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 ϕ0 in the current-phase relation. In particular we show that ϕ0 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.

We consider a Josephson junction where the weak link is formed by a noncentrosymmetric ferromagnet. In such a junction, the superconducting current acts as a direct driving force on the magnetic moment. We show that the ac Josephson effect generates a magnetic precession providing then a feedback to the current. Magnetic dynamics result in several anomalies of current-phase relations (second harmonic, dissipative current) which are strongly enhanced near the ferromagnetic resonance frequency.

There was an error in the sign of the vector product of M and H eff in the Landau-Lifshitz-Gilbert equation, Eq. (6) in the original Letter, which effectively corresponds to the sign change of gyromagnetic ratio γ. Therefore the presented results correspond simply to the inverted direction of the spin precession. This does not affect the qualitative conclusions of the Letter but leads to some sign changes in the formulas.Equation (9) should readEquation (11) should readExpression (12) should readwith Ω AE ¼ ðω AE 1Þ 2 þ α 2 ω 2 . m z -1 -0.5 0 0.5 1 0.7 0.8 0.9 1 1.1 Γ=ω=0.5, α=0.6, r=1 (b) m z -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Γ=ω=1, α=0.1, r=1 (c) m z -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Γ=ω=1, α=0.05, r=5 (d) t m z (t) 0 0.5 1 1.5 2 2.5 3 -1 0 1 Γ=5π, ω=5 α=0.1, r=0.1 (a) analytic numeric m y m x m y FIG. 2. Results of numerical analysis of the magnetic moment dynamics of the φ 0 -junction. (a) Reversal of m z from analytical expression Eq. (15) (dashed curve) and numerical one (normal curve). The other curves are related to the M trajectory: (b) in strong damping case (c) and (d) in the strong coupling regime revealing incomplete and complete magnetic moment reversal, respectively. PHYSICAL REVIEW LETTERS 123, 169901(E) (2019) 0031-9007=19=123(16)=169901(2) 169901-1