Magnetoelectric effects in superconductors due to spin-orbit scattering: Nonlinear σ -model description

Abstract

We suggest a generalization of the nonlinear σ model for diffusive superconducting systems to account for magnetoelectric effects due to spin-orbit scattering. In the leading orders of spin-orbit strength and gradient expansion, it includes two additional terms responsible for the spin-Hall effect and the spin-current swapping. First, assuming a delta-correlated disorder, we derive the terms from the Keldysh path integral representation of the generating functional. Then we argue phenomenologically that they exhaust all invariants allowed in the effective action to the leading order in the spin-orbit coupling (SOC). Finally, the results are confirmed by a direct derivation of the saddle-point (Usadel) equation from the quantum kinetic equations in the presence of randomly distributed impurities with SOC. At this point, we correct a recent derivation of the Usadel equation that includes magnetoelectric effects and does not resort to the Born approximation.

Nonlinear $σ$-model for disordered systems with intrinsic spin-orbit coupling

“…We can recognize that the non-Abelian field strength term S 4,1 = S H is a generalization of the spin-Hall term in Ref. 46 from the extrinsic spin-Hall effect to the intrinsic one, with the correspondence Fij ↔ − p F θ ijk σ k , where θ is the spin-Hall angle. In the equations of motion, this term will lead to conversion between charge and spin currents, a feature that is not present in a model containing only S 2 .…”

“…(35)(36) are not modified on the saddle-point level [60] by the longitudinal corrections, pre-factors in the last two equations have to be renormalized. At this point it is instructive to compare the present covariant theory with intrinsic SOC, and the theory of diffusive systems with SOC of extrinsic origin, such as random impurities [46]. At the level of the nonlinear σ-model, the extrinsic and intrinsic theories can be connected by (i) replacing the usual gradients with the covariant gradients, and (ii) identifying Fij ↔ − p F θ ijk σ k in the spin Hall term.…”

“…However, this does not look obvious if one naively compares the Usadel equation from Ref. [46] and Eq. ( 33) with the current given by Eqs.…”

“…We used this approach previously in Ref. 46 to describe the contribution of extrinsic SOC to magnetoelectric effects in superconductors. However, in the case of intrinsic SOC described by SU(2) gauge fields, the number of terms appearing in the naive expansion of the action with respect to the gradients and fields is immense and cannot be treated manually.…”

Supercurrent-induced long-range triplet correlations and controllable Josephson effect in superconductor/ferromagnet hybrids with extrinsic SOC

“…The quasiclassical theory of superconductivity in the diffusive limit, formulated in terms of the Usadel equation for the Green's functions and taking into account the effect of the extrinsic SOC has been developed in Refs. [72][73][74][75] and has been used for prediction of a number of SOC-related effects in superconducting structures [72,73]. In the framework of this theory the extrinsic SOC enters the Usadel equation via the spin Hall angle θ, the current swapping coefficient κ and the spin-orbit scattering time τ so .…”