Establishing the Fundamental Magnetic Interactions in the Chiral Skyrmionic Mott Insulator ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ by Terahertz Electron Spin Resonance

The recent discovery of Skyrmions in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ has established a new platform to create and manipulate Skyrmionic spin textures. We use high-field electron spin resonance with a terahertz free-electron laser and pulsed magnetic fields up to 64 T to probe and quantify its microscopic spin-spin interactions. In addition to the previously observed long-wavelength Goldstone mode, this technique probes also the high-energy part of the excitation spectrum which is inaccessible by standard low-frequency electron spin resonance. Fitting the behavior of the observed modes in magnetic field to a theoretical framework establishes experimentally that the fundamental magnetic building blocks of this Skyrmionic magnet are rigid, highly entangled and weakly coupled tetrahedra.

Figure 1

Distorted pyrochlore structure of the ${\mathrm{Cu}}^{2+}$ ions in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, showing the proposed alternation of magnetically weak and strong tetrahedra. In total, there are five exchange couplings [23, 25]: two strong, $J_s^{\mathsf{AF}}$ and $J_s^{\mathsf{FM}}$ (solid lines), two weak, connecting strong tetrahedra, $J_w^{\mathsf{AF}}$ and $J_w^{\mathsf{FM}}$ (dashed lines), and a weak longer-range exchange $J_{\mathsf{O}..\mathsf{O}}^{\mathsf{AF}}$ (not shown). The shaded circle indicates a strong tetrahedron with wave function $|\psi {⟩}_t$, which strongly deviates from a semiclassical “3up-1down” state.

Figure 2

(a) ESR spectral data for field along [100] and $T\simeq 6\mathrm{K}$. Lines are calculations (numbers give the degeneracy of each mode); see Fig. 3. Diamonds denote the absorption data reported by Raman [26] and far-infrared [27] studies; see text. Shaded areas show the regions with very low signal intensity due to strong phonon absorptions [27], as shown by the zero-field transmittance data at 4.2 K (b). (c) Examples of ESR spectra (the spectra are offset for clarity).

Figure 3

(a) Spectrum of an isolated strong tetrahedron $t$, $\mathcal{}{H}_0^{(t)}$ labeled by the IRs of ${\mathsf{C}}_{3v}$, the total spin $S$ of $t$, and its projection $M$ along the $\mathbf{}\mathbf{z}$ axis. (b) Spectrum of $\mathcal{}{\mathcal{H}}_{\mathsf{TMF}}^{(t)}=\mathcal{}{\mathcal{H}}_0^t+\mathcal{}{\mathcal{V}}_{\mathsf{MF}}^{(t)}$, where $\mathcal{}{\mathcal{V}}_{\mathsf{MF}}^{(t)}$ includes the exchange fields exerted by neighboring tetrahedra at a mean-field level. Here, $S$ is not a good quantum number. (c) Zero-momentum excitations of the full Hamiltonian from quadratic multiboson theory. Solid lines denote states with $M=0$ or 2, that are accessible by ESR. (d),(e) Semiclassical pictures for two representative modes of different types.