Observation of Magnetic Excitations of Skyrmion Crystal in a Helimagnetic Insulator ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

We have investigated the low-energy dynamics of the triangular lattice of Skyrmions in a helimagnetic insulator ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ in terms of microwave response. We have observed two elementary excitations of the Skyrmion with different polarization characteristics: the counterclockwise circulating mode at 1 GHz with the magnetic field polarization parallel to the Skyrmion plane and the breathing mode at 1.5 GHz with a perpendicular magnetic field polarization. These modes reflect the topological nature of Skyrmions and may play a central role in the Skyrmion dynamics.

Figure 1

(a) Magnetic phase diagram for ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ as determined by the magnetic susceptibility, which is measured with minimal demagnetization field (corresponding to the $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$ configuration in the following microwave experiments). (b) Real-space image of a Skyrmion in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ obtained by Lorentz transmission electron microscopy, reproduced from Ref. 16. (c) Schematic illustration of Skyrmion. (d) Illustrations of magnetic excitations in helimagnetic state. The untransparent and half transparent magnetic moments indicate the magnetic moments in static and excited states, respectively. Bottom panel illustrates the oscillating component (the difference between the excited and static states) of magnetic moments in the coordinate system fixed to the static magnetic moment. The numbers stand for the positions of magnetic moments. (e)–(g) Schematic illustrations of magnetic excitations in Skyrmion crystal.

Figure 2

(a),(b) Microwave absorption spectra $\Delta S_{11}$ at various magnetic fields at 40 K for (a) $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$ and (b) $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$. (c) The magnetic field variation of magnetic susceptibility at 40 K measured in the configuration corresponding to the $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$ microwave measurement. (d) The magnetic field dependence of the resonance peak frequency in the microwave wave absorption spectra for $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$. The closed circles and triangles correspond to the resonant modes in conical magnetic state ($+Q$ and $-Q$ modes) and that in induced ferrimagnetic state, respectively. (e) Microwave absorption spectra $\Delta S_{11}$ at various magnetic fields at 70 K for $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$.

Figure 3

(a) The microwave absorption spectra $\Delta S_{11}$ for $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$ at various magnetic fields at 57.5 K. (b) The magnetic field variation of magnetic susceptibility at 57.5 K measured in the configuration corresponding to the $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$ microwave measurement. (c),(d) The (c) frequency and (d) intensity of the magnetic modes in the microwave absorption spectra in (a). The closed and open circles and closed squares in (c),(d) correspond to the excitations in conical magnetic, Skyrmion crystal, and ferrimagnetic states, respectively. (e) The intensity of the magnetic-resonance mode for $H_{\mathrm{DC}}\perp H_{\mathrm{AC}}$ (counterclockwise rotational mode) in Skyrmion crystal plotted in the $H\mathrm{\text{−}}T$ phase diagram determined by the magnetic susceptibility measurement. Squares stand for the measured (temperatures, magnetic field) points.

Figure 4

(a) The magnetic field dependence of microwave absorption spectra $\Delta S_{11}$ for $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$ at 57.5 K. (b) The magnetic field variation of magnetic susceptibility at 57.5 K measured in the configuration corresponding to the $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$ microwave measurement. (c),(d) The (c) frequency and (d) intensity of the magnetic mode in the microwave absorption spectra for $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$. (e) The intensity of the magnetic-resonance mode for $H_{\mathrm{DC}}\parallel H_{\mathrm{AC}}$ (breathing mode) in Skyrmion crystal plotted in the $H\mathrm{\text{−}}T$ phase diagram. Squares stand for the measured (temperatures, magnetic field) points.