Irreversible thermodynamics of uniform ferromagnets with spin accumulation: Bulk and interface dynamics
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2017 American Physical Society. Using ideas from Landau's Fermi-liquid theory, we apply irreversible thermodynamics to conducting and insulating ferromagnets with magnetic variables M for the quantization axis and for the spin accumulation m of the nonequilibrium excitations; thus the total magnetization is taken to be M=M+m. The resulting theory closely corresponds to the theory of Silsbee et al. [Silsbee, Janossy, and Monod, Phys. Rev. B 19, 4382 (1979)PRBMDO0163-182910.1103/PhysRevB.19.4382]. For the bulk, in addition to confirming the usual Landau-Lifshitz equation for M and a Bloch-like equation for m (with a nonuniform precession term), there are two related cross-relaxation terms between the transverse parts of the nonequilibrium m and M. Unlike the s-d model, where in a field H the equilibrium magnetizations M-s and M-d are both nonzero, for this m-M model in a field H, only the equilibrium magnetization M is nonzero. For the interface, the boundary condition for M is given by micromagnetics, and that for 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. M, m, and H are coupled; in the decoupled approximation, we find the wave vectors for the modes of M and the transverse m. We discuss reciprocity between spin pumping (M driven out of the ferromagnet) and spin transfer torque (M driven into the ferromagnet).