Current-induced non-adiabatic spin torques and domain wall motion with spin relaxation in a ferromagnetic metallic wire

TL;DR

This paper derives a detailed theoretical model for current-induced spin torques in ferromagnetic wires, including non-adiabatic effects and spin relaxation, and applies it to domain wall motion, revealing complex dynamics.

Contribution

It introduces a systematic gradient expansion to compute higher-order non-adiabatic spin torques including spin relaxation effects in ferromagnets.

Findings

Second-order spin torque causes domain wall deformation.

Domain wall velocity depends non-trivially on spin-polarized current.

The model incorporates spin relaxation via a Caldeira-Leggett bath.

Abstract

Within the s-d model description, we derive the current-driven spin torque in a ferromagnet, taking explicitly into account a spin-relaxing Caldeira-Leggett bath coupling to the s-electrons. We derive Bloch-Redfield equations of motion for the s-electron spin dynamics, and formulate a systematic gradient expansion to obtain non-adiabatic (higher-order) corrections to the well-known adiabatic (first-order) spin torque. We provide simple analytical expressions for the second-order spin torque. The theory is applied to current-driven domain wall motion. Second-order contributions imply a deformation of a transverse tail-to-tail domain wall. The wall center still moves with a constant velocity that now depends on the spin-polarized current in a non-trivial manner.