Spin and charge transport induced by gauge fields in a ferromagnet

TL;DR

This paper develops a microscopic theory for spin and charge transport driven by gauge fields in ferromagnets, accounting for spin relaxation, and resolves gauge invariance issues related to spin non-conservation.

Contribution

It introduces a gauge-invariant framework for spin motive force considering spin relaxation effects, extending previous models and clarifying theoretical inconsistencies.

Findings

Linear response current to gauge field is not gauge-invariant with spin-flip processes

Time dependence of spin-relaxation terms restores gauge invariance

Dissipative spin motive force linked to reciprocal spin torque

Abstract

We present a microscopic theory of spin-dependent motive force ("spin motive force") induced by magnetization dynamics in a conducting ferromagnet, by taking account of spin relaxation of conduction electrons. The theory is developed by calculating spin and charge transport driven by two kinds of gauge fields; one is the ordinary electromagnetic field $A^{\rm em}_{\mu}$, and the other is the effective gauge field $A^{z}_{\mu}$ induced by dynamical magnetic texture. The latter acts in the spin channel and gives rise to a spin motive force. It is found that the current induced as a linear response to $A^{z}_{\mu}$ is not gauge-invariant in the presence of spin-flip processes. This fact is intimately related to the non-conservation of spin via Onsager reciprocity, so is robust, but indicates a theoretical inconsistency. This problem is resolved by considering the time dependence of spin-relaxation source terms in the "rotated frame", as in the previous study on Gilbert damping [J. Phys. Soc. Jpn. {\bf 76}, 063710 (2007)]. This effect restores the gauge invariance while keeping spin non-conservation. It also gives a dissipative spin motive force expected as a reciprocal to the dissipative spin torque ("$\beta$-term").