Universal two-stage dynamics and phase control in skyrmion formation

Abstract

We uncover a universal two-stage dynamics during skyrmion formation and establish its connection to equilibrium phases through the introduction of a chiral correlation $χ$. Stage I involves stripe coarsening governed by the exchange-to-DMI ratio $J'$, while stage II entails stripe contraction driven by the synergy between $J'$ and the anisotropy-to-DMI ratio $K'$. The magnetic field-to-DMI ratio $B'$ influences both stages. By combining symbolic regression with neural networks, we model the competition and cooperation among these parameters and derive a skyrmion formation criterion, $0.58 K'J' + μB'J' > 1$. Our model disentangles their distinct roles: $J'$ sets the stripe width, $K'$ primarily controls the skyrmion size, and $B'$ strongly affects the topological charge. This approach provides a general framework for predicting and controlling magnetic phases in chiral magnets.

Figures (4)

• Figure 1: Universal two-stage dynamics of skyrmion formation. (a) Evolution of the normalized chiral correlation $\chi$ (black, left axis) and topological charge $Q$ (pink, right axis) on a logarithmic time scale. Stage I (gray shading) shows coarsening of stripe-like domains, characterized by increasing $\chi$ and oscillatory $Q$, while stage II exhibits stripe contraction into skyrmions, marked by decreasing $\chi$ and saturation of $Q$. Spin snapshots illustrate the morphological evolution during the two stages. (b) Evolution of the average stripe width $d$ (all length values are normalized by the lattice constant $a$) on a logarithmic time scale, highlighting near-linear growth in stage I (inset) and oscillatory deformation in stage II. (c) Zoomed-in view of a stripe at 16 ps (stage II) undergoing rotational contraction into a skyrmion (left). The corresponding map of $\partial \epsilon/\partial\theta$ (right) shows the derivative of the local energy $\epsilon_i$ with respect to the spin polar angle $\theta$, where positive values (purple) indicate the driving torque that aligns spins along the $z$ axis.
• Figure 2: Two-stage dynamics and converged states under parameter variation. Time evolution of the normalized chiral correlation $\chi$ under variation of (a) exchange interaction $J$, (b) anisotropy $K$, and (c) magnetic field $B$. The reference parameters are $J=3$, $D=1$, $K=0.3~\mathrm{meV}$, and $B=1~\mathrm{T}$ (black curve in each panel). Grey shading indicates stage I (coarsening) and stage II (contraction). (d) Representative snapshots of converged states, marked by symbols: labyrinth domain (Lbr, dark triangle), skyrmion (Sk, blue, purple, and green triangles), and ferromagnetic (FM, red triangle) states. (e) Influence of individual energy terms on a labyrinth‐domain segment in (d), evaluated via $\partial \epsilon/\partial\theta$ for the exchange (ex), anisotropy (ani), DMI, and Zeeman (Z) energies. The grey arrow and the dashed arrow denote the initial orientations of the spins, while the green and purple arrows denote the rotational tendencies induced by each energy term, driving the spins away from or towards the $z$ axis, respectively. (f–h) Converged values of $\chi$ and the energy difference relative to the FM state, as functions of $J$, $K$, and $B$. The red solid circle-lines denote $E - E_\mathrm{FM}$, and the intersections with the red dashed lines define the phase boundaries (Lbr–Sk and Sk–FM). The grey dashed line indicates the significant chiral order with $\chi$ = 0.05.
• Figure 3: Machine-learning prediction of magnetic phase diagrams and skyrmion morphology control. (a) Zero-field phase diagram in the $J'–K'$ parameter space. Phase boundaries are marked by red dashed lines (Lbr–Sk) and white dashed lines (Sk–FM). The color scale indicates the chiral correlation $\chi$. (b) Phase diagram in the $J'–B'$ plane at fixed $K' = 0.2$. The dot–dash line marks the skyrmion lattice (SkX) region stabilized at low $J'$ and intermediate $B'$. (c) Confusion matrix of the phase-classification model, achieving 98% accuracy across Lbr, Sk, and FM phases. (d) Spin configurations corresponding to labeled regions in (a) and (b): sparse stripes and small skyrmions with large $K'$ (0.6, 0.8, and 1.0) at low $J'$ (diamond); wide stripes and large skyrmions with low $K'$ (6, 8, and 10) at high $J'$ (star); SkX remains stable with moderate $B'$ (4, 6, and 10) at low $J'$ (triangle), but becomes suppressed with low $B'$ (0, 1, 2) at higher $J'$ (hexagram).
• Figure 4: Symbolic regression predicted magnetic properties across parameter space. Color scale indicates magnitude. Stripe width $d$ (in Lbr phase) and skyrmion radius $r$ (in Sk phase) plotted in the (a) $J'$–$K'$ space at $B'=2$ T/meV and in the (b) $J'$–$B'$ space with $K'=0.2$, respectively. (c, d) Corresponding absolute topological charge $|Q|$ under the same respective conditions.