Spin and charge transport induced by gauge fields in a ferromagnet

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 $A_{\mu }^{\mathrm{em}}$, and the other is the effective gauge field $A_{\mu }^z$ 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 $A_{\mu }^z$ 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 (“$\beta$ term”).

Figure 1

(a) Diagrammatic expression of $K_{\mu \nu }^{\mathrm{cc}}$. The thick (thin) solid line represents an electron line carrying Matsubara frequency $i{\varepsilon }_n+i{\omega }_{\lambda }$ $\left(i{\varepsilon }_n\right)$. The shaded part represents the vertex function, ${\Lambda }_{\nu }^{\sigma }$. (b) Dyson equation for ${\Lambda }_{\nu }^{\sigma }$. The dotted lines represent impurity scattering, either with (${\tilde{\Gamma }}_2$) or without (${\tilde{\Gamma }}_1$) spin-flip scattering.

Figure 2

Diagrammatic expression of ${\chi }_{\mu }^{\alpha \beta }$. The wavy line represents scattering from impurity spins, which are time-dependent in the rotated frame. The shaded part represents the vertex function ${\Lambda }_{\mu }^{\sigma }$.

Figure 3

Diagrammatic expression of ${\chi }_{\mu i}^{\alpha \beta \gamma }$. The gray circle represents the interaction with $A_{\mu }^{\gamma }$.