Spin-Wave Modes and Their Intense Excitation Effects in Skyrmion Crystals

We theoretically study spin-wave modes and their intense excitations activated by microwave magnetic fields in the Skyrmion-crystal phase of insulating magnets by numerically analyzing a two-dimensional spin model using the Landau-Lifshitz-Gilbert equation. Two peaks of spin-wave resonances with frequencies of $\sim 1\mathrm{GHz}$ are found for in-plane ac magnetic field where distribution of the out-of-plane spin components circulates around each Skyrmion core. Directions of the circulations are opposite between these two modes, and hence the spectra exhibit a salient dependence on the circular polarization of irradiating microwave. A breathing-type mode is also found for an out-of-plane ac magnetic field. By intensively exciting these collective modes, melting of the Skyrmion crystal accompanied by a redshift of the resonant frequency is achieved within nanoseconds.

Figure 1

(a) Phase diagram of the Hamiltonian (1) at $T=0$ where HL, SkX, and FM denote helical, Skyrmion-crystal, and ferromagnetic phases, respectively. (b) Spin configuration of the SkX phase with a color map of the spin $z$-axis components $S_{zi}$ at $H_z=3.75\times {10}^{-3}$. Spin vectors at sites ($i_x$, $i_y$) projected onto the $xy$ plane are shown by arrows for $\mathrm{mod}(i_x,6)=\mathrm{mod}(i_y,6)=0$. (c) One Skyrmion is magnified with a color map of the scalar spin chiralities $C_i$.

Figure 2

Imaginary parts of (a) in-plane and (b) out-of-plane dynamical susceptibilities, $\mathrm{Im}\chi (\omega )$, in the SkX phase for several values of $H_z$. The insets show resonant frequencies ${\omega }_R$ as functions of $H_z$.

Figure 3

Spin dynamics of each collective mode in the SkX phase calculated at $H_z=3.75\times {10}^{-3}$. Spins at sites ($i_x$, $i_y$) are represented by arrows when $\mathrm{mod}(i_x,6)=\mathrm{mod}(i_y,6)=0$ with color maps of the $S_{zi}$ components in the left panels, while in the right panels, distributions of the local spin chiralities $C_i$ are displayed. Temporal waveforms of the applied ac magnetic fields, $H_y^{\omega }\sin {\omega }_{R}t$ and $H_z^{\omega }\sin {\omega }_{R}t$, are shown in the uppermost figures where inverted triangles indicate times at which we observe the spin configurations shown here. (a) [(b)] Lower-energy [Higher-energy] rotational mode with ${\omega }_R=6.12\times {10}^{-3}$ (${\omega }_R=1.135\times {10}^{-2}$) activated by the in-plane ac magnetic field. Distributions of the $S_{zi}$ components and the spin chiralities $C_i$ circulate around the Skyrmion core in a counterclockwise (clockwise) fashion. (c) Breathing mode with ${\omega }_R=7.76\times {10}^{-3}$ activated by the out-of-plane ac magnetic field.

Figure 4

Calculated time evolutions of magnetization ($\parallel \mathit{y}$), $m_y(t)=(1/N)\sum _i{S}_{yi}(t)$, in the SkX phase at $H_z=3.75\times {10}^{-3}$ under linearly polarized, left-handed circularly polarized (LHP), and right-handed circularly polarized (RHP) in-pane ac magnetic fields with resonant frequency ${\omega }_R=6.12\times {10}^{-3}$ corresponding to the lower-lying mode.

Figure 5

Melting of the SkX within nanoseconds under irradiating LHP microwave, which excites the rotational spin-wave modes intensively (see text). Color maps of the $S_{zi}$ components are displayed at (a) $t=0$, (b) $t=4000$, and (c) $t=7200$. Figures magnify a partial area with $220\times 220$ sites for clarity, while the calculations are done for $288\times 288$ sites with the periodic boundary condition. Temporal waveform of the microwave is also shown where times corresponding to figures (a), (b), and (c) are indicated by inverted triangles.