We consider a microscopic model of itinerant electrons coupled via ferromagnetic exchange to a local magnetization whose direction vector $\mathit{n}(\mathit{r},t)$ varies in space and time. We assume that to first order in the spatial gradients and time derivative of $\mathit{n}(\mathit{r},t)$, the magnetization distribution function $\mathit{f}(\mathbf{p},\mathbf{r},t)$ of itinerant electrons with momentum $\mathbf{p}$ at position $\mathbf{r}$ and time $t$ has the ansatz form $\mathit{f}(\mathbf{p},\mathbf{r},t)=f_{\Vert }\left(\mathit{p}\right)\mathit{n}(\mathbf{r},t)+f_{1\mathbf{r}}\left(\mathbf{p}\right)\mathit{n}\times {\nabla }_{\mathbf{r}}\mathit{n}+f_{2\mathbf{r}}\left(\mathbf{p}\right){\nabla }_{\mathbf{r}}\mathit{n}+f_{1t}\left(\mathbf{p}\right)\mathit{n}\times {\partial }_t\mathit{n}+f_{2t}\left(\mathbf{p}\right){\partial }_t\mathit{n}$. Using the Landau-Sillin equations of motion approach (Zh. Eksp. Teor. Fiz. 33, 1227 (1957) [Sov. Phys. JETP 6, 945 (1958)]), we derive explicit forms for the components $f_{\Vert }\left(\mathbf{p}\right)$, $f_{1\mathbf{r}}\left(\mathbf{p}\right)$, $f_{2\mathbf{r}}\left(\mathbf{p}\right)$, $f_{1t}\left(\mathbf{p}\right)$, and $f_{2t}\left(\mathbf{p}\right)$ in “equilibrium” and in out-of-equilibrium situations for (i) no scattering by impurities, (ii) spin-conserving scattering, and (iii) spin-nonconserving scattering. The back action on the localized electron magnetization from the out-of-equilibrium part of the two components $f_{1\mathbf{r}}$ and $f_{2\mathbf{r}}$ constitutes the two spin transfer torque terms.

• Received 16 November 2006

DOI:https://doi.org/10.1103/PhysRevB.75.174414