Theory of the spin-galvanic effect and the anomalous phase-shift ϕ 0 in superconductors and Josephson junctions with intrinsic spin-orbit coupling
Résumé
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, i.e. the creation of a supercurrent by an equilibrium spin polarization, may also exist in bulk superconductors. By exploiting the analogy between a linear-in-momentum SOC and a background SU(2) gauge field, we develop a SU(2) gauge-covariant formulation of the quasi-classic transport equations in order to deal with magneto-electric effects in superconductors. General expressions for the direct and inverse Edelstein effects are obtained, which are valid for arbitrary linear-in-momentum SOC and spin-splitting field. For Josephson junctions we find 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 this phase-shift 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. Finally we propose experiments to verify our findings. The magneto-electric effects discussed in this study may find application in the emerging field of coherent spintronics with superconductors.