Hybridized magnon modes in the quenched skyrmion crystal

Magnetic skyrmions have attracted attention as particlelike swirling spin textures with nontrivial topology, and their self-assembled periodic order i.e., the skyrmion crystal (SkX) is anticipated to host unique magnonic properties. In this paper, we investigate magnetic resonance in the quenched SkX state, which is obtained by the rapid cooling of the high-temperature equilibrium SkX phase in the chiral magnetic insulator ${\mathrm{Cu}}_2\mathrm{O}\mathrm{Se}{\mathrm{O}}_3$. At low temperatures, sextupole and octupole excitation modes of skyrmions are identified, which are usually inactive for oscillating magnetic fields $B^{\nu }$ with GHz-range frequency $\nu$ but turn out to be detectable through the hybridization with the $B^{\nu }$-active counterclockwise and breathing modes, respectively. The observed magnetic excitation spectra are well reproduced by theoretical calculations, which demonstrates that the effective magnetic anisotropy enhanced at low temperatures is the key for the observed hybridization between the $B^{\nu }$-active and $B^{\nu }$-inactive modes.

Figure 1

(a) Magnetic field ($B_0$)-temperature ($T$) phase diagram for the field-cooling path shown by the arrows, determined by microwave absorption spectroscopy for ${\mathrm{Cu}}_2\mathrm{O}\mathrm{Se}{\mathrm{O}}_3$ with $B_0\parallel \left[001\right]$. (b) Schematic illustration of a skyrmion crystal (SkX) spin texture. (c) SEM image of the device structure used for the microwave absorption spectroscopy. (d) Schematic side view of the device structure. By injecting an oscillating electric current $I^{\nu }$ of gigahertz frequency (black arrows pointing along the out-of-plane direction) into the Au coplanar waveguide, an oscillating magnetic field $B^{\nu }$ is generated and magnetic resonance is excited.

Figure 2

(a) Magnetic field dependence of microwave absorption spectra in the SkX state for $B_0\parallel \left[001\right]$ at 20 K. The background color represents the microwave absorption intensity $|\mathrm{\Delta }{S}_{11}\left(\nu \right)|$. (b) Theoretically calculated magnetic excitation spectra as a function of $B_0\parallel \left[001\right]$, with the cubic anisotropy parameter $K=0.1D^2/J$ (see main text). The size of each colored point represents the excitation amplitude for $B^{\nu }$. (c) Time ($t$) development of the spatial distribution of $m_z\left(\mathbf{r}\right)$ (the out-of-plane component of local magnetization) calculated for various excitation modes in the SkX state. $\tau$ represents the oscillation period. The polarization direction of $B^{\nu }$ to excite each resonance mode is also indicated at the bottom.

Figure 3

(a), (b) The expanded view of Fig. 2 for the $B_0-\nu$ region representing the hybridization for (a) breathing/octupole modes and (b) CCW/sextupole modes at 20 K. (c), (d) Microwave absorption spectra corresponding to (a) and (b). The peak frequencies (triangles) are extracted from the spectral fitting with the Lorentz function. The fitting curves are drawn by solid lines.

Figure 4

(a) Microwave absorption spectra for $B_0$ giving the minimum value of the frequency difference between two peaks at each temperature. Solid lines are the results of the fitting with the double Lorentz function. (b) Temperature dependence of the hybridization gap $\mathrm{\Delta }$ for the breathing/octupole modes. (c)–(h) The hybridization of breathing/octupole modes at (c) 10 K, (d) 15 K, (e) 20 K, (f) 25 K, (g) 30 K, and (h) 35 K. Peak frequencies determined from the spectral fitting are depicted by triangles.

Figure 5

Theoretically calculated magnetic excitation spectra as a function of $B_0\parallel \left[001\right]$, with the cubic anisotropy parameter (a) $K=0$, (b) $K=0.1D^2/J$, and (c) $K=0.2D^2/J$. The size of each red or blue data point represents the excitation amplitude for $B^{\nu }$.