Domain Wall Dynamics under Nonlocal Spin-Transfer Torque

We study spin-diffusion effects within a continuously variable magnetization distribution, integrating with micromagnetics the diffusive model of Zhang and Li [Phys. Rev. Lett. 93, 127204 (2004)]. Current-driven wall motion is, in the steady velocity regime, shown to be adequately described by an effective nonlocal nonadiabatic parameter. This parameter is found to be 20% larger than its local counterpart for a vortex wall in a NiFe nanostrip and hardly modified for a transverse wall. This may account for the yet unexplained experimental evidence that vortex walls move more easily under current when compared with transverse walls. It is shown that this effective parameter can be derived from the domain wall structure at rest.

Figure 1

Calculation results for an ATW in a NiFe nanostrip with $300\times 5{\mathrm{nm}}^2$ cross section. (a) Magnetization structure of the ATW at rest, colored according to the inplane magnetization angle. (b)–(d) Nonequilibrium spin density for the DW at rest, obtained with ${\tau }_{\mathrm{sd}}/{\tau }_{\mathrm{sf}}=0.04$ and $u=-10\mathrm{m}/\mathrm{s}$, shown by its three components along $x$ (b), $y$ (c), and $z$ (d). The color scale spans $\pm 3\times {10}^{-5}$ for the inplane components and $\pm 5\times {10}^{-4}$ for the $z$ component. (e)–(f) Maps of ${\beta }^{*}$ obtained for the DW at rest (e), and moving at steady velocity (f), the colors spanning the 0–0.1 interval. The color scale for images (b)–(f) is shown in the center, as well as the direction of motion of the carriers. All images have the same 300 nm height. (g) Plot of ${\beta }^{*}$ and $u^{*}$ along two horizontal lines, close to the tip of the ATW, for the DW structure at rest. The line shows the local value $\beta =0.04$.

Figure 2

Same results as in Fig. 1, for the case of a VW. The spin-density color range has become the large one for the $x$ component (b). The plots (g) of ${\beta }^{*}$ and $u^{*}$ now correspond to horizontal and vertical lines across the vortex core. Note that the scales are identical to those of Fig. 1.

Figure 3

Numerically computed steady-state DW velocities $v$ as a function of $u$ and for different values of ${\tau }_{\mathrm{sd}}$ (hence $\beta$), for the ATW (filled symbols) and the VW (open symbols). (a) Comparison of the computed velocities with spin diffusion to the analytical results in the local approximation [Eq. (4), solid lines]. (b) Relative increase of velocity due to spin diffusion. The horizontal lines display the evaluation of ${\beta }_{\mathrm{diff}}$ from the weighted average of ${\beta }^{*}$ computed for the DW structures at rest.