Nonequilibrium dynamics of magnetic hopfions driven by spin-orbit torque

TL;DR

This work investigates the nonequilibrium dynamics of magnetic hopfions under spin-orbit torque by solving the Landau-Lifshitz-Gilbert equation for Hopf numbers $H=1$ to $4$ in a frustrated cubic lattice. It reveals helicity-dependent translational and precessional motion for $H=1$ and identifies an effective tension mechanism that forcibly splits $H\ge2$ hopfions into lower-$H$ constituents, with a hierarchical steady-state phase diagram guiding predictions for higher $H$. The authors develop a quantitative framework linking SOT strength and magnetic field to splitting thresholds, and demonstrate that time-dependent SOT can repeatedly induce splitting and recombination, enabling controllable topology switching. These findings point to the potential for multilevel, topology-based spintronic devices and provide a practical route toward manipulating 3D topological textures in frustrated magnets.

Abstract

Hopfions--three-dimensional topological solitons with knotted spin texture--have recently garnered attention in topological magnetism due to their unique topology characterized by the Hopf number $H$, a topological invariant derived from knot theory. In contrast to two-dimensional skyrmions, which are typically limited to small topological invariants, i.e., skyrmion numbers, hopfions can, in principle, be stabilized with arbitrary Hopf numbers. However, the nonequilibrium dynamics, especially interconversion between different Hopf numbers, remain poorly understood. Here, we theoretically investigate the nonequilibrium dynamics of hopfions with various Hopf numbers by numerically solving the Landau-Lifshitz-Gilbert equation with spin-orbit torque (SOT). For $H=1$, we show that SOT induces both translational and precessional motion, with dynamics sensitive to the initial orientation. For $H=2$, we find that intermediate SOT strengths can forcibly split the hopfion into two $H = 1$ hopfions. This behavior is explained by an effective tension picture, derived from the dynamics observed in the $H=1$ case. By comparing the splitting dynamics across different $H$, we identify a hierarchical structure governing SOT-driven behavior and use it to predict the dynamics of hopfions with general $H$. Furthermore, we show that by appropriately scheduling the time dependence of the SOT, it is possible to repeatedly induce both splitting and recombination of hopfions. These results demonstrate the controllability of hopfion topology via SOT and suggest a pathway toward multilevel spintronic devices based on topology switching.

Figures (12)

• Figure 1: Schematics of the magnetic hopfions and the setup for this study. (a) Magnetic hopfion with Hopf number $H = 1$ (left) and $H=2$ (right). The gray arrows represent the toroidal moments [Eq. \ref{['eq:toroidal']}]. The lower right insets in left and right panels display their two preimages linked with each other once and twice, respectively. The colors represent spins, with their $S^x$ and $S^y$$(S^z)$ components indicated by color (grayscale), as shown in the upper right inset. We use this color scheme throughout this paper. (b) A typical setup for this study. An electric current in the lower heavy metal layer with strong spin-orbit coupling (green arrow) generates a spin current in a perpendicular direction (blue and orange arrows), injected into the upper magnetic layer. The spin current exerts the SOT on the magnetic moments in the magnetic layer. The schematic shows a splitting of the hopfion with $H=2$ into two individual hopfions with $H=1$ by the SOT.
• Figure 2: SOT-driven dynamics of the hopfion with $H = 1$ under $\zeta = 0.001$ and $B = 0.003$, starting from the initial state with (a) the helicity $\eta=0$ and (b) $\eta=\pi$. The insets show the 2D spin structures on the slice at $z=0$ of the initial state. The colored and gray curves denote the real-space trajectory and its projections onto each boundary plane up to $\tau = 10000$, respectively. In (a) and (b), the hopfion reaches (-29, -13, -36) and (28, 12, -36) at $\tau=10000$, respectively. The lower panels display time evolutions of the hopfion's position, velocity, toroidal moment, and precessional axis of the toroidal oscillation during the dynamics up to $\tau=50000$.
• Figure 3: (a) Steady-state phase diagram of the hopfion with $H = 1$ under the magnetic field $B$ and the SOT with strength $\zeta$. The points at $\zeta = 0$ correspond to the initial states obtained by following the procedures in Secs. \ref{['sec:ansatz']} and \ref{['sec:optimization']}, while the other points correspond to the steady states of the LLG analyses performed using those initial states up to $\tau=50000$. The hatched region for $B \geq 0.0035$ is not accessible due to the instability of the initial state in the absence of the SOT. (b) Terminal velocity $v$ of the hopfions as a function of $\zeta$ in the stable region, given by the norm of Eq. \ref{['eq:velocity']} with $\tau=50000$ and $\Delta \tau = 1000$. (c) Averaged magnetization as a function of $\zeta$ at $B=0.003$ and $\tau = 50000$. The colored symbols and the gray lines correspond to the numerical results and the trivial solution in Eq. \ref{['eq:forcedFM']}, respectively. The background colors represent the stable and unstable regions estimated from (a).
• Figure 4: Snapshots of the hopfion with $H=2$ under $B=0.003$ and $\zeta = 0.002$. All panels except the lower-right one include insets showing the preimages on which $S^x \simeq +1$ (blue) or $S^x \simeq -1$ (red). For $\tau \gtrsim 19000$, the hopfion with $H=2$ splits into two hopfions, each with $H=1$, whose helicities $\eta$ differ by $\Delta \eta = \pi$. The viewpoint is varied with time for better visibility.
• Figure 5: (a) Steady-state phase diagram of the hopfion with $H = 2$. The notations are common to Fig. \ref{['fig3:H1_diagram']}(a). The splitting of the hopfion into two individuals with $H=1$ occurs in the blue shaded region. (b) Terminal velocity $v$ of the hopfions as a function of $\zeta$ in the stable region, obtained by the same manner as in Fig. \ref{['fig3:H1_diagram']}(b). The dashed lines represent the $\zeta$ range where splitting occurs.