Spin Accumulation and Longitudinal Spin Diffusion of Magnets

TL;DR

This paper develops a theoretical framework for understanding spin accumulation and longitudinal spin diffusion in magnets, introducing new thermodynamic parameters and equations to describe non-equilibrium spin transport.

Contribution

It extends spintronics theory to include longitudinal spin variables, deriving macroscopic equations that account for diffusion and decay of spin accumulation.

Findings

Derived new thermodynamic exchange constant $\lambda_{M}$.

Identified complex wavevector dependence on frequency.

Predicted two decay constants at fixed wavevector.

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 $\vec{M}$. 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.