Nonequilibrium Rashba field driven domain wall motion in ferromagnetic nanowires

We study the effects of spin-orbit interaction (SOI) on the current-induced motion of a magnetic (Bloch) domain wall in ultrathin ferromagnetic nanowires. The conspiracy of spin relaxation and SOI is shown to generate a strong nonequilibrium Rashba field, which can dominate even for weak SOI. This field causes intricate spin precession and a transition from translatory to oscillatory wall dynamics with increasing SOI. We show that current pulses of different lengths can be used to efficiently control the domain wall motion.

Figure 1

Velocity of the DW center vs propagation time for (a) different ${\alpha }_R$ and (b) different $I_c$ (${10}^{12}\mathrm{A}/{\mathrm{m}}^2$). An increase of either ${\alpha }_R$ or $I_c$ increases the Rashba field $H_R$. At the critical value $H^{\mathrm{WB}}$, we find a current-induced Walker breakdown and the velocity starts to oscillate. Further increase of ${\alpha }_R$ or $I_c$ increases the oscillation frequency.

Figure 2

Momentary Rashba field $H_R$ (in Tesla) close to the DW center (taken at $x_{\mathrm{DW}}=0$). We put $\alpha ={\beta }^{\left(i\right)}={\beta }^{\left(ii\right)}\equiv \beta$ and show the behavior for (a) different $\beta$ (with $I_c={10}^{12}$ A$/$m${}^2$) and (b) for different current densities $I_c$ (in ${10}^{12}$A$/$m${}^2$), with $\beta =0.06$. For comparison, the $n_y$ component of the magnetization is shown as a dashed curve. The last two panels illustrate the Rashba field $H_R\left(x_{\mathrm{DW}}\right)$ and the contribution $H_R^{\left(1\right)}$ directly at the DW center: (c) $H_R$ vs $\beta$ and (d) $H_R$ vs $I_c$.

Figure 3

Velocity of the DW center vs propagation time for five current pulses of different length $t_p$. After the pulse, the DW drifts for a certain time in either positive or negative direction (depending on the oscillation phase at the pulse end). The dashed curve shows the corresponding result for a steady-state current.

Figure 4

(a) Momentary velocity of the DW center vs propagation time for a steady-state current. (b) Average DW velocity $\langle v_{\mathrm{DW}}\rangle =x_{\mathrm{DW}}(t\to \infty )/t_p$ vs current pulse duration $t_p$ with $I_c$ (${10}^{12}\mathrm{A}/{\mathrm{m}}^2$). Depending on the momentary velocity at the pulse end, the DW drifts in different directions. Especially for short pulses, this may strongly affect $\langle v_{\mathrm{DW}}\rangle$.

Figure 5

Global average velocity of the DW center $\langle v_{\mathrm{DW}}\rangle$ vs current density $I_c$ for fixed pulse duration $t_p$. Since $I_c$ determines the oscillation frequency, the pulse can end up in different oscillation states, causing positive or negative drift. This can strongly change $\langle v_{\mathrm{DW}}\rangle$ and causes DW motion against the current direction for small $I_c$.

Figure 6

The magnetization profile in the vicinity of the DW center, $x\approx x_{\mathrm{DW}}$, at different times $t$ (in nanoseconds, from the bottom to the top). Arrows indicate $n_{x,z}(x-x_{\mathrm{DW}},t)$, while $n_y$ is color coded. For clarity, we show the magnetization profile right at the DW center, $x=x_{\mathrm{DW}}$, in a separate column on the right of the profile. Due to the finite Rashba field, the magnetization precesses around the $y$ axis, i.e., the direction of the Rashba field. Parameters are as in Fig. 3 for constant current.