Spin accumulation and longitudinal spin diffusion of magnets

Abstract

We extend to the longitudinal component of the magnetization the spintronics idea that a magnet near equilibrium can be described by two magnetic variables. One is the usual magnetization M. The other is the nonequilibrium quantity, called the spin accumulation, by which the nonequilibrium spin current can be transported. M-represents a correlated distribution of a very large number of degrees of freedom, as expressed in some equilibrium distribution function for the excitations; we therefore forbid M to diffuse, but we permit M to decay. On the other hand, we permit m, due to spin excitations, to both diffuse and decay. For this physical picture, diffusion from a given region occurs by decay of M to m, then by diffusion of m, and finally by decay of mto M in another region. This somewhat slows down the diffusion process. Restricting ourselves to the longitudinal variables M and m with equilibrium properties Meq=M0+χM H and meq=0, we argue that the effective energy density must include a new, thermodynamically required exchange constant λM=-1/χM. We then develop the appropriate macroscopic equations by applying Onsager's irreversible thermodynamics and use the resulting equations to study the space and time response. At fixed real frequency ω there is, as usual, a single pair of complex wave vectors ±k but with an unusual dependence on ω. At fixed real wave vector, there are two decay constants, as opposed to one in the usual case. Extending the idea that nonequilibrium diffusion in other ordered systems involves a nonequilibrium quantity, this work suggests that, in a superconductor, the order parameter Δ can decay but not diffuse, but a nonequilibrium gap-like δ, due to pair excitations, can both decay and diffuse.