Spin relaxation torque in metallic ferromagnets

TL;DR

This paper microscopically calculates the spin relaxation torque in metallic ferromagnets with impurity-induced spin-orbit interaction, revealing its relation to electric field derivatives and the non-conservation of spin current.

Contribution

It provides a microscopic derivation showing that the spin relaxation torque depends on electric field derivatives and challenges the conventional divergence form, highlighting non-conservation of spin current.

Findings

Spin relaxation torque relates to spatial derivatives of electric fields.

Spin current is not conserved due to orbital angular momentum dissipation.

The $eta$ term in spin transfer torque can be asymmetric.

Abstract

Spin relaxation torque ${\cal T}$ in uniform metallic ferromagnets with the spin-orbit interaction arising from random impurities is calculated microscopically. ${\cal T}$ is shown to be written by spatial derivatives of the electric field, but is not always given as a divergence of a torque dipole density. The result is due to the fast dissipation of the orbital angular momentum, and thus a conserving spin current cannot be defined. It is discussed that the $\beta$ term in the spin transfer torque can also be asymmetric.