Functional Keldysh theory of spin torques

Abstract

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 $\ensuremath{\alpha}$ and the nonadiabatic spin transfer torque parameter $\ensuremath{\beta}$ for a model ferromagnet. We find that $\ensuremath{\beta}\ensuremath{\ne}\ensuremath{\alpha}$, 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 $s\text{\ensuremath{-}}d$ 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 $\ensuremath{\beta}∕\ensuremath{\alpha}$ 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 $\ensuremath{\beta}∕\ensuremath{\alpha}$ 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.