Spin Accumulation and Longitudinal Spin Diffusion of Magnets

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 $\vec{M}$. The other is the non-equilibrium quantity $\vec{m}$, called the spin accumulation, by which the non-equilibrium spin current can be transported. $\vec{M}$ 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 $\vec{M}$ to diffuse, but we permit $\vec{M}$ to decay. On the other hand, we permit $\vec{m}$, due to spin excitations, to both diffuse and decay. For this physical picture, diffusion from a given region occurs by decay of $\vec{M}$ to $\vec{m}$, then by diffusion of $\vec{m}$, and finally by decay of $\vec{m}$ to $\vec{M}$ in another region. This somewhat slows down the diffusion process. Restricting ourselves to the longitudinal variables $M$ and $m$ with equilibrium properties $M_{eq}=M_{0}+\chi_{M\parallel}H$ and $m_{eq}=0$, we argue that the effective energy density must include a new, thermodynamically required exchange constant $\lambda_{M}=-1/\chi_{M\parallel}$. We then develop the macroscopic equations by applying Onsager's irreversible thermodynamics, and use the resulting equations to study the space and time response. At fixed real frequency $\omega$ there is, as usual, a single pair of complex wavevectors $\pm k$ but with an unusual dependence on $\omega$. At fixed real wavevector, there are two decay constants, as opposed to one in the usual case. Extending the idea that non-equilibrium diffusion in other ordered systems involves a non-equilibrium quantity, this work suggests that in a superconductor the order parameter $\Delta$ can decay but not diffuse, but a non-equilibrium gap-like $\delta$, due to pair excitations, can both decay and diffuse.