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 $\stackrel{P\vec}{M}$. The other is the nonequilibrium quantity $\stackrel{P\vec}{m}$, called the spin accumulation, by which the nonequilibrium spin current can be transported. $\stackrel{P\vec}{M}$ 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 $\stackrel{P\vec}{M}$ to diffuse, but we permit $\stackrel{P\vec}{M}$ to decay. On the other hand, we permit $\stackrel{P\vec}{m}$, due to spin excitations, to both diffuse and decay. For this physical picture, diffusion from a given region occurs by decay of $\stackrel{P\vec}{M}$ to $\stackrel{P\vec}{m}$, then by diffusion of $\stackrel{P\vec}{m}$, and finally by decay of $\stackrel{P\vec}{m}$ to $\stackrel{P\vec}{M}$ in another region. This somewhat slows down the diffusion process. Restricting ourselves to the longitudinal variables $M$ and $m$ with equilibrium properties ${M}_{eq}={M}_{0}+{\ensuremath{\chi}}_{M\ensuremath{\parallel}}H$ and ${m}_{eq}=0$, we argue that the effective energy density must include a new, thermodynamically required exchange constant ${\ensuremath{\lambda}}_{M}=\ensuremath{-}1/{\ensuremath{\chi}}_{M\ensuremath{\parallel}}$. 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 $\ensuremath{\omega}$ there is, as usual, a single pair of complex wave vectors $\ifmmode\pm\else\textpm\fi{}k$ but with an unusual dependence on $\ensuremath{\omega}$. 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 $\mathrm{\ensuremath{\Delta}}$ can decay but not diffuse, but a nonequilibrium gap-like $\ensuremath{\delta}$, due to pair excitations, can both decay and diffuse.