Irreversible thermodynamics of uniform ferromagnets with spin accumulation: Bulk and interface dynamics

Abstract

Using ideas from Landau's Fermi-liquid theory, we apply irreversible thermodynamics to conducting and insulating ferromagnets with magnetic variables $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ for the quantization axis and for the spin accumulation $\stackrel{P\vec}{m}$ of the nonequilibrium excitations; thus the total magnetization is taken to be $\stackrel{P\vec}{\mathcal{M}}=\stackrel{P\vec}{M}+\stackrel{P\vec}{m}$. The resulting theory closely corresponds to the theory of Silsbee et al. [Silsbee, Janossy, and Monod, Phys. Rev. B 19, 4382 (1979)]. For the bulk, in addition to confirming the usual Landau-Lifshitz equation for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ and a Bloch-like equation for $\stackrel{P\vec}{m}$ (with a nonuniform precession term), there are two related cross-relaxation terms between the transverse parts of the nonequilibrium $\stackrel{P\vec}{m}$ and $\stackrel{P\vec}{M}$. Unlike the s-d model, where in a field $\stackrel{P\vec}{H}$ the equilibrium magnetizations ${\stackrel{P\vec}{M}}_{s}$ and ${\stackrel{P\vec}{M}}_{d}$ are both nonzero, for this m-M model in a field $\stackrel{P\vec}{H}$, only the equilibrium magnetization $\stackrel{P\vec}{M}$ is nonzero. For the interface, the boundary condition for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ is given by micromagnetics, and that for $\stackrel{P\vec}{m}$ is given by irreversible thermodynamics, where the current of transverse spins crossing the interface is proportional to the discontinuity in the transverse part of the vector spin chemical potential. $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}, \stackrel{P\vec}{m}$, and $\stackrel{P\vec}{H}$ are coupled; in the decoupled approximation, we find the wave vectors for the modes of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ and the transverse $\stackrel{P\vec}{m}$. We discuss reciprocity between spin pumping ($\stackrel{P\vec}{\mathcal{M}}$ driven out of the ferromagnet) and spin transfer torque ($\stackrel{P\vec}{\mathcal{M}}$ driven into the ferromagnet).