Dynamics and Pinning for Skyrmions in Altermagnets

Abstract

We examine the dynamics, Hall angle, and pinning for N{é}el skyrmions in an altermagnet. Using an atomistic model, we show that skyrmion velocity and Hall angle dependence is anisotropic with respect to the direction of the drive, due to the fourfold symmetry implied by the two sublattices of the altermagnet. The skyrmion Hall angle and velocity at fixed drive show strong variations for increasing ratios of the exchange constant of the sublattices, $J_2/J_1$. This fourfold anisotropy of altermagnetic (ATM) skyrmions also leads to anisotropic pinning effects for an ATM skyrmion interacting with isotropic circular pinning sites. We also propose a simple particle model for this system that takes into account this anisotropy and find that it captures both the variations of the ATM skyrmion Hall angle and velocity as a function of drive direction, as also found in the atomistic simulations. Using this particle model, we examine ATM skyrmions interacting with a periodic array of pinning sites. For increasing ratios of $J_2/J_1$, we find a strongly non-monotonic ATM skyrmion velocity, where there is a minimum in the velocity where the skyrmion locks to different symmetry directions of the periodic pinning lattice. For a random array, we find that ATM skyrmions show strongly anisotropic depinning thresholds and velocity responses for different drive directions, and that the Hall angle is nearly constant with drive. In comparison, for the same parameters, the depinning threshold for a ferromagnetic (FM) skyrmion is lower, and the skyrmion Hall angle shows a strong velocity dependence. The lower depinning threshold for FM skyrmions is due to stronger Magnus forces.

Figures (10)

• Figure 1: (a) Schematic showing ATM skyrmions on sublattices $A$ and $B$. Sublattice $A$ ($B$) skyrmions have $Q=+1$ ($Q=-1$), so the sample has no net topological charge. Inset: colormap used to represent skyrmions throughout this work. The colors in the main panel correspond to $m_z=0$. As a consequence of the anisotropic exchange interactions, the skyrmions become elliptical. (b) Schematic of sublattice $A$ showing the square arrangement of atoms with lattice constant $a$ and the magnetic moment ${\bf m}_{i, j}^A$ (green). Bonds along $y$ have exchange constant $J_2$ (red) and bonds along $x$ have exchange constant $J_1$ (blue), giving an anisotropic exchange interaction. (c) The corresponding schematic of sublattice $B$ with magnetic moments ${\bf m}_{i, j}^B$ (orange). The exchange constants are swapped: bonds along $y$ have exchange constant $J_1$ (blue) and bonds along $x$ have exchange constant $J_2$ (red). (d) Three-dimensional view of a portion of our sample showing sublattices $A$ (green) and $B$ (orange) along with exchange constants $J_1$ (blue) and $J_2$ (red). A third exchange constant $J_3$ (black) couples sublattices $A$ and $B$ along $z$.
• Figure 2: Simulation results for an ATM system with $J_2/J_1=3$. (a) The skyrmion Hall angle $\theta_\mathrm{Hall}$ vs current direction $\phi$ at $j=1.0\times 10^{10}$ (black), $7.0\times 10^{10}$ (red), $13.0\times 10^{10}$ (blue), and $19.0\times 10^{10}$ (green). (b) The maximum Hall angle $\theta_\mathrm{Hall}^\mathrm{max}$ across all $\phi$ values vs $j$ (black) along with an exponential fit (red). (c) ATM skyrmion velocity, $v=\sqrt{v_x^2+v_y^2}$, vs $\phi$ at the same $j$ values from panel (a). (d) $v$ vs $j$ at different values of $\phi$. (e) Schematic of the current direction (black arrows) and the corresponding ATM skyrmion velocity (red arrows). Red arrows are in scale with respect to one another. The inset shows the two skyrmions on each sublattice, with nodes occurring where the curves cross.
• Figure 3: Simulation results for an ATM system with $j=5\times10^{10}$ A m$^{-2}$. (a) $\theta_\mathrm{Hall}$ vs $\phi$ at $J_2/J_1$=1.75 (black), 2.25 (red), 2.75 (blue), and 3.00 (green). (b) The maximum Hall angle $\theta_\mathrm{Hall}^\mathrm{max}$ across all $\phi$ values vs $J_2/J_1$ (black) along with an exponential fit (red). (c) Skyrmion velocity $v$ vs $\phi$ at the $J_2/J_1$ values from (a). (d) $v$ vs $J_2/J_1$ for different $\phi$ values. (e) Real space images of different ATM skyrmion configurations in the two layers with different $J_2/J_1$.
• Figure 4: Results of the ATM skyrmion behavior using the toy model from Eq. \ref{['eq:4']} with parameters $G=\mathcal{D}=v_e=1$, $\alpha=0.5$, $\beta=0.01\alpha$ and different $\kappa=1.5$ (black), 2.5 (red), 3.5 (blue), and 4.5 (green). (a) ATM skyrmion Hall angle $\theta_\mathrm{Hall}$ vs applied current angle $\phi$. (b) ATM skyrmion absolute velocity $v$ vs applied current angle $\phi$.
• Figure 5: (a) Atomistic simulation calculations of the interaction energy of ATM skyrmions as a function of $y$ vs $x$ position relative to the pinning site center for circular pinning sites with a radius of $1.5$ nm. The maximum energy occurs just before the skyrmion becomes core pinned. For $r < 15$ nm, the pinning energy is anisotropic due to the anisotropic shape of the skyrmion. (b) Cross-sections of $\Delta E$ vs $r$ for $\theta = 0.0^\circ$ (black), $32^\circ$ (red), $60^\circ$ (blue), and $88^\circ$ (green) taken along the dashed lines in panel (a).