# Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory of magneto-electric effects in normal and superconducting systems

## TL;DR

This paper derives a generalized Usadel equation for materials with linear spin-orbit coupling, unifying the description of magneto-electric effects in both normal and superconducting states.

## Contribution

It introduces a unified theoretical framework linking magneto-electric effects in normal and superconducting systems through an extended Usadel equation.

## Key findings

- Derivation of Usadel equation for gyrotropic materials with spin-orbit coupling
- Connection of magneto-electric effects to superconducting counterparts
- Application example on bulk spin-galvanic effect

## Abstract

The Usadel equation is the standard theoretical tool for the description of superconducting structures in the diffusive limit. Here I derive the Usadel equation for gyrotropic materials with a generic linear in momentum spin-orbit coupling. It accounts for the spin-charge/singlet-triplet coupling and in the normal state reduces to the system of spin-charge diffusion equations describing various magneto-electric effects, such as the spin Hall effect (SHE), the spin-galvanic effect (SGE) and their inverses. Therefore the derived Usadel equation establishes a direct connection of these effects to their superconducting counterparts. The working power of the present formalism is illustrated on the example of the bulk SGE.