Internal modes of a skyrmion in the ferromagnetic state of chiral magnets

A spin texture called skyrmion has been recently observed in certain chiral magnets without inversion symmetry. The observed skyrmions are extended objects with typical linear sizes of 10 to 100 nm that contain ${10}^3$ to ${10}^5$ spins and can be deformed in response to external perturbations. Weak deformations are characterized by internal modes, which are localized around the skyrmion center. Knowledge of internal modes is crucial to assess the stability and rigidity of these topological textures. Here, we compute the internal modes of a skyrmion in a ferromagnetic background state by numerical diagonalization of the dynamical matrix. We find several internal modes below the magnon continuum, such as the mode corresponding to the translational motion and different kinds of breathing modes. The number of internal modes is larger for lower magnetic fields. Indeed, several modes become gapless in the low-field region indicating that the single skyrmion solution becomes unstable, although a skyrmion lattice remains thermodynamically stable. On the other hand, only three internal modes exist at high fields and the skyrmion texture remains locally stable even when the ferromagnetic state becomes thermodynamically stable. We also show that the presence of out-of-plane easy-axis anisotropy stabilizes the single skyrmion solution. Finally, we discuss the effects of damping and possible experimental observations of these internal modes.

Figure 1

(a) Schematic view of the spin profile of a skyrmion. (b) Sketch of the effective magnon potential produced by the skyrmion. The potential is drawn along the radial direction of the skyrmion.

Figure 2

Magnetic field dependence of the eigenfrequencies of different modes in the absence of easy-axis anisotropy ($A_z=0$). The modes are assigned to the $l$th order breathing mode and the translational mode. We show only several eigenfrequencies of the magnon continuum (shaded region) because only the lowest 30 eigenmodes have been obtained with the Lanczos method.

Figure 3

Deformation of the skyrmion associated with different internal modes. The spin configuration is obtained from the eigenvector ${\tilde{\mathbf{n}}}_v$ by using $\mathbf{n}={\mathbf{n}}_s+F{\tilde{\mathbf{n}}}_{v}sin\left(\omega t\right)$ with $F=10$. The spin configuration is plotted at $t=0$ (left column), $t=\pi /\left(2\omega \right)$ (middle column), and $t=3\pi /\left(2\omega \right)$ (right column). The white arrows in the first row denote the direction of the translational motion and in the second row represent the direction of the uniform breathing. The vectors in the plots denote the $n_x$ and $n_y$ components, and the contour plot denotes the $n_z$ component with red representing $n_z=1$ and blue representing $n_z=-1$. Here, $D/J_{\mathrm{ex}}=0.1$ and $A_z=0$.

Figure 4

Same as Fig. 2 but with $A_z=0.5D^2/J_{\mathrm{ex}}$ in (a) and $A_z=1.0D^2/J_{\mathrm{ex}}$ in (b). The eigenfrequencies for $D/J_{\mathrm{ex}}=0.1$ and $0.2$ are almost identical in normalized units.

Figure 5

Dependence of the eigenfrequencies $\omega$ on the damping coefficient $\alpha$. Symbols are from numerical diagonalization and lines are given by Eq. (4). For clarity, not all lines are shown.

Figure 6

Definition of the local coordinate system. The local $\tilde{z}$ axis is along the spin direction of the unperturbed skyrmion. The local $\tilde{x}$ axis is in the $x$-$y$ plane of the original coordinates.