Driving magnetic skyrmions with microwave fields

We show theoretically by numerically solving the Landau-Lifshitz-Gilbert equation with a classical spin model on a two-dimensional system that both magnetic skyrmions and skyrmion lattices can be moved with microwave magnetic fields. The mechanism is enabled by breaking the axial symmetry of the skyrmion, for example, through application of a static in-plane external field. The net velocity of the skyrmion depends on the frequency and amplitude of the microwave fields as well as the strength of the in-plane field. The maximum velocity is found where the frequency of the microwave coincides with the resonance frequency of the breathing mode of the skyrmions.

Figure 1

(a) Skyrmion configuration in the presence of an in-plane field $H_y=0.006$ with $D=0.18$, $J=1$, and $H_z=0.02$. (b) The symmetric topological charge density distribution of a skyrmion when $H_y=0$. (c) The corresponding topological charge density for the skyrmion shown in (a) when $H_y>0$. (d) The skyrmion lattice with 12 skyrmions in a sample of size $N=174\times 150$ sites, with an in-plane field $H_y=0.004$.

Figure 2

Imaginary parts of the $H_y$-dependent dynamical susceptibility ${\chi }_{zz}$ as a function of frequency for (a) a single skyrmion and (b) a skyrmion lattice. The spectra are obtained by applying a sinc-field pulse $h=h_0\mathrm{sinc}\left({\omega }_{0}t\right)$ to the system with $h_0=1\times {10}^{-5}$ and ${\omega }_0=0.1\pi$ in the $z$ axis; the magnetization dynamics is recorded every $dt=5$ for 8000 steps.

Figure 3

(a) The displacements of the guiding center $(X,Y)$ for a single skyrmion. The simulation parameters are ${\omega }_0=0.017$, $h_0=2\times {10}^{-4}$, and the in-plane field $H_y=0.004$. (b)–(d) The velocities $v_x$ and $v_y$ as functions of (b) the microwave frequency ${\omega }_0$, (c) the microwave amplitude $h_0$, and (d) the in-plane field $H_y$. The fixed simulation parameters are the same as in (a).

Figure 4

The total spatial force density $f_i={\tilde{\mathbf{m}}}_s·[{\partial }_i{\tilde{\mathbf{m}}}_s\times 〈\mathbf{m}\times {\mathbf{H}}_{\mathrm{eff}}〉]$ for (a) $H_y=0$ and (b) $H_y=4\times {10}^{-3}$, where we have used ${\tilde{\mathbf{m}}}_s=\langle \mathbf{m}\rangle$. The microwave frequency is ${\omega }_0=0.017$.

Figure 5

(a) The velocities $v_x$ and $v_y$ as a function of an in-plane field $H_y$ for the skyrmion lattice at frequency ${\omega }_0=0.023$. (b) The velocities as a function of frequency ${\omega }_0$ with $H_y=4\times {10}^{-3}$. The microwave amplitude is $h_0=2\times {10}^{-4}$.