θ2ϵkja{σa,∂jˇg}+iκ2ϵkja[σa,ˇg∂jˇg]). (4)

Here is the diffusion coefficient. In addition to the usual diffusion current, Eq. (4) contains the expected SH-angle and the swapping term first described in Ref.(Lifshits and Dyakonov, 2009). From the equation for 0th moment of the full GF we find that the isotropic component of the GF subjected to the normalization condition , satisfies the generalized Usadel equation,

 τ3∂tˇg+∂t′ˇgτ3+∂kˇJk+i[hστ3+ˇΔ,ˇg] (5) =−18τso[σaˇgσa,ˇg]+14Dθϵkja[σa,ˇg∂kˇg∂jˇg]

where . Finally, the Kupriyanov-Lukichev boundary conditions (Kupriyanov and Lukichev, 1988) at the interface between a conventional BCS superconductor and a metal with extrinsic SOC can be easily generalized by using the matrix current of Eq.(4),

 νkˇJk=D2Rbσ0[ˇgBCS,ˇg], (6)

where is a unit vector normal to the interface, is the barrier resistance per area, the conductivity of the normal region and is the bulk superconductor GF.

Equations (4)-(6) are the main results of this paper. They describe the proximity effect in materials with extrinsic SOC. Despite the derivation relies on Born approximation, where only the side-jump contribution to the SH-angle appears (Raimondi and Schwab, 2010; Shen et al., 2014), the final set of Eqs.(4)-(6) is expected to be quite general with and being the material parameters accounting for all extrinsic and intrinsic (in cubic materials) contributions to the charge-spin coupling. In fact, Eq.(4) can be viewed as a symmetry based gradient expansion of the current.

In the normal state the terms proportional to and vanish from Eq.(5). These nonlinear in terms do appear only if superconducting correlations are present and may lead to new interesting unexplored phenomena.

Non dissipative magnetoelectric effects.- We now discuss physical effects predicted by Eqs.(5)-(6). For clarity we assume a weak superconducting proximity effect and linearize the Usadel equation. Moreover, we focus here on non-dissipative physics, and switch to the equilibrium Matsubara formalism by replacing in Eq.(5) , the Matsubara frequency. After linearization the Usadel equation in non-superconducting regions reads

 D∇2^f−{[|ω|+ihσsgn(ω)],^f}=3^f−σa^fσa4τso, (7)

where is the anomalous GF which describes the induced superconducting condensate and consists of the singlet and odd-frequency triplet components. The linearized boundary condition (6) now reads

 νk(∂kfs−θϵkja∂jfat)=iγfBCS (8) νk(∂kfat−θϵkja∂jfs−κ[∂afkt−δka∂jfjt]) = 0 (9)

where and As we can see from Eqs.(7)-(9) the effect of SOC is twofold. On the one hand, the extrinsic SOC leads to the known additional relaxation of the condensate (via the Elliot-Yaffet mechanism), described by the right hand side of Eq.(7), if the triplet component is non vanishing. On the other hand, the SOC induces, out of the singlet, the triplet component at the hybrid interfaces, even in the absence of the exchange field . The term in Eqs. (8), (9) proportional to the SH-angle describes the singlet-triplet conversion, which is the analog to the charge-spin conversion in normal metals. This conversion can be understood as a consequence of inversion asymmetry at the hybrid interface. Due to the antisymmetric tensor in the SH term the singlet-triplet conversion occurs only in setups with currents flowing parallel to the interfaces, as for example lateral Josephson junctions that will be discussed below.

As a first example we consider a superconducting film with extrinsic SOC in the absence of the exchange field, . The film occupies the region and is infinite in the -plane. The region is occupied by vacuum and hence the boundary condition at is obtained by assuming zero current, i.e. the r.h.s of Eq.(6) vanishes. We assume a small gradient of the superconducting phase along , so that the singlet component of the anomalous GF is given by . The triplet component can be easily obtained from Eq.(7) and Eq.(9) which for the present geometry read . Despite the film is nonmagnetic (), the -component of the triplet is generated due to a finite SH-angle , and this leads to a finite magnetic moment :

 my(z)=μBθT∑ωjx(ω)Dksinhkzcoshkd/2, (10)

where is the spectral supercurrent, and with and . The induced magnetization Eq.(10) is opposite at opposite sides of the film so that the net magnetic moment is zero, which is a clear consequence of the inversion symmetry. The supercurrent-induced accumulation of the odd-frequency triplet component and the spin density at the film edges is the non-dissipative analog of extrinsic SHE.

Let us now consider a normal metal layer (N) of thickness and finite SH-angle , in contact with a bulk superconductor. The N and S layers occupies the region and respectively. We assume a supercurrent flowing within the S layer due to a small phase gradient . Because of the proximity effect the singlet component penetrates N where it is converted to a triplet component due to the SH term in the boundary conditions. Both singlet and triplet components can be easily determined from Eq.(7) and the boundary conditions at the S/N interface Eqs.(8), (9). The induced magnetic moment is then given by:

 my(z) = μBγ2θT∑ωjx(ω)coshkω(z−d)Dk2ωksinh2kωdsinhkd × [coshkz−coshkωdcoshk(z−d)]

Thus the supercurrent flowing in the S layer induces a spin density over the whole N layer. In contrast to our previous example, now the net magnetization is nonzero. In other words, the supercurrent generates a global spin, which is allowed due to the structure inversion asymmetry of the S/N bilayer. Phenomenologically this can be described as a non-dissipative Edelstein effect (EE). The important difference with the usual EE (Edelstein, 1995; Bergeret and Tokatly, 2015; Konschelle et al., 2015) is that it originates solely from the extrinsic SOC and the macroscopic asymmetry of the structure.

Experimentally it might be easier to detect the inverse of this effect. Namely, the generation of supercurrents by a combination of SOC and exchange field, which is our third example. We consider a multi-terminal lateral S/F structure (Fig. 1) which resembles lateral structures used in experiments on SFS structures(23); (25); (24); (26) The -th S terminal is infinite in -direction and has a width , while F is a ferromagnet with an exchange field along . The current density flowing through the -th S/N interface is readily obtained from Eq.(6):

 j(n)z(x)=πTeRbn∑ωIm[f(n)0(x)f∗s(x,0)], (11)

here is the GF of the -th S electrode with the phase , and is the singlet component induced in N at . If all phases are identical, e.g. for all , only the real part of the singlet GF in N contributes to the current as in this case . From Eqs.(7)-(9) we find that only for simultaneously non-vanishing and the component can be generated as follows: Due to the proximity effect a purely imaginary “primary” is induced in F, where it is converted, via the exchange coupling term in Eq.(7), into the real triplet . Finally, is converted into via the SH term, , in Eq.(8). Since the SH singlet-triplet coupling involves gradients, it is clear that will be generated near inhomogeneities – the edges of the S terminals. In general the function can be written as follows

 fRes(x)=k2hθM∑n=1γn[s(x−xn)−s(x−xn−Wn)] (12)

where is the number of terminals, , , and is a function localized near the origin and describing the singlet component induced at the left/right edges of each S electrode. In the limit of thick, formally semi-infinite F layer we find (see SM for details)

 s(x)fBCS = k2−−k2ωk+(k2+−k2−)2e−k+|x|+k2+−k2ωk−(k2+−k2−)2e−k−|x| − 2k2soπ2(k2+−k2−)2∫∞−∞dx′K0(k+|x−x′|)K0(k−|x′|)

where , and is the modified Bessel function of the second kind.

In the one-terminal case () the right hand side in Eq. (47) is antisymmetric with respect to the center of the terminal. Therefore the current Eq.(11), being also antisymmetric, averages to zero after the integration over . In other words, in a one-terminal S/N lateral structure, the combination of the extrinsic SOC and the exchange field generates circulating currents as sketched in Fig. 1b.

In the two-terminal case, shown on Fig. 1c, the total current flowing through S1-terminal is nonzero due to induced from S2-terminal:

 I1 = πk2hθTeRb1Rb2σ0∑ωfBCS × ∫x1+W1x1dx[s(x−x2)−s(x−x2−W2)]

Therefore besides currents circulating around each interface, there is a finite Josephson current induced by mutual effect of extrinsic SOC and the exchange field (see Fig. 1c). This supercurrent at resembles the anomalous current in a -junction studied in the context of intrinsic SOC in polar crystals (Bergeret and Tokatly, 2015; Konschelle et al., 2015; Buzdin, 2008). Here we show that -junction can be built out of the most common inversion symmetric materials provided they show a finite exchange spin-splitting and a SH-angle. The anomalous current is proportional to , which in turn is proportional to the anomalous Hall conductivity in ferromagnets (Nagaosa et al., 2010). Hence F materials with large are good candidates for showing an anomalous supercurrent in lateral SFS structures. If for example one uses a ferromagnet with strong exchange field such that , the amplitude of the anomalous current is according to our theory proportional to times the critical current of the junction.Thus, for materials with the anomalous phase current can be detected by using quantum interferometer devices as done for example in Ref.(30) for nanowires.

In the right panel of Fig. 2 we show the anomalous current through the S1-terminal as a function of the field . The current starts from zero at , reaches a maximum and finally decays for large fields because of the usual suppression of superconductivity by the field (Buzdin, 2005; Bergeret et al., 2005). Inversely, if a finite Josepshon current () between the two S electrodes induces a finite magnetic moment (see SM for details) similar to the situation found in the S and S/N layered systems. In the left panel of Fig. 2 we show the -dependence of the magnetic moment induced at .

In conclusion, we have presented a new theoretical framework that describes diffusive superconducting hybrid structures with extrinsic SOC. We have derived equations that contain hitherto unknown terms proportional to the SH-angle, responsible in the normal state for the SHE, and the Lifshits-Dyakonov spin-currents swapping parameter. Our equations pave the way to explore numerous novel effects in the field of superconducting spintronics (27); (28); (29) and open up numerous opportunities for the control of charge and spin currents in the non-dissipative regime. As illustrative examples we demonstrate the existence of magnetoelectric effects in different superconducting structures. We show that these effects are proportional to the SH-angle and hence can be observed by combining materials with known large , like Pt or Co, with superconducting electrodes.

###### Acknowledgements.
The work of F.S.B. was supported by Spanish Ministerio de Economía y Competitividad (MINECO) through the Project No. FIS2014-55987-P and the Basque Government under UPV/EHU Project No. IT-756-13. I.V.T. acknowledges support from the Spanish Grant FIS2013-46159-C3-1-P, and from the “Grupos Consolidados UPV/EHU del Gobierno Vasco” (Grant No. IT578-13)

## Appendix A Supplementary Material

### a.1 Derivation of the Usadel equation in the presence of extrinsic spin-orbit coupling

In this section we derive the generalised Usadel equation to account for magnetoelectric effects [Eq. (5) in the main text]. We consider a diffusive conventional superconductor described by the Hamiltonian

 H=HBCS−hσ+^W(r), (13)

where is the usual mean field BCS Hamiltonian, is the exchange field, the Pauli matrices, and is a random impurity potential

 ^W(r)=V(r)+^VSOC(r). (14)

It consists of the usual (scalar) elastic scattering and the spin-orbit part

 ^VSO=λ2(∇V(r)×^p)σ, (15)

where the coupling constant is proportional to the effective Compton wavelength squared and the momentum operator. In order to derive the quantum diffusion equation we introduce the Keldysh matrix Green functions (GF) which is the matrix

 ˇG=(GRGK0GA),

consisting of the retarded, advanced and Keldysh 44 matrices () in the Nambu-spin space. obeys the equationLangenberg and Larkin (1986)

 [τ3i∂t+12m∂2r+μ+hσ+ˇΔ−ˇΣ]ˇG(r,t;r′,t′)=δ(r−r′)δ(t−t′), (16)

where is the chemical potential, the superconducting order parameter and is the self-energy due to the impurity scattering, Eq. (14). We treat the latter within the self-consistency Born approximation, i.e.

 ˇΣ(r1,r2)=⟨^W(r1)ˇG(r1,r2)^W(r2)⟩=ˇΣ0+ˇΣ1+ˇΣ2,

where denotes average over the impurity configuration. The three terms on the r.h.s correspond to the following contributions:

The usual elastic scattering term

 Σ0=⟨V(r1)ˇG(r1,r2)V(r2)⟩, (17)

the spin relaxation term which, quadratic in the SOC potential

 Σ2=⟨^VSO(r1)ˇG(r1,r2)^VSO(r2)⟩ (18)

and the “mixed” term

 Σ1=⟨V(r1)ˇG(r1,r2)^VSO(r2)⟩+⟨^VSO(r1)ˇG(r1,r2)V(r2)⟩ (19)

which is responsible for the coupling between charge and spin degrees of freedom and leads to the SHE and AHE. As usual, we assume for the random potential

 ⟨V(r1)V(r2)⟩=12πNFτδ(r1−r2), (20)

where is the momentum relaxation time.

We follow the usual steps in order to obtain the quantum kinetic equation from Eq. (16)Langenberg and Larkin (1986): (1) One subtracts from Eq. (16) its conjugate, (2) performs the Wigner transform and then (3) the gradient expansion. After these steps are carried out one obtains the kinetic-like equation [Eq. (1) in the main text]:

 pkm∂kG+iτ3∂tG−i∂t′Gτ3+i[hστ3,G]=I, (21)

where is the collision term. It consists of three contributions corresponding to the three self-energy terms (17-19). and can be treated in the lowest order of the gradient expansion. In contrast and in order to catch consistently the charge-spin coupling we need to include linear terms of in the gradient expansion:

 I1(r,p)=−i[ˇΣ1p,ˇGp]+12{∂rˇΣ1p,∂pˇGp}−12{∂pˇΣ1p,∂rˇGp}…. (22)

The derivation of the quasiclassical expressions for follows the standard steps, and hence those terms will be added straightforwardly in the end equation. Here we focus on the term and how to include it in the quasiclassical formalism.

We start by writing explicitly the self-energy Eq. (19):

 Σ1=−i⟨Aaj(r1)V(r2)⟩∂rj1σaG(r1,r2)+i⟨Aaj(r2)V(r1)⟩∂rj2G(r1,r2)σa, (23)

where

 Aaj(r1)=λ2ϵkja∂rk1V(r1),

is the Levi-Civita tensor, and sum over repeated indices is implied.

By using Eq. (20) one obtains:

 Σ1=−iλ22πNFτϵkja∂rk1δ(r1−r2)σa∂rj1G(r1,r2)+iλ2ϵkja2πNFτ∂rk2δ(r1−r2)∂rj2G(r1,r2)σa.

Now we Wigner-transform this expression. This implies to go over the relative and center of mass coordinates and to Fourier- transform with respect to :

 Σ1(r,p)=−iλ2ϵkja2πNFτ∑p′∫dρe−iρ(p−p′)∂ρkδ(ρ)(ip′j[σa,G(p′)]+12∂rj{σa,G(p′)}). (24)

By noticing that and that the Green’s functions are peaked at the Fermi level we can express in terms of the quasiclassical GFs as:

 Σ1=λ2ϵkja2τ[σa,pk⟨pjg⟩]−iλ2cϵkja2τ∂rj{σa,pk⟨g⟩−⟨pkg⟩}. (25)

In this last expression the brackets denote average over the momentum direction. It is important to note that the second term contains a gradient and hence it is, in principle, of smaller order than the first one in the gradient expansion. As noticed before, the description of the spin-charge coupling compels to keep these higher order terms.

We substitute now Eq. (25) into the expression for the collision term Eq. (22) and keep terms up to linear order in the gradients:

 Missing or unrecognized delimiter for \right (26)

where are the first and second term in Eq. (25) respectively. We emphasise once again that in order to get the next-leading order correction correctly it is crucial to keep all terms in the expansion Eq. (22).

The collision term described by Eq. (26) does not allow for a straightforward integration over the quasiparticle energy and hence one cannot derive a closed differential equation (Eilenberger equation) for the quasiclassical . In order to overcome this difficulty we consider the diffusive limit and derive equations for the zeroth and first moments of .

From Eq. (26) we obtain :

 iπNF∑pI1=ϵkjaλ24τ(i{σa,[⟨pkg⟩,⟨pjg⟩]}+∂k{σa,[⟨g⟩,⟨pjg⟩]}−[σa,{⟨g⟩,∂k⟨pjg⟩}]), (27)

for the zeroth moment of and

 iπNF∑ppkI1=−ϵkjaλ2p2F2τ13(i[[σa,⟨pjg⟩],⟨g⟩]+12[{σa,∂j⟨g⟩},⟨g⟩]) (28)

for the first moment.

In the diffusive limit one assumes that and ( is any energy involved in the kinetic equation) and expands in spherical harmonics: such that , , and , . In this limit one can simplify expressions (27-28) and get:

 iπNF∑pI1≈ϵkjaλ2pF4τ(∂k{σa,[g0,gj]}−[σa,{g0,∂kgj}]) (29)

and

 iπNF∑ppkI1≈−ϵkjaλ2p2F4τ13(i[[σa,gj],g0]+12[{σa,∂jg0},g0]). (30)

By switching to the Matsubara representation in Eq. (16) we obtain for the zero and first moments

 Missing or unrecognized delimiter for \left (31)

and

 τvF3∂kg0+ϵkjaλ2pF413[{σa,∂jg0},g0]=−12[g0,gk]+ϵkjaλ2p2F2i3[[σa,gj],g0], (32)

where , and in the second equation we only took leading order terms in the diffusive expansion. At this stage and before writing the Usadel equation, it is worth to make two remarks: (i) The first term in Eq. (31) is the divergence of the matrix current

 ˇJk=vFgk−ϵkjaλ2pF4τ{σa,[g0,gj]}. (33)

The last term of this expression stems form the SOC and described the coupling between the charge and spin currents. (ii) The structure of Eq. (32), , ensures the validity of the normalization condition

 g20=1. (34)

The final step is to get an expression for the anisotropic component in terms of the isotropic one from Eq. (32). In leading order with respect to the parameters and , where is the characteristic length over which varies, the anisotropic component reads

 gk=−τvFg0∂kg0+ϵkjaλ2pF2{σa,∂jg0}−ϵkjaλ2p2Fi3[σa,τvFg0∂jg0]. (35)

This can be checked by substituting Eq. (35) into Eq. (32), using the normalization condition (34), and by keeping only leading order terms. If we now substitute this expression for into Eq. (33) we obtain the expression of the matrix current in terms of :

 Missing or unrecognized delimiter for \left (36)

Here is the spin-Hall angle defined as and the “swapping” term Lifshits and Dyakonov (2009).

Finally, by substituting Eq. (LABEL:gani) into Eq. (31) we obtain the Usadel equation:

 ∂kJk+[(ω−ihσ)τ3,g0]=−18τSO[σag0σa,g0]+ϵkjaλ2cpF4vF[σa,{g0,13∂kg0∂jg0}]. (37)

Terms with two derivatives acting on the same , i.e. after summation over indices vanish because of the antisymmetric tensor . By substitution of Eq. (36) into Eq. (37) and going back to the real times representation one obtains Eq. (5) of the main text.

The generalisation of the Kupriyanov-Lukichev boundary condition at hybrid interfaces is straightforward from the current expression Eq. (36) (we omit here the index in ):

 ˇg∂kˇg−θSHϵkja{σa,∂jˇg}+iκswϵkja[σa,ˇg∂jˇg]=−12RbσF[gBCS,^g], (38)

Observables like the charge current and magnetic moment can be expressed in terms of the quasiclassicla Green’s functions:

 jk=iπT16eσF∑ωTrτ3ˇJk (39)

and

 ma=μBiπN0T4∑ωTrτ3σaˇg. (40)

We should notice that in the normal case Eq. (37) simplifies drastically: First the retarded and advanced GFs equals to respectevely and hence there is only on equation for the Keldysh component which in such a case consist on the charge and spin distribution functions . Second the equation can be straightfoirwarly integrated over energies and hence instead of writing the equations for , one write them for the charge and the spin density and . In particular by simple

### a.2 Solution of the Usadel equation for a lateral multi-terminal S-F structure

Let us consider the geometry shown if Fig. 1 of the main text and calculate the current through the -th S/F interface, which is given by Eq. (11) in the main text. Thus, we need to determine the real part of the singlet component of the condensate induced in N. In the geometry under consideration with an exchange field in direction, the anomalous GFs depends on two coordinates and . It is convenient to introduce the Fourier component with respect to ,

 ^f(x,z)=∫dqeiqx^f(q,z)

The singlet and triplet components then satisfy the following equations,

 ∂2zzfs(q,z)−(k2ω+q2)fs(q,z)−ik2hft(q,z) = 0 (41) ∂2zzft(q,z)−(k2ω+q2+k2so)ft(q,z)−ik2hft(q,z) = 0 (42)

with boundary conditions at

 ∂zfs(q,0)−iqθft(q,0) = ifBCSF0(q) (43) ∂zft(q,0)−iqθfs(q,0) = 0, (44)

where is the Fourier transform of the r.h.s of the boundary condition at the S-electrodes described by

 F0(x)=∑γneiφn[Θ(x−xn)−Θ(x−xn−Wn)],

and . Let us assume that for all S terminals. According to Eq. (11) in the main text, to obtain the current through the nth S/N boundary we only need to calculate the real part of the singlet component, at the S/F interface (). One can straightforwardly verify from Eqs. (41)-(44) that in the linear order in the Fourier component of is given by

 fRes(q)=ik2hθqF0(q)s(q),

where

 s(q)fBCS=1(k2+−k2−)2⎡⎢ ⎢⎣k2−−k2ω2(q2+k2+)+k2+−k2ω2(q2+k2−)−2k2SO√q2+k2+√q2+k2−⎤⎥ ⎥⎦, (45)

and . Thus, can be obtained by transforming back

 fRes(x)=k2hθ∫dx1[∂x1F0(x1)]s(x−x1) (46)

Since is a combination of step functions its spatial derivative gives a sum of delta-functions, thus:

 fRes(x)=k2hθ∑nγn[s(x−xn)−s(x−xn−Wn)]. (47)

The inverse Fourier transform of the function , Eq. (45) can be written explicitly as

 s(x)fBCS=k2−−k2ωk+