Functional Keldysh theory of spin torques

We present a microscopic treatment of current-induced torques and thermal fluctuations in itinerant ferromagnets based on a functional formulation of the Keldysh formalism. We find that the nonequilibrium magnetization dynamics is governed by a stochastic Landau-Lifschitz-Gilbert equation with spin-transfer torques. We calculate the Gilbert damping parameter $\alpha$ and the nonadiabatic spin transfer torque parameter $\beta$ for a model ferromagnet. We find that $\beta \ne \alpha$, in agreement with the results obtained using imaginary-time methods of Kohno et al. [J. Phys. Soc. Jpn. 75, 113706 (2006)]. We comment on the relationship between $s\text{−}d$ and isotropic-Stoner toy models of ferromagnetism and more realistic density-functional-theory models, and on the implications of these relationships for predictions of the $\beta ∕\alpha$ ratio which plays a central role in domain-wall motion. Only for a single-parabolic-band isotropic-Stoner model with an exchange splitting that is small compared to the Fermi energy does $\beta ∕\alpha$ approach 1. In addition, our microscopic formalism naturally incorporates the fluctuations needed in a nonzero-temperature description of the magnetization. We find that to first order in the applied electric field, the usual form of thermal fluctuations via a phenomenological stochastic magnetic field holds.

Figure 1

(a) Feynman diagram for the transverse spin-density–spin-density response function. This diagram may equivalently be thought of as the spin-wave, or magnon, propagator. The magnon frequency is denoted by $\omega$ and its momentum by $\mathbf{q}$. (b) Photon–two-magnon interaction vertex that ultimately gives rises to spin-transfer torques. (Note that the photons correspond to the external electric field used in our theory to induce transport current and hence to terms proportional to the current within linear response.) This vertex describes the interaction of spin waves with frequency $\omega$ and momentum $\mathbf{q}$ with the electric current that is generated by an external electric (photon) field of frequency ${\omega }_p$. (The theory of spin-transfer torques is related to the ${\omega }_p\to 0$ limit of this diagram and this limit is taken in the Feynman diagram.)

Figure 2

(a) Lowest-order diagram and (b) first vertex correction to the spin-density–spin-density response function. The momentum of the spin wave is denoted by $\mathbf{q}$. Impurity scattering is indicated by the thin dashed line where the × indicates the position of the impurity.

Figure 3

Feynman diagrams that contribute to the spin-wave current interaction that gives rise to spin-transfer torques. (a) Lowest-order diagram, and [(b) and (c)] leading-order vertex corrections.