Breathing mode of a skyrmion on a lattice

The breathing modes of a skyrmion, corresponding to coupled oscillations of its size and chirality angle, are studied numerically for a conservative classical-spin system on a $500\times 500$ lattice. The dependence of the oscillation frequency on the magnetic field is computed for a model with Dzyaloshinskii-Moriya interaction. In accordance with previous works, it is linear at small fields, reaches maximum on increasing the field, then sharply tends to zero as the field approaches the threshold above which the skyrmion loses stability and collapses. Physically transparent analytical model is developed that explains the results qualitatively and provides the field dependence of the oscillation frequency that is close to the one computed numerically. Dissipation of a breathing motion in which the skyrmion chirality angle $\gamma$ is rotating in one direction depends on the initial amplitude. Below a certain threshold the mode is stable, while above that threshold it becomes strongly damped by the reservoir of spin waves and quickly ends with the skyrmion collapse. To the contrary, smaller-amplitude breathing motion in which $\gamma$ oscillates is undamped in the absence of other interactions. Adding perpendicular anisotropy and removing the applied field makes the breathing mode of any amplitude very slow and undamped.

Figure 1

Spin field in a Bloch-type skyrmion studied in the paper. Its breathing mode corresponds to the coupled oscillations of the skyrmion size and spin angles.

Figure 2

Skyrmion size vs the applied magnetic field computed on $200\times 200$ and $500\times 500$ lattices for $A/J=0.02$ and PMA values $D=0$ and $0.001J$.

Figure 3

Numerically obtained oscillations of $m_z$ and of the skyrmion size $\lambda$ in the breathing mode computed on a $500\times 500$ lattice for $A/J=0.02,H/J=-0.001$, and $\mathrm{\Delta }\gamma =-0.1^{\circ }$. Time is measured in units of $\hslash /J$.

Figure 4

Field dependence of the oscillation frequency of the breathing mode computed numerically on lattices of different sizes, analytically for a BP skyrmion shape with logarithmic accuracy, and semianalytically in a model with the skyrmion shape corrected (see Sec. 3 for the definition of parameter $l)$. The slope of the straight-line part of the spectrum computed analytically in models that ignore spin waves roughly coincides with the ferromagnetic resonance (FMR) slope. The slope computed by the numerical method that accounts for all excitations in the system is close to 0.8 of the FMR slope, which agrees with Refs. [26, 27].

Figure 5

Frequency of the breathing mode in the linear-$H$ regime computed on lattices of different size for different values of the DMI constant $A$.

Figure 6

Breathing-mode frequency vs the applied magnetic field for $A/J=0.02$ and different PMA values.

Figure 7

Coexisting breathing and bulk precession modes after rotating spins around $z$ and and then around $y$ axes out of the equilibrium-skyrmion state, computed for $A/J=0.02,H/J=-0.001$, and $\mathrm{\Delta }\gamma =0.1^{\circ }$. These modes oscillate at close but distinctly different frequencies.

Figure 8

Qualitative dependence of the skyrmion energy on skyrmion size $\lambda$ for $\gamma =\pi /2$. The minimum corresponding to the equilibrium size disappears for $H>H_c$. At $H=H_c$ the energy has an inflection point where both first and second derivatives are zero.

Figure 9

Equipotential lines in the $\left\{\lambda ,\gamma \right\}$ plane corresponding to Eq. (12) with $l=\mathrm{const}$.

Figure 10

Modified skyrmion shape given by Eq. (21) with the equilibrium size ${\lambda }_{\mathrm{eff}}$ obtained by the numerical minimization of the energy for two values of the magnetic field $H/J=0.001$ and $H/J=0.002$ below the critical field $H_c/J=0.0026$. Comparison with the BP shape and numerical results obtained on the lattice are also shown.

Figure 11

Equipotential lines in the $\left\{{\lambda }_x,{\lambda }_y\right\}=\left\{\lambda cos\gamma ,\lambda sin\gamma \right\}$ plane corresponding to Eq. (12) with $l=\mathrm{const}$. Dashed circle corresponds to changing the chirality angle $\gamma$ of the equilibrium skyrmion.

Figure 12

Dynamics of the breathing mode after rotating of the spins by the angle $\mathrm{\Delta }\gamma$ in the system of $500\times 500$ spins with $A/J=0.02$ and $H/J=-0.001$. Upper panel: no PMA; for $\mathrm{\Delta }\gamma \gtrsim {50}^{\circ }$, the skyrmion collapses fast. Lower panel: $D/J=0.002$; for $\mathrm{\Delta }\gamma \gtrsim {10}^{\circ }$, the skyrmion collapses.

Figure 13

Time dependence of the skyrmion size $\lambda$ in the model with $A/J=0.02,D/J=0.002$, and $H/J=-0.001$ after the initial rotation of spins $\mathrm{\Delta }\gamma ={180}^{\circ }$. After four oscillations, the skyrmion collapses.

Figure 14

Dynamics of the breathing mode after rotation of the spins by the angle $\mathrm{\Delta }\gamma$ in the system of $500\times 500$ spins with $A/J=0.02,D/J=0.002$, and $H=0$. Here, breathing mode collapses only in the range ${40}^{\circ }\lesssim \mathrm{\Delta }\gamma \lesssim {70}^{\circ }$.

Figure 15

Time dependence of the skyrmion size $\lambda$ in the model with $A/J=0.02,D/J=0.002$, and $H=0$ after the initial rotation of spins $\mathrm{\Delta }\gamma ={180}^{\circ }$. This large-amplitude breathing mode is very slow and undamped.