Year: 2007
Type: article
Abstract: We consider a microscopic model of itinerant electrons coupled via ferromagnetic exchange to a local magnetization whose direction vector $\mathbit{n}(\mathbit{r},t)$ varies in space and time. We assume that to first order in the spatial gradients and time derivative of $\mathbit{n}(\mathbit{r},t)$, the magnetization distribution function $\mathbit{f}(\mathbf{p},\mathbf{r},t)$ of itinerant electrons with momentum $\mathbf{p}$ at position $\mathbf{r}$ and time $t$ has the ansatz form $\mathbit{f}(\mathbf{p},\mathbf{r},t)={f}_{\ensuremath{\Vert}}(\mathbit{p})\mathbit{n}(\mathbf{r},t)+{f}_{1\mathbf{r}}(\mathbf{p})\mathbit{n}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\nabla}}_{\mathbf{r}}\mathbit{n}+{f}_{2\mathbf{r}}(\mathbf{p}){\ensuremath{\nabla}}_{\mathbf{r}}\mathbit{n}+{f}_{1t}(\mathbf{p})\mathbit{n}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\partial}}_{t}\mathbit{n}+{f}_{2t}(\mathbf{p}){\ensuremath{\partial}}_{t}\mathbit{n}$. 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 ${f}_{\ensuremath{\Vert}}(\mathbf{p})$, ${f}_{1\mathbf{r}}(\mathbf{p})$, ${f}_{2\mathbf{r}}(\mathbf{p})$, ${f}_{1t}(\mathbf{p})$, and ${f}_{2t}(\mathbf{p})$ 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.
Source: Physical Review B
Authors: Frédéric Piéchon, André Thiaville
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2041199379