Theory of the spin-galvanic effect and the anomalous phase-shift $\varphi_{0}$ in superconductors and Josephson junctions with intrinsic spin-orbit coupling

Due to the spin-orbit coupling (SOC) an electric current flowing in a normal metal or semiconductor can induce a bulk magnetic moment. This effect is known as the Edelstein (EE) or magneto-electric effect. Similarly, in a bulk superconductor a phase gradient may create a finite spin density. The inverse effect, also known as the spin-galvanic effect, corresponds to the creation of a supercurrent by an equilibrium spin polarization. Here, by exploiting the analogy between a linear-in-momentum SOC and a background SU(2) gauge field, we develop a quasiclassical transport theory to deal with magneto-electric effects in superconducting structures. For bulk superconductors this approach allows us to easily reproduce and generalize a number of previously known results. For Josephson junctions we establish a direct connection between the inverse EE and the appearance of an anomalous phase-shift $\varphi_{0}$ in the current-phase relation. In particular we show that $\varphi_{0}$ is proportional to the equilibrium spin-current in the weak link. We also argue that our results are valid generically, beyond the particular case of linear-in-momentum SOC. The magneto-electric effects discussed in this study may find applications in the emerging field of coherent spintronics with superconductors.

All the above mentioned phenomena in S/F structures originate from the interaction between the superconducting correlations and the exchange field of the ferromagnet. However it has recently been shown that spin-orbit coupling (SOC) in S/F structures will also lead to, for example, a long-range triplet component [45,46] and peculiarities in the density of states [47][48][49]. On the other hand, transport properties of non-superconducting structures with strong SOC are being intensively studied because of their potential application in a novel direction of spintronics, which exploits the coupling between spin and charge currents [50][51][52][53].
In particular, the SOC in semiconductors and normal metals is at the root of a number of interesting phenomena that originate from the coupling between the charge and spin degrees of freedom. Prototype of these phenomena is the spin Hall effect (SHE) [54][55][56][57][58][59][60][61][62][63][64] which consists in the creation of a spin polarized current by an electric field. Reciprocally, by means of the inverse SHE a spin current can create an electric field [65][66][67]. These effects allow to generate and detect spin polarized currents in non-magnetic materials [68][69][70][71][72].
There is also another relevant effect in normal systems related to the SOC. It consists in creating a stationary spin density S a , along the a-direction in response to an electric field E k applied in k-direction [73,74]. Within arXiv:1506.02977v3 [cond-mat.mes-hall] 5 Oct 2015 linear response, this effect is described by S a (ω) = σ a k (ω) E k (ω) , (1.1) where the sum over repeated indexes is implied here, and throughout this paper. In particular, in 2D systems with Rashba SOC, the applied electric field and the generated spin density are perpendicular to each other. This magneto-electric effect, also called the Edelstein effect (EE), has been observed in experiments [60,75]. The Edelstein conductivity σ a k (ω) in Eq.(1.1) is related to the Kubo correlator χ a k (ω) = Ŝ a ;ĵ k ω of the spin and current operators via σ a k (ω) = χ a k (ω) /iω [76]. Because of the gauge invariance in normal systems the function χ a k (ω) should vanish in the limit ω → 0 reflecting the fact that there is no response to a static vector potential. Therefore the σ a k (0) = σ a k remains finite and describes the dc EE. It has been pointed out in Ref. [76] that this property, together with the Onsager reciprocity principle, implies that the inverse dc EE, also referred to as the spin-galvanic effect, consists in generating a charge current j k by a steady spin generation induced by a timedependent magnetic field via the paramagnetic effect: with the Landé g-factor, µ B the Bohr magneton, anḋ B a the time derivative of the magnetic field along the a-axis. The inverse EE effect has also been observed in experiments [77,78]. Similar magneto-electric and spin-galvanic effects should also exist in superconductors [79,80]. However, there the physical situation is different because in the presence of the superconducting condensate the gauge invariance does not forbid the existence of a finite static current-spin response function χ a k . In contrast to the normal case, in a superconductor an equilibrium electric (super-)current can flow in the absence of an external electric field. The supercurrent j = n s v s (here n s is the density of superconducting electrons and v s the superfluid velocity) is proportional to the gradient of the macroscopic gauge-invariant phase ∇φ = ∇ϕ − eA ∼ v s , which is the physical field coupled to the current operator in the Hamiltonian of a superconductor. The existence of such a gauge-invariant field implies that the static response function χ a k = Ŝ a ;ĵ k ω=0 can be nonzero without violating the gauge invariance. In principle, a supercurrent can thus generate an equilibrium spin polarization according to the general linear response relation: where ∂ k = ∂/∂x k . This effect has been indeed theoretically demonstrated by Edelstein for a 2D superconductor with Rashba SOC, who calculated the proportionality tensor χ a k at temperatures T close to the critical superconducting temperature T c , in both pure ballistic [79] and diffusive [80] limits.
Because in the superconducting state the response function χ a k at ω → 0 is finite, the reciprocity of the EE effect becomes complete. In contrast to the normal case, in superconductors a static Zeeman field B can induce a supercurrent j k . Therefore, instead of Eq. (1.2) the following relation holds where h a = gµ B B a . An explicit expression of this type has been obtained in a particular case of a 2D ballistic superconductor with intrinsic Rashba SOC [81,82]. It is then clear that the free energy of a superconductor with a SOC must have a term of the Lifshitz-type and equations (1.3)-(1.4) follow directly from the general thermodynamic definitions of the spin and current densities, S = δF/δh and j = −δF/δA. In principle, equations (1.3) and (1.4) apply for bulk superconductors, but one can expect similar effects to occur also in a S-X-S Josephson junction, between two massive superconductors (S) and a normal or ferromagnetic bridge X with an intrinsic SOC. In a Josephson junction the supercurrent depends on the phase difference ϕ between the superconducting electrodes. In the particular cases of a weak proximity effect between the S and the X, or in the high-temperature regime (T T c ), the current phase relation is given by j = j c sin ϕ, where j c is the critical Josephson current.
When the SOC competes with a Zeeman effect, the natural conjectures following Eqs. (1.3)-(1.4) are: (i) In accordance with Eq. (1.3), the flow of a supercurrent may generate a spin polarization in the X bridge (the Edelstein effect); (ii) In turn, from Eq. (1.4), a Zeeman (spin-splitting) field may induce a supercurrent through the junction, even if the phase difference between the electrodes vanishes (the inverse Edelstein effect).
In the present work we develop a complete theory of the magneto-electric and spin-galvanic effects in hybrid superconducting structures and confirm the above conjectures. We focus on systems with linear in momentum SOC that can be conveniently described in terms of an effective background SU(2) gauge field. This allows us to use the SU(2) covariant quasiclassical equations for the Green's functions derived in Ref. [45,46,123]. We establish a connection between the tensor χ a k in Eqs. (1.3-1.4) and the equilibrium spin-current J a j [124,125]. We show that in a generic S-X-S Josephson junction the condition for a nontrivial anomalous phase ϕ 0 to appear is that J a j h a = 0, where h a can be either an external Zeeman field or the internal exchange field of a ferromagnet. Our SU(2) covariant formulation results in a simple and tractable system of equations to describe hybrid structures with arbitrary linear in momentum SOC, temperatures, degree of disorder, and quality of the hybrid interfaces. We also show that qualitatively our results are generically valid beyond the particular case of the linear in momentum SOC.
The structure of the paper is the following: In the next section we present a qualitative discussion of the superconducting proximity effect in structures with SOC and its connection with the spin diffusion in normal systems. This qualitative analysis allows us to guess the form of the quasiclassical equations for superconducting structures in the presence of generic spin fields, and in particular to explicitly show the analogy between the charge-spin coupling in normal systems and the singlet-triplet coupling in superconducting ones. In section III we present our model, discuss the associated symmetries, and derive microscopically the quasiclassical equations for generic linear in momentum SOC. In section IV we use the derived equations to explore the magneto-electric effects in bulk superconductors. We generalize the previously known results for the EE and its inverse obtained for 2D Rashba SOC [79][80][81]126] to generic linear-in-momentum SOC, and relate them to the spin current and the SU(2) gauge-fields. In section V we explore the Josephson effect through a S-X-S diffusive junction and in section VI through a ballistic one. In both cases we show that the anomalous phase ϕ 0 is proportional to J a i h a and determine its dependence on other parameters of the structure, like temperature and length. We finally present our conclusions and discuss possible experimental setups to verify our predictions in Section VII.

II. DIFFUSION OF SUPERCONDUCTING CONDENSATE IN THE PRESENCE OF SPIN-ORBIT COUPLING: HEURISTIC ARGUMENTS
Before presenting the full quantum kinetic theory it is instructive to discuss at the qualitative level the main features of the proximity induced superconductivity in the presence of an intrinsic SOC. For this sake we present a simple heuristic derivation of the equations describing the coupled motion of the singlet and triplet components induced in a ferromagnet from a bulk s-wave superconductor.
Let us consider a S-X-S junction, where X is a diffusive ferromagnet. We assume that the system is at equilibrium, and that the proximity effect between S and X is weak. In such a case the junction is fully described by the quasiclassical anomalous Green functionf (r), which describes the superconducting condensate in X. In general f (r) is a 2 × 2 matrix in the spin spacef = f s1 + f a t σ a . Here the scalar f s and the vector with components f a t describe the singlet and the triplet components of the condensate, respectively. In this section we show, that the functions f s (r) and f t (r) are reminiscent of the charge and spin density in the normal systems.
In the absence of SOC, but in the presence of the exchange field h the diffusion of the condensate is described by the well known linearized Usadel equations (see e.g. Ref. [2]), where D is the diffusion constant, and ω n is the Matsubara frequency. The terms proportional to 2 |ω n | are responsible for the decay of the superconducting correlations in the normal metal. The last terms in the left hand sides of Eqs. (2.1)-(2.2) describe the usual singlettriplet coupling coming from the exchange field. It is worth emphasizing the presence of imaginary unit i in the exchange field terms, which reflects the breaking of the time reversal symmetry. Because of this, the singlettriplet conversion due to the exchange field is always accompanied with a phase shift of π/2. This point will be of primary importance in the following for understanding the origin of the anomalous phase ϕ 0 . To understand how the Usadel equations (2.1)-(2.2) are modified in the presence of SOC we recall the description of the diffusion of spin S(r) and charge n(r) densities in normal systems. The general spin diffusion equation in a normal conductor with SOC takes the form, where T a is a so called spin torque. In the absence of SOC, T a = 0 and hence spin is a conserved quantity which satisfies the usual spin diffusion equation. In non-centrosymmetric materials SOC acts as an effective momentum-dependent Zeeman field that causes precession of spins of moving electrons. This precession breaks conservation of the average spin, and shows up formally as a finite torque T a = 0 in Eq. (2.3). In the diffusive regime the motion of the electrons consists of a random motion superimposed on an average drift caused by the density gradients. The spin precession related to these types of motion generate the corresponding contributions to the spin torque. To the lowest order in gradients the general expression for the torque can be written as follows [127][128][129], Here the first term describes the Dyakonov-Perel (DP) spin relaxation that originates from the spin precession of randomly moving electrons [54]. The positive definite matrix Γ ab is the DP relaxation tensor with the eigenvalues equal to the inverse squares of the DP spin relaxation lengths. The other two contributions to the torque are related to the average motion of spins. In particular, the second term in the right hand side of Eq. (2.4) originates from the diffusive motion of spins caused by inhomogeneities of the spin density distribution. The corresponding spin precession is described by antisymmetric (spin rotation) matrices P ab k = −P ba k with P ∼ 1/ so , where so is the spin precession length.
The last term in Eq. (2.4), which is proportional to the charge density gradient, can be called the spin-Hall torque. The charge density gradient generates the charge current which is then transformed to the spin current via the spin Hall effect. Precession of the spins driven by the charge density gradient, via the spin Hall effect, is the origin of the spin-Hall torque in Eq. (2.4). The spin-Hall torque is parameterized by the tensor C a k which is proportional to θ sH / so , where θ sH is the spin Hall angle -the conversion coefficient between the charge and the spin currents.
Equation (2.3) with the spin torque of Eq. (2.4) is commonly used in spintronics context to describe spin dynamics in semiconductors with intrinsic SOC [127][128][129] (for a discussion between intrinsic and extrinsic SOC, see e.g. [64]). In the stationary case the diffusion equations for the spin and charge densities reduce to It is important to emphasize here that spin-charge coupling mediated by the spin-Hall torque (C a k ) is responsible for the EE. This can be seen directly from Eq. (2.6): A uniform charge density gradient produces a uniform spin density given by S a = (Γ −1 ) ab C b k ∂ k n. We can now construct the Usadel equations in the presence of SOC in analogy to the normal case. Since SOC does not violate the time reversal symmetry it acts in exactly the same way on the time-reversal conjugated states composing the Cooper pair. Therefore the diffusion of the singlet and the triplet condensates should be modified by SOC in complete analogy with the diffusion of the charge and spin densities in normal systems. The formal connection between the diffusion of the triplet condensate function f a t in superconductors and the spin density S a in normal metals has been discussed recently in Ref. [46], and it has been also noticed in Ref. [86]. Hence, in order to include the effects of SOC in the Usadel equations all we need to do is to replace the diffusion operators (the Laplacians) in Eqs. (2.1) and (2.2) with the diffusion operators entering Eqs. (2.5) and (2.6), respectively. The result is the following system of equations describing a coupled diffusion of the singlet and triplet condensates in the presence of SOC, In contrast to the normal case, in addition to the DP relaxation, both the f s and f t experience an additional decay proportional to the inverse decay length κ ω = 2|ω n |/D, due to the finite lifetime of the superconducting condensate in the normal metal.
The most important novel feature of Eqs. (2.7)-(2.8) is the presence of two mechanisms for the singlet-triplet coupling which are described by the two terms in the square brackets. The first mechanism is the above discussed Zeeman coupling related to the modification of the internal structure of the Cooper pair by the spinsplitting field h [see Eqs. (2.1)-(2.2)]. The second channel of singlet-triplet coupling comes from the spin-Hall torque, which converts the gradient of f s into f t and vise versa, in a complete analogy with the EE in normal systems. The corresponding singlet-triplet "conversion amplitudes" have a relative phase shift of π/2, which is related to the different transformation properties of the Zeeman and spin-orbit fields with respect to the time reversal. We will see in the next sections that the interference of these two singlet-triplet conversion channels is indeed responsible for the magneto-electric/spin-galvanic effects in superconductors, and, in particular, for the appearance of the intrinsic anomalous phase ϕ 0 in Josephson junctions.
Although the present heuristic derivation of Eqs. (2.7)-(2.8) may seem imprecise, it uncovers a simple, but deep connection between the physics of inhomogeneous superconductors with SOC and the well known spintronics effects, such as the spin Hall effects and direct and inverse magneto-electric effects (EE). In Sec. III we present a rigorous derivation of the quasiclassical kinetic equations for superconductors with a linear in momentum SOC, which in the diffusive limit confirms the correctness of Eqs. (2.7)-(2.8). In the rest of the article we study in detail the physical consequences of the interference of the two singlet-triplet conversion channels and their connection with the theory of ϕ 0 -Josephson junctions.

III. THE MODEL AND BASIC EQUATIONS
In this section we introduce our model and discuss the symmetries associated with superconducting systems in the presence of spin-orbit coupling (SOC). We also present the derivation of the quasiclassical equations in the presence of linear in momentum SOC.
A. The Hamiltonian in the presence of generic SOC and symmetry arguments for the appearance of an anomalous phase Our starting point is a general Hamiltonian describing a metal or a semiconductor with a linear in momentum SOC, an exchange field and superconducting correlations where ψ ↑,↓ (r) are the annihilation operators for spin up and down at position r, and Ψ † = (ψ † ↑ , ψ † ↓ ) is the spinor of creation operators. H 0 is the free electron part [149] where µ the chemical potential and V imp the potential induced by non-magnetic impurities. The magnetic interactions appear in two places: as a SU(2) scalar potential A 0 ≡ A a 0 σ a /2, describing for example the intrinsic exchange field in a ferromagnet or a Zeeman field in a normal metal, and as a SU (2) vector potential A i ≡ A a i σ a /2, describing the SOC. The latter is associated to the momentum operator [150] In practice, all the linear-in-momentum SOC can be represented as a gauge potential (see e.g. [130] or [131] and references therein). In the widely studied case of a free electron gas with Rashba SOC, in the second term of the r.h.s of Eq. (3.1) describes the coupling strength which gives rise to superconductivity in some regions of space.
In analogy to electrodynamics one can define the fourpotential A µ , with space components (µ = 1, 2, 3 or µ = x, y, z) given by the SOC and the time component (µ = 0) by the Zeeman field. Following the analogy one can define the strength tensor and the electric and magnetic SU(2) fields where ε ijk is the Levi-Civitta symbol.
In normal metals and semiconductors, the SHE and EE are consequences of the existence of a finite SU(2) magnetic field. For a pure-gauge vector potential the SOC can be gauged out [46], the SU(2) magnetic field is zero, and hence the SHE and EE do not appear [151]. Following our analogy, in the superconducting case an anomalous phase can only appear if the SU(2) magnetic field is finite. This explains why S-F-S junction without SOC do not present any magneto-electric effect, or equivalently, no anomalous phase. As it is well known, the ground state of S-F-S junctions corresponds to a phase difference either equal to 0 or to π [1,2,132].
A simple way to describe qualitatively magneto-electric effects in a superconductor is to provide simple symmetry arguments. Let us consider a ballistic superconductor at T close to its critical temperature T c and focus on the Ginzburg-Landau free energy. In such an expansion, a SOC is responsible for the presence of a first-order derivative of the order-parameter, the so-called Lifshitz invariant which describes most of the original phenomenology of non-centrosymmetric superconductors [82]. Assuming that the amplitude of the order parameter is constant but its phase position-dependent, the Lifshitz invariant reads F L ∝ T i ∂ i ϕ where T i is a vector which has to be odd with respect to the time-reversal operation, and SU (2) invariant. As discussed in Ref. [92], to the lowest order in SOC the Lifshitz invariant for a superconductor can be expressed in terms of the SU(2) fields: If we focus on the static case, the electric field is given by F 0j = −∂ j A 0 . Moreover we define the equilibrium spin current [125] in terms of the SU(2) magnetic field as If A 0 is spatially homogenous, for example induced by an external magnetic field, Eq. (3.5) reads [92] This Lifshitz invariant agrees with the ones derived from microscopic considerations [133] or quasi-classic expansions [134] for a particular sort of SOC. Eq. (3.6) confirms our guessed Eq. (1.5) and demonstrates that the Edelstein response tensor χ a k behaves like the spin current tensor J a i . The form of F L in Eq. (3.6), in terms of the equilibrium spin current, suggests that our results remain valid for any momentum dependence of the SOC. We now proceed to derive the quasiclassical equations and provide a microscopic description of the magneto-electric effects in superconductors.

B. The quasiclassical equations in the presence of SOC
In order to describe the transport properties of hybrid structures containing superconducting, normal (N) and/or ferromagnetic (F) layers with interfaces, arbitrary temperature and degree of disorder we have to go beyond the Ginzburg-Landau limit. We present here the quasiclassical equations [135][136][137][138] for the Green's functions in the presence of a non-Abelian gauge-field [45,46,123] (for a similar discussion in normal metal, see [139]).
We follow here the derivation presented in Ref. [46]. The basic transport equation derived from Hamiltonian (3.1) for the Wigner-transformed covariant Green func-tionsǦ (p, r) in the time-independent limit reads : where ω n = 2T π (n + 1/2) is the fermionic Matsubara is the (s-wave) gap parameter of amplitude ∆ and phase ϕ. The scattering at impurities is described within the Born approximation, where τ is the elastic scattering time, ǧ is the GF matrix integrated over the quasiparticle energy, and · · · describes the average over the Fermi momentum direction. After integration of (3.7) over the quasiparticle energy and by using the fact thatǦ is peaked at the Fermi level one obtains the generalized Eilenberger equation [46,92]: where n i , i = x, y, z are the components of the Fermi velocity vector. When deriving (3.8) we have neglected corrections to the exchange term A 0 of the order of |A j | /p F 1. In fact, one sees from (3.7) that , and so it renormalizes the term −i τ 3 A 0 ,Ǧ already present in (3.7). The correction to A 0 is of the order A j /p F 1 and we neglect them from now on.
In the Nambu spaceǧ readš where the g, f are matrices in the spin space which depend on the spaces coordinates x i , the momentum direction n i and the Matsubara frequency. The time-reversal conjugateḡ andf are defined asḡ(n) = σ y g * (−n)σ y and f = σ y f * (−n)σ y . The latter is the anomalous GF which describes the superconducting correlations.
From the knowledge ofǧ one can calculate the charge current (density) with e the electron charge and N 0 the normal density of states for each spin. Whereas the spin polarization is given by C. Linearized quasiclassical equations in diffusive and pure-ballistic limits In the present work we mainly consider two limiting cases: the pure ballistic one in which τ → ∞ and the diffusive limit where τ is a small parameter. The transport equation in the ballistic limit is directly obtained from (3.8) by neglecting the right-hand side. The diffusive limit is a bit more puzzling. Because of the anticommutator in the l.h.s of Eq. (3.8), the normalization conditionǧ 2 = 1 does not hold directly and therefore the usual derivation of the Usadel equations cannot be carried out [140]. There is, however, a way out of this puzzle if one assumes that the amplitude of the anomalous GF's, f in (3.9) is small. Then the matrix GF (3.9) can be written asǧ ≈ sgn (ω n ) This linearization procedure is justified in two cases: either for temperatures close to the critical temperature T c when the amplitude of the order parameter ∆ is small, or in S-X structures when the proximity effect is weak due to a finite interface resistance for arbitrary temperature.
In the diffusive limit one can expand f ≈ f 0 + n k f k + · · · , in angular harmonics where f = f 0 f k . We first average (3.12) over the momentum direction: where dim = 1, 2, 3 is the dimension of the system. Next we multiply Eq.(3.12) by n k and average over the momentum direction to obtain Eqs. (3.13) and (3.14) constitute a closed set of coupled differential equations for f 0 and f k . In particular from Eq. (3.14) we can write f k in terms of f 0 up to terms of second order in τ : Note that the Usadel equation was obtained in several works in the absence of gauge-fields, where one skipped the terms of the order τ 2 . We keep here these terms since they are crucial for the description of magneto-electric effects [86,92,141].
The equations can be further simplified by noticing that the anti-commutator in the second line of Eq. (3.15) can be written as In virtue of (3.13), the first term in the right-hand-side of the last equation is in fact of order τ and so this term in (3.15) is of order τ 3 and can be neglected. The second term reads∇ k A 0 = −i [A k , A 0 ] = F k0 for a spaceindependent gauge-potential. This electric field renormalizes the paramagnetic effects A 0 , and is neglected in the following. Finally, we replace (3.15) into (3.13) to obtain the Usadel equation for f 0 : (3.18) at an interface located at position x 0 between a bulk superconductor described by the anomalous GF f BCS and the X bridge. The interface is characterized by the transparency γ and normal vector of component N i . For a fully transparent interface, we impose the continuity of the GFs.
We now need to write the current and spin density in terms of the isotropic anomalous GFs. It is easy to verify, by checking its conservation, that in the linearized case the electric current, Eq. (3.10), is given by: 19) and correspondingly in the diffusive limit (3.20) The spin polarization (3.11) is more subtle to deal with in the linearized approximation, since the normalization condition do not apply in our case. In accordance with the case without SOC, one may assume that it can be expressed in terms of the isotropic anomalous f as: with σ a ff = σ a f 0 f 0 in the diffusive limit. In the next section we will show a posteriori that these expressions leads to the known results in bulk systems in the presence of Rashba SOC. For the following discussions it is convenient to write the anomalous GF f as the sum of singlet (scalar) and triplet (vector in spin space) f = f s + f a t σ a , and to expand all the spin variables in term of Pauli matrices: F ij = F a ij σ a /2, A µ = A a µ σ a /2. From Eqs. (3.12) we obtain the equations for the singlet and triplet components in the ballistic case: ∂ ∂n j f s .
(3.23) Equivalently, from Eq. (3.17) one obtains the equations for the isotropic part of the singlet f s0 and triplet f t0 components in the diffusive case (for simplicity we skip the subindex 0) : We write the covariant derivative as∇ In particular the form of Eq. (3.25) proves the full analogy between singlet-triplet and charge-spin coupling in diffusive systems. [cf. Eqs. (2.5-2.6)]. In Ref. [46], the analogy between the diffusion of spin in normal systems and the triplet components was discussed. Here we can extend this result and find that the tensor C a k , responsible for the SHE in normal systems, is an additional source for the singlet-triplet conversion and, as we will see in the next sections, is at the root of magneto-electric effects and the anomalous phase. Equations (3.22 -3.23) and (3.24-3.25) are the central equations of this work, which we now solve for different situations. In section VI B we go beyond this linear approximation.

IV. THE EDELSTEIN EFFECT IN BULK SUPERCONDUCTORS FOR T → Tc
In order to illustrate the usefulness of the SU(2) covariant quasiclassical equations presented above, we study here the magneto-electric effect and its inverse in bulk superconductors with an intrinsic SOC linear in momentum and derive the response coefficients in (1.3) and (1.4).
We assume that the superconducting order parameter ∆ is constant in magnitude but has a spatially dependent phase ∆ (r) = |∆|e iϕ(r) , where ∇ϕ is assumed to be a constant vector.
Let us first consider a diffusive superconductor. From (3.24) in the lowest order of ∇ϕ one obtains From Eq. (3.21) it becomes clear that the spin density is determined by the product of the singlet (4.1) and triplet (4.2) components which results in S a = χ a i ∂ i ϕ with This is the Edelstein result generalized for arbitrary linear in momentum SOC. With the help of Eqs. (3.24-3.25) we can also describe the inverse EE, the so-called spin-galvanic effect. We now assume a finite and spatially homogenous A a 0 and a zero phase gradient. In such a case one can obtain f t directly from Eq. (3.25), which is now proportional to A a 0 . By substitution of this result into the expression for the current, Eq. (3.20), and by noticing that only the second line contributes to the current we obtain j i = eχ a i A a 0 , with χ a i given by Eq. (4.3) in agreement with Onsager reciprocity.
In short, we are able to derive in a few lines the tensor (4.3), which describes the EE and inverse EE in superconductors. Moreover, the expression (4.3) is valid for arbitrary linear in momentum spin-orbit effect and generalizes the result obtained in Ref. [80] for the particular case of a Rashba SOC. If one assumes the same here, i.e. A y x = −α = −A x y and all the other components equal to zero, one obtains from Eq. (4.3) n α 3 2 |ω n | + Dα 2 (4.4) that coincides with the expression obtained in Ref. [80].
If we neglect in Eq. (4.2) the Dyakonov-Perel relaxation, then the triplet component is simply proportional to f a t ∼ A a 0 (r). By substituting this into the expression for the current Eq. (3.20) one can easily show that This expression suggests that a spatially inhomogenous magnetization together with SOC may also induce a finite supercurrent. In this case the spin-galvanic effect scales with the square of the SOC parameter, in contrast to the α 3 dependency found previously for spatially uniform magnetization. The same effects can be explored in the pure ballistic limit, for which Eqs. (3.22-3.23) apply. The singlet component in the lowest order in the SOC is given by whereas the triplet component can be obtained easily from Eq. (3.23) (4.7) By using Eq. (3.21) we obtain the Edelstein result S a = χ a j ∂ j ϕ but now for an arbitrary linear in momentum SOC Identically, we find j i = eχ a i A a 0 . In the particular case of a 2D systems with Rashba SOC we recover the Edelstein result for a ballistic superconductor [79]: (4.9) The agreement between our and Edelstein results proves the validity of the expression (3.21) in the linearized approximation.
To conclude this section we note that for Rashba SOC in both cases, diffusive (4.4) and ballistic (4.9), χ a i is proportional to α in the strong SOC limit (see also [134]), and to α 3 for weak spin-orbit interaction (see also [92]). So, the quasi-classic formalism is able to recover in an elegant way some well established results obtained after cumbersome diagrammatic [79,80], and it also allows some easy generalizations of them.

V. MAGNETO-ELECTRIC EFFECTS IN DIFFUSIVE JOSEPHSON JUNCTIONS
We now turn to the central topic of the present work which is the description of magneto-electric effects in S-X-S Josephson junctions and demonstrate their connection to the anomalous phase problem. We first consider the diffusive limit and postpone the discussion of ballistic junctions for the next section.
In particular we consider a S-X-S Josephson junction with an interlayer X of length L. We assume that the magnetic interactions are only finite in X and vanish in the S electrodes. Moreover, we assume that the structure has infinite dimensions in the y − z plane and therefore the GFs only depend on the x coordinate. The superconducting bulk solutions in the leads are written as f L = f BCS e −iϕ/2 and f R = f BCS e iϕ/2 , in the left (x ≤ −L/2) and right (x ≥ L/2) electrodes respectively, with whereas the normal metal fills the region −L/2 ≤ x ≤ L/2. We will consider both the highly resistive and the perfectly transparent interfaces between the S and X parts. When the barrier transparency is low, the linearized approximation is justified for all temperatures, whereas for transparent barriers, one is limited to temperatures close to the critical temperature of the junction.
For the particular case of Rashba SOC in the X region and an in-plane exchange field the Josephson current has been calculated in Ref. [92]. It has been shown explicitly that the current-phase relation is given by I = I c sin(ϕ − ϕ 0 ). The anomalous phase ϕ 0 was calculated as a function of the strength of the spin fields, the temperature and the junction parameters. Here instead we focus in a generic linear-in-momentum SOC and we derive the expressions for the anomalous Josephson current in the lowest order of the spin fields. This will allow us to understand the link between the inverse EE and the ϕ 0 -junctions.

A. Diffusive junction with low transparency interfaces
We first consider a S-X-S diffusive Josephson junction with highly resistive S-X interfaces. In this limit the linearization of the quasiclassical equations is justified for all temperatures. Our goal here is to determine the Josephson current through the junction, which in the linearized regime is given by Eq. (3.20). The components of the condensate function f s entering this expression, has to be obtained by solving the system (3.24)-(3.25) in the normal metal which couples the singlet with the triplet component. For the specific S-X-S geometry considered here this equations read: and the boundary conditions for the resistive interface (cf. Eq. (3.18)): The expression Eq. (3.20), can be simplified by calculating the current at the right interface (x = L/2) and by using the boundary condition (5.4): It is clear from this equation that the correction to the current due to the spin-fields (the anomalous current) is proportional to Im [f * R δf s (L/2)], where δf s is the first correction to the singlet component due to the gauge potentials. In the absence of a phase difference between the S electrodes f R is real and the anomalous current is proportional to the imaginary part of the singlet component. According to Eq. (5.2), in the absence of spin-fields (exchange and SOC), there is no triplet component and the singlet component is real. Therefore no supercurrent flows at zero phase difference.
In the presence of spin-fields there are two sources for singlet-triplet conversion, as seen from the second term in the l.h.s of Eq. (5.3). The first one is the extensively studied mechanism for singlet-triplet conversion in S/F junctions via the intrinsic exchange field A 0 [1? ]. Inclusion of SOC leads to an additional singlet-triplet conversion mechanism described by the last term in the l.h.s of Eq (5.3). As discussed in section II, the singlettriplet conversion in this case corresponds to the chargespin conversion in normal systems with SOC. Conversely, once the triplet component is created, both mechanisms will convert it back to singlet, as can be seen in Eq.(5.2).
The singlet-triplet-singlet conversion at the lowest orders in perturbation with respect to the spin-fields is schematized in Fig. 1. The black arrows represent the singlet-triplet conversion due to the exchange field which implies a π/2 phase shift due to the i factor in front of A 0 in Eqs. (5.2-5.3). The red arrows represent the singlettriplet conversion due to the SOC, specifically due to the coupling term in Eqs. (5.2-5.3) proportional to J a i ∂ i . No additional phase is associated with this latter process. If one follows the black path, i.e. the singlet-triplet-singlet conversion only due to the exchange field, the resulting contribution to the singlet component acquires a minus sign (a π shift) and it is proportional to A 2 0 . This means that there is no anomalous phase 0 < ϕ 0 < π induced and hence no Josephson current flows when ϕ = 0. Similarly, if one follows the red path the resulting singlet component also remains real with no change of sign. From Fig.1 it becomes clear that a nontrivial ϕ 0 only appears from the "cross-term" path that consist in one black and one red arrow. In other words, the mutual action of exchange field and SOC leads to a finite ϕ 0 and hence to a supercurrent even at zero phase difference. In this case the contribution to this current in the lowest order of the spin fields, is proportional to A a 0 J a i ∂ i f s between the exchange field and the spin-current tensor, as anticipated in the introduction.
In order to quantify this effect and calculate ϕ 0 in the S-X-S junctions it is convenient to introduce the singlet and triplet propagators associated with Eqs. (5.2-5.4): and Thus, Eqs. (5.2-5.4) can be re-written as a set of integral equations Figure 1: Schematic representation of the singlet-tripletsinglet conversion process at the lowest order with respect to the spin-fields. Black arrows represent the action of the exchange field, whereas red arrows encode the effect of the singlet-triplet coupling term due to the SOC. Only mixed redblack paths lead to the appearance of an anomalous phase ϕ0 in the singlet component and hence to a supercurrent in a S-X-S junction even without a phase bias between the S electrodes and f a t (x) = sgn (ω n ) the second term in Eq. (5.8) takes into account the boundary condition (5.4) The K s propagator can be obtained from Eq. (5.6) whereas the equations for the triplet kernel, Eqs.(5.7), can be written in the form of an integral equation which is convenient for the subsequent perturbative analysis: In the lowest order of the gauge potentials one can obtain the correction δf s to the singlet component by substituting the result (5.10) into Eqs. (5.8)-(5.9). We consider here only the "cross-term" correction δf s proportional to both the exchange field A 0 and the spincurrent J i and which is responsible for the anomalous phase-shift: In principle, one has all the elements to solve Eqs. (5.8-5.9), for example recursively by performing a perturbative expansion in the gauge potentials. Here, in order to get analytical compact expressions we restrict our analysis to the short junction limit, i.e. L min(κ −1 ω , |A k | −1 ). In this case K s ≈ κ −2 ω L −1 (cf. Eq. (5.10)) and from Eq. (5.11) it is easy to verify that K t readŝ We are interested in calculating the anomalous phase ϕ 0 which can be obtained by noticing that the current (5.5) can be written as for a small ϕ 0 . The anomalous phase ϕ 0 can be obtained by setting ϕ = 0 and dividing by the critical current j c in the absence of SOC. In the short junction limit this is given by: We follow this procedure and from Eq.(5.5) and Eqs. (5.12-5.13) we obtain (5.16) This expression clearly shows the relation between the appearance of the anomalous phase, ϕ 0 , and the inverse Edelstein effect in bulk systems. Both, the Josephson current (proportional in the linearized case to ϕ 0 ) and the bulk supercurrent are proportional to A 0 J x , i.e. both are generated from the mutual action of the exchange field and the SOC.
It is worth noticing that in the present case of low transparent interfaces, the anomalous phase grows linearly with L, the length of the junction (5.16). In the next subsection we show that in the case of a transparent barrier the anomalous phase behaves like L 3 .
In the particular case of a 2D situation, with a SOC coupling of Rashba (described by the parameter α) and Dresselhaus (β) type we obtain from Eq. (5.16): Besides controlling the anomalous phase and hence the Josephson current by tuning the external magnetic field, this expression also suggests that the current can be controlled by tuning the Rashba SOC by means of an external gate. In the particular case that α = β the anomalous phase is zero and no supercurrent will flow.

B. Diffusive junction with transparent interfaces
We now briefly consider the limit of a full transparent barrier. In that case one assumes continuity of the quasiclassical GFs at the S-X interfaces. The problem is then formally the same as in the previous section, except that the second equations in (5.6) and (5.7), for the propagators K s andK t are replaced by: respectively. In this case one should remove the second term in Eq.
(5.8) and f (0) Now the singlet propagator is given by: In the short junction limit K s is proportional to L and it is temperature independent. From Eq. (5.11)K t ∼ K s . Thus, in this case the anomalous phase-shift is also temperature independent and proportional to In contrast to the case of finite barrier resistance, Eq. (5.16), the anomalous phase scale with L 3 . This means that in short junctions a finite barrier resistance between the S and the normal metal favors the growth of ϕ 0 . These results generalize those presented recently in Ref. [92] for the particular case of Rashba SOC. We can then conclude that the anomalous phase, at lowest order in the gauge potentials, is proportional to A a 0 J a x , independently of the type of interface.

VI. MAGNETO-ELECTRIC EFFECTS IN BALLISTIC JOSEPHSON JUNCTIONS
In this section we consider a pure ballistic S-X-S junction,i.e. we solve (3.8) in the limit τ → ∞. As before, the junction is along the x-axis and the two superconducting electrodes at position x ≤ −L/2 and x ≥ L/2. The spin fields, both exchange and SOC, are only finite in the X region. We assume that the the transverse dimensions of the junction are very large, and therefore the GFs depends on xand only weakly on y, z. We also assume that the interfaces between X and S are perfectly transparent.
In the next subsection we first analyze the Josephson current for temperatures close to the superconducting critical temperature T c , and make a connection with the diffusive structures studied in the previous section. In the second subsection we derive analytical expressions for the anomalous current at arbitrary temperature for the case of small spin fields.

A. Ballistic junction at T → Tc
In the case of large enough temperatures we analyze the linearized Eilenberger equation The solutions for the singlet and triplet components in equations (3.22) and (3.23) can be written as propagation in two di- and In the opposite propagation direction f > s,t are found from f < s,t by substituting L/2 → −L/2.
In analogy with the diffusive case (cf. Fig 1), expressions (6.1) and (6.2) show explicitly the effect of the SOC on the condensate function. In the absence of SOC the exchange field A 0 is the only source for singlet-triplet conversion. The manifestation of the triplet component in S-F-S junctions has been extensively studied in the past (see [1,2] for reviews). As discussed in section II, the imaginary unit i in front of the A 0 terms leads to a π/2 phase shift. In the case of a finite SOC the gaugefield, F ij , is an additional source of triplet correlations. Notice that in the ballistic case, F ij not only couples the singlet and triplet components, but also the s-p-wave components of the condensate [143]. Moreover, the term e −Pinix/nx in (6.2) leads to a momentum dependent rotation of the triplet component in the spin-space.
The origin of the anomalous phase ϕ 0 can be easily understood in the lowest order in the spin fields. Assuming a vanishing phase difference between the superconductors and combining Eqs.(6.1)-(6.2) with the expression for the current (3.19), one obtains for the first nontrivial contribution to the current: F a ij ∂ nj e −Pinix/nx This correction is proportional to J b i A b 0 and coincides with those obtained in bulk superconductors with SOC(section IV) and a diffusive S-X-S junctions (section V).
Quantitatively, a compact analytical solution for the current at zero-phase difference can be obtained from Eqs. (6.1-6.2) in the short junction limit, i.e. for L v F /2ω n : where F a xi F a i0 = J a x A a 0 . Thus the anomalous current is generated by the spin-polarization A a 0 via the spincurrent J a i . This is the spin-galvanic effect, discussed in the previous sections, for a ballistic S-X-S junction.

B. Arbitrary temperatures
The previous result for the current has been obtained at temperatures close to the critical one. We now consider an arbitrary temperature and calculate the current up to the lowest order in the gauge field F ij . In order to calculate the current from Eq.(3.10) to the lowest order in F ij we need to compute the first two components matrixǧ =ǧ (0) +ǧ (1) + · · · .
At zeroth order in F ij the ballistic equation reduces to 4) which admits for solutioň withǧ (0) 0 a constant matrix found from the boundary conditions. The propagatorǔ (x) in Eq. (6.5) is given by when we assume that neither A 0 nor∆ nor A j depend on the position.ǔdescribes how the functionǧ 0 "propagates" from its value at x = 0,ǧ 0 , to any point x. The constantǧ ∞ in (6.5) satisfies and describes the bulk contribution deep inside the superconductor. Notice that according to Eqs.(6.7)-(6.6), [ǧ ∞ ,ǔ] = 0, henceǧ ∞ can not be obtained by the application of (6.6). As we will see below [see (6.13)], the solutions ofǔ in a superconductor are evanescent waves, so the contributionǔǧ (0) 0ǔ −1 vanishes deep inside the superconductor, whereas the contributionǧ ∞ remains finite.
The current can be written in powers of F ij , j x = j (0) x + j (1) x + · · · with [see (3.10)] Tr n xǧ (0) 0 τ 3 (6.11) and the first order correction Tr n xǧ1 τ 3 (6.12) Notice that the second line in (6.10) vanishes after the angular average. We then need to obtainǧ (0) 0 andǧ 1 to determine the current through the S-X-S Josephson junction.
In the normal region, the solution readš 0 (x)ǔ † 0 (x) + · · · , (6.16) whereǧ (1) 0 (x) =ǧ 1 + x 0Ǧ (z) dz , (6.17) whereǔ 0 =ǔ (∆ = 0) [see (6.6)] is a unitary matrix that can be written aš The spin matrices u andū are defined as The matricesǧ 0 andǧ 1 in Eq. (6.16) are obtained from the boundary conditions, assuming continuity of the GFs at the left and right boundaries. At zeroth order we obtainǧ whereasŪ (x) =ū (x) u (−x) is its time-reversal conjugate. We here give only the contribution corresponding to the positive projection of the Fermi velocity, the negative projection can be found straightforwardly. The matrices entering the first-order correction, Eq. (6.17), have the following form in Nambu spacě After multiplication by n x and taking the angular average the first line of this equation vanishes. The second line of (6.25) can be simplified using the normalization condition g 2 0 − f 0f0 = 1 available for the zeroth order correction. We obtain andf ± (z, n) = σ y f * ± (z, −n) σ y its time reversal conjugate. If becomes independent of the SOC [see the definitions (6.18)], and the contribution (6.26) vanishes. Therefore we expect (6.26) to be proportional to F ij [A 0 , A j ] ∝ F ij F 0j at the smallest order in the gaugefields.
By expanding the expression (6.26) in the gaugepotentials, up to the term proportional to the electric-like field one obtains where Tr {U (L)} ≈ 2 in the small gauge-field limit. Besides the terms proportional to A 0 -only, responsible for the oscillations of S/F proximity effect, the SOC A j only appears in the electric-field construction (the last term on each line), due to symmetry with respect to the timereversal. After angular averaging only the last contributions of the two lines are non-zero. This leads to tanh ω n L v F |n x | + arcsinh ω n ∆ + i ϕ 2 (6.31) (note that the sum over j applies inside the angular averaging as well). As in all previous examples the anomalous current is proportional to Tr {F xk F k0 } = J a x A a 0 , where the later form suggests our expressions are valid beyond the linear-in-momentum-SOC approximation, given any spin current J a i and paramagnetic interaction A a 0 . Close to the critical temperature M ≈ ∆ 2 sin ϕ ωn≥0 e −2ωL/v F |nx| /2ω 2 and we recover (6.3).
Commonly, the concept of a ϕ 0 -junction is defined for junctions with a sinusoidal current-phase relation. This is valid at temperatures close to the critical temperature or in the case of a weak proximity effect between the S electrodes and the X bridge. However, in several cases the current-phase relation is more complex and higher harmonics are involved [132]. This is the case of the ballistic junction studied here with a current-phase relation given by sum of Eqs. (6.29) and (6.30). In such cases the ϕ 0 is defined as the phase difference across the junction that minimize the energy, or equivalently, as the phase difference imposed to the junction in order to get a zero current state, i.e. j (ϕ 0 ) = 0. In our perturbative analysis ϕ 0 is small and hence , and from Eqs. (6.29) and (6.30) we obtain (6.33) In Fig. 2 we show the temperature dependence of ϕ 0 for the ballistic junction for a 2D system when only F a xy F a y0 is non-zero. We assume a circular Fermi surface, n x = cos θ and n y = sin θ. We plot the anomalous phase for different junction lengths.

VII. DISCUSSION AND CONCLUSIONS
In order to verify our findings and prove the existence of the anomalous ϕ 0 phase one can design a superconducting ring interrupted by a semiconducting link with a strong SOC, similar to the one used recently in Ref. [144] for the characterization of the current phase relation of a Nb/3D-HgTe/Nb junction or in [145] for the observation of a spontaneous supercurrent induced by a ferromagnetic π-junction. A schematically view of the proposed setup is shown in Fig.3: It consists in a superconducting ring (green) grown on top of a semiconductor or a metallic substrate with strong SOC (grey). In order to isolate electrically the S ring from the semiconductor one can for example add an insulating barrier (blue) under the ring.
If a magnetic field is applied in the plane of the ring, it will act as a Zeeman field and hence, according to our previous results, it will create a spontaneous circulating supercurrent, see [7,146] for more details. This supercurrent will generate a magnetic flux that in principle can be measured by a second loop [144] or a micro-Hall sensor [145].
In the case when the bridge is made of a 2D semiconductor with a generic SOC described by a combination of Rashba and Dresselhaus terms: A x = −ασ y + βσ x and A y = ασ x − βσ y the generated supercurrent should be proportional to j s ∝ α 2 − β 2 (h x β + h y α) .
Thus the current depends on the direction of the applied magnetic field. In particular, for a field perpendicular to the 2D gas the effect should vanishes. In addition by applying a gate voltage one could modify the ratio between Dresselhaus and Rashba interactions and hence control the supercurrent flow. We thus expect that the dependency of the spontaneous supercurrent with respect to the orientation of the magnetic field and/or the gate voltage realizes a clear demonstration of the spin-galvanic effect in Josephson systems.
Instead of using a semiconducting bridge one could grow the superconducting loop on top of a metallic substrate. Metals with strong SOC, like Pt and Ta, are good candidates to observe the ϕ 0 -junction behavior, but also an ultra-thin layer of Pb might be used [147]. In such a case probably one cannot control the ϕ 0 -shift using a gate, but a spontaneous circulating current might still be controlled by switching the in-plane external field on and off.
Eventually, the existence of a magneto-electric phaseshift ϕ 0 can be probed by measuring the Shapiro steps in S-X-S Josephson junctions as suggested in Ref. [148]. In order to isolate electrically part of the S loop from the conducting substrate we assume an insulating layer between them (in blue). By applying an in-plane magnetic field a circulating supercurrent might be generated which in turn induces a magnetic flux that can be measured via an extra pick-up coil (shaded grey). For more details see discussion in the main text.
In conclusion, we have demonstrated that the inverse Edelstein effect, also called spin-galvanic effect, and the appearance of an anomalous phase-shift ϕ 0 in Josephson junctions are the two sides of the same coin. We presented a full SU(2) covariant quasi-classic formalism that allows to study these magneto-electric phenomena in bulk and hybrid superconducting structures with arbitrary linear-in-momentum SOC (section III).
With the help of our quasi-classic transport formalism we derived the Edelstein effect close to the critical temperature of a bulk superconductor, recovering the Edelstein's result in a very compact way (section IV) and generalizing it for the case of an arbitrary linearin-momentum SOC. We have shown that the Edelstein effect and its inverse are reciprocal in the sense of the Onsager relations, both in ballistic and diffusive superconducting systems: A static supercurrent can induce a finite magnetization due to the presence of a spin-orbit coupling, and reciprocally a finite magnetization produces a finite supercurrent in a bulk system. We have demonstrated that the linear-response tensor is directly proportional to the equilibrium spin-current tensor J a i . We have also generalized this result to inhomogeneous systems. In particular we have studied the current-phase relation of a Josephson junction consisting of two superconductors coupled via a normal metal with both SOC and spin-splitting field. We have demonstrated that a supercurrent can flow even if the phase difference between the S electrodes is zero. This current is associated to an anomalous phase-shift ϕ 0 . This result holds for both ballistic (section VI) and diffusive systems (sec-tion V), for arbitrary linear-in-momentum spin-orbit coupling, and for arbitrary barrier resistance between the superconductor and the normal metal. For all these situations we have demonstrated that SU(2) gauge-fields are the only objects of relevance in the phenomenology of the ϕ 0 -shift, and in particular we have shown that ϕ 0 ∝ A a 0 J a i = F a 0j F a ji , i.e. the anomalous phase-shift is proportional to the SU(2) electric and magnetic fields, or equivalently to the spin-current tensor. We thus directly linked the anomalous phase-shift in superconducting systems to the inverse Edelstein effect (also known as the spin-galvanic effect) extensively studied in normal systems.