Spin and charge transport induced by gauge fields in a ferromagnet

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 ${A}_{\ensuremath{\mu}}^{\mathrm{em}}$, and the other is the effective gauge field ${A}_{\ensuremath{\mu}}^{z}$ 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 ${A}_{\ensuremath{\mu}}^{z}$ is not gauge invariant in the presence of spin-flip processes. This fact is intimately related to the nonconservation 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 [H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007)]. This effect restores the gauge invariance while keeping spin nonconservation. It also gives a dissipative spin motive force expected as a reciprocal to the dissipative spin torque (``$\ensuremath{\beta}$ term'').