Dynamics of antiskyrmion shrinking

TL;DR

We develop a continuum collective-coordinate theory for the shrinking of antiskyrmions in isotropic bulk DMI using a triangular elliptical ansatz, yielding four coupled dynamics for the semi-axes $a_0,b_0$, helicity $\varphi_0$, and rotation $\omega$. Without DMI, the semi-axes decouple from helicity and rotation, driving ellipticity to zero (toward circularity) with an exponential-to-square-root collapse, while helicity grows linearly and then logarithmically near collapse; with finite DMI, the semi-axes couple to helicity and $\omega$, producing quadrupole-like oscillations, a helicity that grows linearly and diverges logarithmically, and a rotation $\omega$ that follows half the helicity slope with pitchfork-like phase behavior. Numerical LLG simulations on a lattice qualitatively confirm these predictions, showing isotropic shrinking at $D=0$ and ellipticity-driven dynamics with quadrupolar breathing and $\omega$-locking-and-unlocking behavior at finite $D$. The results illuminate the complex shrinking pathways of antiskyrmions and their potential manipulation as information carriers in spintronic platforms. Overall, the work provides a tractable continuum framework that captures the essential nonlinear couplings between shape, helicity, and in-plane rotation in antiskyrmion shrinking.

Abstract

Antiskyrmions are unstable in ferromagnetic systems with isotropic bulk or interfacial Dzyaloshinskii-Moriya interaction (DMI). We develop a continuum model for the shrinking dynamics of antiskyrmions in bulk DMI systems, using the Landau-Lifshitz-Gilbert equation for the time derivative of the magnetization field. Owing to the structure of their azimuthal angle, or helicity, elliptic antiskyrmions are energetically favored over circular ones. To capture this feature, we parametrize the magnetization field with a triangular radial profile and an elliptic in-plane shape. This ansatz yields four coupled dynamical equations governing time evolution of the semi-axes, helicities, and rotation angles. In the absence of the DMI, circular antiskyrmions shrink isotropically, exhibiting a crossover from exponential decay to square-root collapse. Initially elliptic antiskyrmions are driven towards circularity. For finite DMI, the semi-axes dynamics couples to the helicity and rotation, where the theory predicts a rotation angle following by half of the slope of the helicity evolution which is linear in time. Only at small semi-axes a cross-over to a logarithmic divergence occurs. The shrinking dynamics of the antiskyrmion size is found to be accompanied by quadrupole-like oscillations. Numerical simulations on the lattice support the predictions from the continuum model.

Figures (15)

• Figure 1: Color map of the $z$-component of Eq. \ref{['eq:vectorfield']} parametrized via the elliptic triangular approach, highlighting the semi-axes $a_0$, $b_0$, and the rotation angle $\omega$.
• Figure 2: Color maps of the total energy $E(\varphi_0, \omega)$ for elliptic antiskyrmions obtained from numerical integration of Eq. \ref{['eq:totalEnergy']}. Panel (a) shows the case $\tilde{D}=0.02$, $\tilde{B}=0.02$, $a_0=20$, $b_0=17$, and panel (b) shows $\tilde{D}=0.04$, $\tilde{B}=0.02$, $a_0=15$, $b_0=4$. The color maps illustrate the $\pi$-periodicity of the energy landscape with respect to $\varphi_0$ and $\omega$, with energy minima along paths given by Eq. \ref{['eq:minimumEnergyCondition']}. Energy maxima occur along lines shifted by $\pi/2$ relative to the minima. While the absolute energies vary with $a_0$, $b_0$, and $\tilde{D}$, the general structure of the color maps remains the same.
• Figure 3: (a) Color map of the total energy $E(a_0, b_0)$ for elliptic antiskyrmions, with cuts indicated by colored dashed lines. (b) Energy along these cuts $E(b_0)$, at fixed values $a_0$ of the major semi-axis. Smaller antiskyrmions generally have lower energy than larger ones. Along cuts for $a_0=$const., the energy exhibits a minimum at $b_0 < a_0$ for a finite $b_0>0$. The magenta line in (a) indicates the minimum energy path on the energy surface, while the gray line corresponds to $a_0 = b_0$. Elliptical antiskyrmions are energetically favorable compared to circular ones. The parameters are $\tilde{D}=0.02$, $\tilde{B}=0.02$, $\varphi_0=0$, and $\omega=\pi/2$.
• Figure 4: Energy density of an elliptic triangular antiskyrmion for two cases of the in-plane rotation angle, $\omega = 0$ (panels (a), (c), (e), (g)) and $\omega = \pi/2$ (panels (b), (d), (f), (h)). Panels (a) and (b) show the total energy density including all energy contributions, with the vector field configuration indicated by the pictogram ($\varphi_0 = \pi/2$). Panels (c) and (d), (e) and (f), and (g) and (h) show the contributions from the exchange, DMI, and Zeeman terms, respectively. While the exchange and Zeeman contributions are rotationally invariant, the DMI contribution varies with $\omega$, leading to a higher total energy density for $\omega = 0$ compared to $\omega = \pi/2$. Parameters: $\tilde{D}=0.2$, $\tilde{B}=0.02$, $a_0=12$, $b_0=8$, $\varphi_0=\pi/2$.
• Figure 5: Time evolution of the semi-axes and helicity obtained from the numerical solution of the coupled differential equations following the triangular ansatz. Panels (a) and (b) show the circular case with $a_{0,0}=b_{0,0}=100$. Here, the semi-axes dynamical equation reduces to Eq. \ref{['eq:dota0circular']}, and the helicity follows Eq. \ref{['eq:dotphi0circular']}. Panels (c) and (d) show the elliptical case with $a_{0,0}=100$, $b_{0,0}=40$. Both semi-axes shrink exponentially at large semi-axes with the same rate determined by the Zeeman term. The exchange term dominates at small semi-axes and drives $\chi=a_0/b_0 \to 1$, pushing the system towards the circular limit and the subsequent time evolution follows a square root-like collapse of the semi-axes. The helicity exhibits a linear growth at large semi-axes and a logarithmic divergence as $t\to t_c$ at small semi-axes. Parameters: $\tilde{B}=0.02$, $\alpha=0.1$ and $\varphi_{0,0}=\pi/2$.