Universality of annihilation barriers of large magnetic skyrmions in chiral and frustrated magnets

Magnetic skyrmions are whirls in the magnetization whose topological winding number promises stability against thermal fluctuations and defects. They can only decay via singular spin configurations. We analyze the corresponding energy barriers of skyrmions in a magnetic monolayer for two distinct stabilization mechanisms, i.e., Dzyaloshinskii-Moriya interaction (DMI) and competing interactions. Based on our numerically calculated collapse paths on an atomic lattice, we derive analytic expressions for the saddle-point textures and energy barriers of large skyrmions. The sign of the spin stiffness and the sign of fourth-order derivative terms in the classical field theory determines the nature of the saddle point and thus the height of the energy barrier. In the most common case for DMI-stabilized skyrmions, positive stiffness and negative fourth-order term, the saddle-point energy approaches a universal upper limit described by an effective continuum theory. For skyrmions stabilized by frustrating interactions, the stiffness is negative and the energy barrier arises mainly from the core of a singular vortex configuration.

Figure 1

Minimal energy path for the creation or annihilation of a DMI-stabilized skyrmion as obtained from the GNEB method, see Appendix pp2-s1. The upper panels [(a)–(f)] show the real-space magnetic texture in the proximity of the center of the skyrmion for various states along the minimal energy path. The color encodes the out-of-plane component of the magnetization. Panel (f) is a close-up of the saddle-point texture in panel (c). The lower panel shows the energy $E/J$ along the minimal energy path as a function of the reaction coordinate $\mathrm{\Lambda }$, see Appendix pp2-s1. The energy is evaluated with respect to the polarized phase (e). The arrows indicate the location of the upper panels in the minimal energy path. The results were obtained for $D/J=0.2, {\mu }_{s}H/J=0.06, K=0$, and $a=1$.

Figure 2

Energy barriers for the annihilation of an isolated skyrmion in the polarized state, cf. Fig. 1. The barrier is defined as the energy difference between the saddle point of the minimal energy path and the isolated skyrmion. The plot shows the results for a triangular lattice with a magnetic field $H$ (red) and a square lattice with a magnetic field $H$ (blue) or a uniaxial anisotropy $K$ (yellow), each for various values of Dzyaloshinskii-Moriya interaction $D$ and $a=1$. Data of the same color collapse onto the same lowest-order effective theory for large skyrmions, see Eq. (4).

Figure 3

Energy of an isolated skyrmion relative to the polarized state. The plot shows the results obtained for the lattice model Eq. (1), evaluated on a triangular lattice with a magnetic field $H$ (red triangles) and a square lattice with a magnetic field $H$ (blue squares) or a uniaxial anisotropy $K$ (yellow squares), each for various values of Dzyaloshinskii-Moriya interaction $D$. The results are plotted in the units of the effective continuum theory, Eq. (3). The parabolic lines are the results of this effective theory with the leading-order perturbative corrections, see Eq. (7). The dashed red line and the solid blue line with ${\mu }_{s}H=1.00D^2/J$ are identical within this approximation.

Figure 4

Schematic plot of the skyrmion energy as function of the skyrmion radius based on Eq. (12). For ${\overline{\mathcal{K}}}_4<0$ the maximum with an energy smaller than $4\pi \mathcal{J}$ occurs for $r_{\text{BP}}>a$ and is described by the continuum theory in Eq. (8). For ${\overline{\mathcal{K}}}_4>0$, in contrast, the barrier is typically larger than $4\pi \mathcal{J}$ and is determined by physics at the scale of the lattice constant $a$ where the continuum theory breaks down.

Figure 5

Creation barrier, i.e., energy of the saddle point, for the creation of a skyrmion in the polarized state, cf. Fig. 1. The plot shows the results obtained for the lattice model Eq. (1), evaluated on a triangular lattice with a magnetic field $H$ (red triangles) and a square lattice with a magnetic field $H$ (blue squares) or a uniaxial anisotropy $K$ (yellow squares), each for various values of Dzyaloshinskii-Moriya interaction $D$. The results are plotted as a function of $a\mathcal{D}/\mathcal{J}=D/J=a/\xi$, see Eq. (3), where $\xi$ is the skymrion size. The dashed black line indicates the analytical upper limit $\mathrm{\Delta }E/\mathcal{J}=4\pi$, see Sec. 2b2, approached for $\xi \to \infty$. The gray line is the perturbative analytical result, see Eq. (13), which is valid for $a\mathcal{D}/\mathcal{J}\ll 1/6$.

Figure 6

Creation barrier, i.e., energy of the saddle point, for the creation of a skyrmion in the polarized state, cf. Fig. 1, in the presence of frustrating interactions. The plot shows the results obtained for the lattice model Eq. (14), evaluated on a triangular lattice for $h=1$ and $\kappa =0$ in the dimensionless units of Eq. (4) for three different values of $J_2/J_1$. While ${\tilde{\mathcal{K}}}_4=-\frac{a^2{\mathcal{D}}^2}{16{\mathcal{J}}^2}<0$ for $J_2=0$ (red triangles, same data as in Fig. 5), we obtain ${\tilde{\mathcal{K}}}_4=\frac{a^2{\mathcal{D}}^2}{16{\mathcal{J}}^2}>0$ for $J_2/J_1=1/6$ (yellow triangles) and ${\tilde{\mathcal{K}}}_4=5\frac{a^2{\mathcal{D}}^2}{16{\mathcal{J}}^2}>0$ for $J_2/J_1=1/4$ (blue triangles). In the latter case, the creation barrier is substantially larger than $4\pi \mathcal{J}$.

Figure 7

Minimal energy path for the creation or annihilation of a skyrmion in the symmetric model system, Eq. (14), as obtained from the GNEB method, see Appendix pp2-s1. The upper panels [(a)–(f)] show the real-space magnetic texture in the proximity of the center of the skyrmion for various states along the minimal energy path. The color encodes the out-of-plane component of the magnetization. Panel (f) is a close-up of the saddle-point texture in panel (c). The lower panel shows the energy $E/J_1$ along the minimal energy path as a function of the reaction coordinate $\mathrm{\Lambda }$, see Appendix pp2-s1. The energy is evaluated with respect to the polarized phase (e). The arrows indicate the position of the upper panels in the minimal energy path. These results were obtained for $J_2/J_1=31/90, {\mu }_{s}H/J_1=1/450$, and $K/J_1=1/900$.

Figure 8

Energy of an isolated skyrmion relative to the polarized state. The plot shows the results obtained for the model Eq. (14) on a triangular lattice. The magnetic field ${\mu }_{0}H/J_1$ and uniaxial anisotropy $K/J_1$ is chosen such that dots of the same color map to the same rescaled continuum field $h$ and anisotropy $\kappa$ but are evaluated for different values of $J_2/J_1$. The data are plotted as a function of the resulting frustration ${\mathcal{I}}_1(J_2/J_1)$, see Eq. (16). The solid lines are fits for the expected linear behavior at vanishing frustration, see Eq. (19), using only the point with lowest frustration for fitting.

Figure 9

Saddle-point energy of large skyrmions (see Fig. 7) in a frustrated magnet on a triangular lattice, Eq. (1). The magnetic field ${\mu }_{0}H/J_1$ and uniaxial anisotropy $K/J_1$ are chosen to obtain the fixed values of $h$ and $\kappa$ shown in the legend of the figure. The data are plotted as a function of the resulting frustration ${\mathcal{I}}_1(J_2/J_1)$, see Eq. (16). The solid lines are fits using Eq. (20) with $e_1(h,\kappa )$ as the only fitting parameter $e_1=2.84$ (blue), 1.27 (red), $-0.20$ (yellow), and 1.85 (green)]. The black dashed line denotes the asymptotic value of the energy for ${\mathcal{I}}_1\to 0$, Eq. (21) (gray dashed line: approximate value obtained for a unrelaxed, symmetric vortex, $e_0\approx 2.8482$, see text).

Figure 10

Minimal energy path for the transition of a skyrmion into an antiskyrmion as obtained from the GNEB method, see Appendix pp2-s1. The upper panels [(a)–(e)] show the real-space magnetic texture in the proximity of the center of the (anti-)skyrmion for various states along the minimal energy path (lower panel, red data). The color encodes the out-of-plane component of the magnetization. Panel (f) shows the maximum of the initialized path (lower panel, blue data) with two vortices which is unstable. These results are obtained for $J_2/J_1=9/25, {\mu }_{s}H/J_1=3/250, K/J_1=3/500$ in model Eq. (14).

Figure 11

Minimal energy path for the creation or annihilation of a skyrmion in the symmetric model system, Eq. (14), as obtained from the GNEB method, see Appendix pp2. The upper panels [(a)–(f)] show the real-space magnetic texture in the proximity of the center of the skyrmion for various states along the minimal energy path. The color encodes the out-of-plane component of the magnetization. Panel (f) is a close-up of the saddle-point texture in panel (c). The lower panel shows the energy $E/J_1$ along the minimal energy path as a function of the reaction coordinate $\mathrm{\Lambda }$, see Appendix pp2. The energy is evaluated with respect to the polarized phase (e). The arrows indicate the position of the upper panels in the minimal energy path. These results were obtained for $J_2/J_1=31/90, {\mu }_{s}H/J_1=1/1800$, and $K/J_1=1/360$.