Spin relaxation torque in metallic ferromagnets

Abstract

Spin relaxation torque $\mathcal{T}$ in uniform metallic ferromagnets with the spin-orbit interaction arising from random impurities is calculated microscopically. $\mathcal{T}$ 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.