Definition of the spin current: The angular spin current and its physical consequences

TLDR

Spin transport is described by two currents: a linear current for translational motion and an angular current for rotational precession, both arising naturally in the spin continuity equation. The study aims to define the linear and angular spin current densities and examine their physical properties. The authors derive explicit expressions for the linear and angular spin current densities and analyze their properties within the spin continuity framework. They find that introducing an angular spin current is necessary to fully describe spin transport, that it can induce an electric field analogous to the linear spin current, and that this field scales as 1/r² compared to 1/r³ for the linear current.

Abstract

We find that in order to completely describe the spin transport, apart from spin current (or linear spin current), one has to introduce the angular spin current. The two spin currents, respectively, describe the translational and rotational motion (precession) of a spin. The definitions of these spin current densities are given and their physical properties are discussed. Both spin current densities appear naturally in the spin continuity equation. In particular, we predict that the angular spin current (or the spin torque as called in previous works), similar to the linear spin current, can also induce an electric field $\stackrel{P\vec}{E}$. The formula for the induced electric field $\stackrel{P\vec}{E}$ by the angular spin current element is derived, playing the role of ``Biot-Savart law'' or ``Ampere law.'' When at large distance $r$, this induced electric field $\stackrel{P\vec}{E}$ scales as $1∕{r}^{2}$, whereas the $\stackrel{P\vec}{E}$ field generated from the linear spin current goes as $1∕{r}^{3}$.