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 spi
n 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).
We theoretically examine the spin-transfer torque in the presence of spin-orbit interaction (SOI) at impurities in a ferromagnetic metal on the basis of linear response theory. We obtained, in addition to the usual spin-transfer torque, a new contrib
utioin $sim {bm j}_{rm SH}^{phantom{dagger}} cdot abla {bm n}$ in the first order in SOI, where ${bm j}_{rm SH}^{phantom{dagger}}$ is the spin Hall current driven by an external electric field. This is a reaction to inverse spin Hall effect driven by spin motive force in a ferromagnet.
