Current-induced torques in textured Rashba ferromagnets

In systems with small spin-orbit coupling, current-induced torques on the magnetization require inhomogeneous magnetization textures. For large spin-orbit coupling, such torques exist even without gradients in the magnetization direction. Here, we consider current-induced torques in ferromagnetic metals with both Rashba spin-orbit coupling and inhomogeneous magnetization. We first phenomenologically construct all torques that are allowed by the symmetries of the system, to first order in magnetization-direction gradients and electric field. Second, we use a Boltzmann approach to calculate the spin torques that arise to second order in the spin-orbit coupling. We apply our results to current-driven domain walls and find that the domain-wall mobility is strongly affected by torques that result from the interplay between spin-orbit coupling and inhomogeneity of the magnetization texture.

Figure 1

Average velocity of a Bloch ($z$) wall as a function of the applied field. The dashed (blue) line is the situation without SO coupling (${\lambda }_R=0$), the dot-dashed (blue) line represents current-driven domain-wall motion without torques induced by the SO coupling but with the parameters renormalized by the SO coupling, and the dotted (black) line shows the results with only the homogeneous SO torques, i.e., ${\mathit{\tau }}^{\left(1\right)}$ and ${\mathit{\tau }}^{(1\perp )}$ added to the STTs. The solid (red) line shows the result of the solution of the equations of motion including all spin torques. The parameters used to obtain these results are given in Table . The inset is an illustration of the tilted washboard potential of the ${\varphi }_{\mathrm{DW}}$ coordinate for increasing values of applied field. Due to the SO coupling there are inequivalent local minima. The angle points in the direction of the first local minimum, with slightly more field the angle makes a fast rotation to the second minimum. Above Walker breakdown there are no more local minima and the angle rotates in time.

Figure 2

Average velocity of a Néel ($x$) wall as a function of applied electric field in the $x$ direction. Lines are as in Fig. 1. The equations of motion can be found in the Appendix.

Figure 3

Average velocity of a Bloch ($y$) wall as a function of applied electric field in the $x$ direction. Lines are as in Fig. 1. The equations of motion can be found in the Appendix.