Calculation of current-induced torque from spin continuity equation

Current-induced torque is formulated based on the spin continuity equation. The formulation does not rely on the assumption of separation of local spin and charge degrees of freedom, in contrast to approaches based on the $s\text{−}d$ model or mean-field approximation of itinerant ferromagnetism. This method would be thus useful for the estimation of torques in actual materials by first-principles calculations. As an example, the formalism is applied to the adiabatic limit of the $s\text{−}d$ model in order to obtain the analytical expression for torques and the corresponding $\beta$ terms arising from spin relaxation due to spin-flip scattering and spin-orbit interaction.

Figure 1

Diagrammatic representation of $D_{\mu \nu }^{\beta \gamma \delta }$. Double-dashed, dotted, and wavy lines denote interaction with impurity spin, applied electric field $\mathit{E}$, and gauge field $\mathit{A}$, respectively.