Intrinsic staggered spin-orbit torque for the electrical control of antiferromagnets -- application to CrI$_3$

Abstract

Spin-orbit torque enables the electrical control of the orientation of ferromagnets' or antiferromagnets' order parameter. In this work we consider antiferromagnets in which the magnetic sublattices are connected by inversion+time reversal symmetry, and in which the exchange and anisotropy energies are similar in magnitude. We identify the staggered dampinglike spin-orbit torque as the key mechanism for electrical excitation of the Néel vector for this case. To illustrate this scenario, we examine the 2-d Van der Waals antiferromagnetic bilayer \ch{CrI3}, in the $n$-doped regime. Using a combination of first-principles calculations of the spin-orbit torque and an analysis of the ensuing spin dynamics, we show that the deterministic electrical switching of the Néel vector is the result of dampinglike spin-orbit torque which is staggered on the magnetic sublattices.

Figures (8)

• Figure 1: (Color online) (a) shows a top-down view of one layer of CrI3. The second layer (not shown) is displaced along the $x$-direction by a nearest-neighbor distance. (b) Side view of CrI3. Note a lack of symmetry with respect to $x\rightarrow -x$. Other symmetries depend on the spin configuration: for a purely antiferromagnetic state, the system is invariant under inversion+time-reversal. For a state with canting in the $y$ direction, the system has a 2-fold rotational symmetry about the $y$-axis. (c) Spin configurations on magnetic sublattices A and B considered in this work, with finite canting in the $y$-direction. (d) Mixed representation of system spin in $\hat{{\bf N}}=(L_x,M_y,L_z)$ space, showing the spin-orbit torque switching trajectory of $\hat{{\bf N}}$ for applied electric field in the $y$-direction.
• Figure 2: (Color online) Angular dependence of the dampinglike (a) and fieldlike (b) torkance on the $\hat{\mathbf{N}}$ direction $(\theta,\phi)$ for one layer of bilayer CrI3 under an external electric field along the $\hat{y}$ direction at Fermi level $\mu=50~\rm{meV}$ above the conduction band minimum. The arrow (color) on the sphere indicates the direction (magnitude) of the torkance at the given ${\bf N}$ direction. We use $k_BT=3~\rm{meV}, \eta=25~\rm{meV}$.
• Figure 3: (Color online) Torkance as a function of chemical potential relative to the conduction band edge. The applied electric field is in $\hat{{\bf y}}$ direction. The $\hat{\mathbf{N}}$ vector is in $\hat{{\bf z}}$ (a) and $\hat{{\bf x}}$ (b). Red and blue lines represent staggered time-reversal even torkance and uniform time-reversal odd torque, respectively. The torkance for Fermi energies in the valence band are substantially smaller and not shown here.
• Figure 4: (Color online) Magnetization dynamics under spin-orbit torque, for applied electric field in the $\hat{\bf y}$ direction. (a) and (b) show the Néel and magnetization vector components as a function of time with applied electric field strength $-1.2$ V/µm and $-3.5$ V/µm, respectively. The initial configuration is $L_z=1$. Red, black, and blue lines represent the dynamics of $L_x$,$M_y$, and $L_z$ respectively. (c) and (d) show the final steady state of $\hat{\mathbf{N}}$ as a function of applied field with staring point at the $L_z=\pm1$ respectively. The spread in the $y$ coordinate indicates the oscillation amplitude, and the color of the spread represents the oscillation frequency.
• Figure 5: (Color online) Final steady state of $\hat{\mathbf{N}}$ as a function of applied field with staring point at the $L_z=+1$ for various chemical potentials respectively. The spread in the $y$ coordinate indicates the oscillation amplitude, and the color of the spread represents the oscillation frequency.