Domain wall motion driven by an electric current

We have recently proposed [1] that an electric current in a ferromagnetic film generates two mutually orthogonal spin torques, $\mbox{\boldmath $\tau$}_1 = b_J {\bf M}\times ({\bf M} \times \frac{\partial \bf M}{\partial x})$ and $\mbox{\boldmath $\tau$}_2 = c_J {\bf M}\times \frac{\partial \bf M}{\partial x} $ where ${\bf M}$ is the magnetization vector and the constants $b_J$ and $c_J$ are proportional to the current density. By including these two spin torques in the Landau-Lifshitz-Gilbert equation; we have simulated the domain motion in a number of experimentally accessible geometries. We have found that the current-driven domain wall motion displays many unique features compared to that driven by an external field. One particular example is to predict the critical current as a function of the applied magnetic field in a ``constriction'' geometry where the domain wall is originally trapped before applying an electric current. The calculated critical current densities are compared to the existing experimental data. [1] S. Zhang and Z. Li, Phys. Rev. Lett. {\bf 93}, 127204 (2004).