Antiferromagnetic domain walls under spin-orbit torque

TL;DR

The paper addresses how antiferromagnetic domain walls respond to spin-orbit torques, revealing dynamics that differ from ferromagnets and depend on current polarization. Using a continuum nonlinear sigma-model with damping and spin-torque terms, it analyzes three regimes: precessional walls under perpendicular polarization, steady-propagating walls under in-plane polarization, and oscillatory walls when the polarization has both components. Key results include a spontaneously precessing wall with frequency $\omega = -\beta/\alpha$ and width $\ell = 1/\sqrt{1-\omega^2}$, a low-current linear velocity $v = (\pi/2)(\beta/\alpha)$ for propagating walls, and an oscillatory motion with $v = (\pi/2)(\beta_2/\alpha)\cos(\beta_3\tau/\alpha)$ and $\omega = \beta_3/\alpha$. The study shows how damping breaks Lorentz invariance and yields observable signatures such as current-dependent wall magnetization, offering guidance for AFM spintronic devices and texture imaging.

Abstract

Domain walls in antiferromagnets under a spin-polarized current present rich dynamics that is not observed in ferromagnets, and it is tunable by the current polarization. Precessional dynamics is obtained for perpendicular spin polarization, in agreement with expectations in older works. Propagating walls are obtained for an in-plane polarization. We obtain the velocity as a function of current by a perturbation method for low velocities, and the wall profile is found to lack a definite parity. For high velocities, the main features of the wall profile are obtained by a direct solution of an equation that is valid in a limiting case. We discuss the magnetization of the dynamical walls and find that this can become large, providing a potential method for observations. Oscillatory motion of domain walls is obtained for spin polarization that has both perpendicular and in-plane components, and an analytical description is given.

Figures (5)

• Figure 1: (a) Traveling domain walls with a constant velocity found numerically for damping parameter value $a=1$ (the large value of $\alpha$ is used to amplify some features of the wall). The applied spin torques are $\beta = 0.1$, $\beta= 0.3$ and $\beta= 0.498$ (the domain wall for $\beta=0$ is shown for comparison) and the corresponding velocities are $v\approx 0.156$, $v\approx 0.45$ and $v\approx 0.75$. The Néel vector at the ends of the wall is tilted with respect to the south ($\Theta=\pi$) and the north pole ($\Theta=0$). The wall profile for $v\neq 0$ is not symmetric with respect to the center of the wall. (b) The numerically obtained domain wall profile (solid line) for $\beta=0.498$ is compared with the analytical result in Eq. \ref{['eq:DW_high_spintorque']} (dashed line). We have power-law decay for $\xi\to-\infty$ as predicted in \ref{['eq:DW_high_spintorque']} but exponential decay for $\xi\to\infty$.
• Figure 2: The velocity $v$ of the propagating domain wall as a function of the parameter $\beta$ for various values of the damping parameter $\alpha$. The spin current is polarized in $\bm{\hat{e}}_2$. The solid lines give the theoretical formula for low currents, $v=\frac{\pi}{2}\,\frac{\beta}{\alpha}$.
• Figure 3: Propagating domain wall width, defined as $\ell=1/\Theta'$ at the point with $\Theta=\pi/2$, as a function of the velocity $v$ for damping parameter values $\alpha=0.25$ (red) and $\alpha=0.8$ (blue). The maximum velocity is below the value $v=1$ for both values of $\alpha$. The solid line (black) gives the domain wall width in the Lorentz invariant case, $\ell=\sqrt{1-v^2}$.
• Figure 4: Under spin current with $\beta\bm{\hat{p}} = \beta_2\bm{\hat{e}}_2 + \beta_3\bm{\hat{e}}_3$, the domain wall undergoes oscillatory motion, moving periodically between two end positions, $-x_{\rm min} < x < x_{\rm max}$. The domain wall is shown at approximately (a) the left end position $x=x_{\rm min}$, (b) at a central position, and (c) at the right end position $x=x_{\rm max}$ of the motion. The parameters are $\beta_2=\beta_3=0.05$ and $\alpha=1$. The in-plane components indicate that the Néel vector is precessing during the motion.
• Figure 5: (a) The position $x$ for an oscillating domain wall as a function of time given by a simulation for parameter values $\beta_2=\beta_3=0.05$ and $a=1$. (b) The velocity $v$ in the simulation is shown by blue points. This is compared with the result \ref{['eq:velocity_oscillations']} shown by a red solid line. It is seen that the system quickly achieves the oscillating steady state.