Skyrmions Driven by Intrinsic Magnons

We study the dynamics of a Skyrmion in a magnetic insulating nanowire in the presence of time-dependent oscillating magnetic field gradients. These ac fields act as a net driving force on the Skyrmion via its own intrinsic magnetic excitations. In a microscopic quantum field theory approach, we include the unavoidable coupling of the external field to the magnons, which gives rise to time-dependent dissipation for the Skyrmion. We demonstrate that the magnetic ac field induces a super-Ohmic to Ohmic crossover behavior for the Skyrmion dissipation kernels with time-dependent Ohmic terms. The ac driving of the magnon bath at resonance results in a unidirectional helical propagation of the Skyrmion in addition to the otherwise periodic bounded motion.

Figure 1

A magnetic Skyrmion with topological number $Q=-1$ placed in a nanowire strip in the $xy$ plane with a time-oscillating magnetic field $\mathbf{b}(\mathbf{r},t)$ applied along the perpendicular $z$ direction.

Figure 2

Frequency dependence of the diagonal part of the damping kernel $z{\widehat{\gamma }}_{yy}(T/4,z)/N_A$ with $T=2\pi /{\omega }_{\mathrm{ext}}$ and various values of ${\omega }_{\mathrm{ext}}$. The super-Ohmic to Ohmic transition manifests itself as different power law exponents of kernels ${\widehat{\gamma }}_{yy}(t,z)$ and ${\widehat{\gamma }}_{yy}^0(z)$ for finite external ${\omega }_{\mathrm{ext}}$. Inset: The colored surface represents the quantity $z{\widehat{\gamma }}_{yy}(t,z)/N_A$ for times $t$ up to $4T$ and for ${\omega }_{\mathrm{ext}}={\omega }_{\mathrm{res}}/2$. Here ${\omega }_{\mathrm{res}}={ϵ}_0/\overline{S}$, with ${ϵ}_0=0.1358$ being the energy of the breathing mode and $\overline{S}=S(J/D{)}^2$ with $S=1$ and $J/D=4$.

Figure 3

(a) Frequency dependence of the off-diagonal part of the damping kernel $z{\widehat{\gamma }}_{xy}(T/2,z)/N_A$ with $T=2\pi /{\omega }_{\mathrm{ext}}$ and various values of ${\omega }_{\mathrm{ext}}$. The super-Ohmic to Ohmic transition manifests itself as different power law exponents of kernels ${\widehat{\gamma }}_{yy}(t,z)$ and ${\widehat{\gamma }}_{yy}^0(z)$ depicted in (c). In (b) the colored surface represents the quantity $z{\widehat{\gamma }}_{xy}(t,z)/N_A$ for times $t$ up to $4T$ and for ${\omega }_{\mathrm{ext}}={\omega }_{\mathrm{res}}/2$. Here ${\omega }_{\mathrm{res}}={ϵ}_0/\overline{S}$, with ${ϵ}_0=0.1358$ being the energy of the breathing mode and $\overline{S}=SJ/D$ with $S=1$ and $J/D=4$.

Figure 4

(a) Helical path for the Skyrmion’s collective coordinate $R_i(t)$ calculated by a numerical solution of the equation of motion (10). The inclusion of the time-dependent dissipation induces a unidirectional motion to the otherwise $T$-periodic bounded motion (black dashed line), where $T=2\pi /{\omega }_{\mathrm{ext}}$. The displacement of the Skyrmion can be visualized by the sequential coloring starting from $t=100T$ (yellow line) up to $t=130T$ (black line). The black arrow shows the direction of motion. (b) $T$-periodic time dependence of the collective coordinates $R_i(t)$ and (c) mean position $⟨R_x{⟩}_k$ and velocity $⟨{\dot{R}}_x{⟩}_k$ as a function of consecutive periods $k$.