Magnonic Weyl states in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$

The multiferroic ferrimagnet ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ with a chiral crystal structure has attracted a lot of recent attention due to the emergence of a magnetic skyrmion order in this material. Here, the topological properties of its magnon excitations are systematically investigated by linear spin-wave theory and inelastic neutron scattering. When considering Heisenberg exchange interactions only, two degenerate Weyl magnon nodes with topological charges $\pm 2$ are observed at high-symmetry points. Each Weyl point splits into two as the symmetry of the system is further reduced by including into consideration the nearest-neighbor Dzyaloshinskii-Moriya interaction, crucial for obtaining an accurate fit to the experimental spin-wave spectrum. Also, one additional pair of Weyl points appears near the $\mathbf{R}$ point. The predicted topological properties are verified by surface state and Chern number analysis. Additionally, we predict that a measurable thermal Hall conductivity can be associated with the emergence of the Weyl points, the position and number of which can be tuned by modifying the Dzyaloshinskii-Moriya interaction in the system.

Figure 1

(a) The structure of ferrimagnetic ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ is shown together with five different interaction paths, marked by $J_{\text{s}}^{\text{FM}}, J_{\text{s}}^{\text{AFM}}, J_{\text{w}}^{\text{FM}}, J_{\text{w}}^{\text{AFM}}$, and $J_{\text{O..O}}$, where $J_{\text{O..O}}$ represents the antiferromagnetic long-range interaction. (b), (c) Monopole distribution of the absolute magnitude of the Berry curvature corresponding to the two lowest magnon bands in the $k$ planes marked in the inset. In (b), the DMI was not taken into account, while (c) shows the result with the DMI included. The $k$ planes are chosen so that they include the Weyl points. (d) The magnon dispersion of the lowest four bands of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. The dotted blue line represents the dispersion without the effect of the DMI, with the Weyl points emerging at $\mathbf{R}$ and at $\mathrm{\Gamma }$. The black line represents the dispersion upon including the DMI, with six Weyl points emerging at ${\mathbf{R}}^1, {\mathbf{R}}^2, {\mathbf{R}}^3, {\mathbf{R}}^4, {\mathrm{\Gamma }}^1$, and ${\mathrm{\Gamma }}^2$. The exact positions of the Weyl points are shown in (e): ${\mathbf{R}}^1=(-0.39,0.47,-0.23), {\mathbf{R}}^2=(-0.38,0.34,-0.25), {\mathbf{R}}^3=(0.40,0.22,-0.33), {\mathbf{R}}^4=(-0.40,0.18,0.49), {\mathrm{\Gamma }}^1=(0.01,-0.03,0.04), {\mathrm{\Gamma }}^2=(0.01,-0.11,0.03)$, and their unit-direction projections are indicated by blue open circles.

Figure 2

Momentum-energy cuts along (a) $\left(11L\right)-\left(HH2\right)$, (b) $\left(1.45KK\right)$, (c) $\left(1.5KK\right)$, (d) $\left(1.8KK\right)$, (e) $\left(HH\frac{3}{2}\right)$, and (f) $\left(\frac{3}{2}\frac{3}{2}L\right)$ directions in reciprocal space. All measurements were done at 2 K without applying magnetic field. Data in (a), (e), and (f) were measured at the triple-axis spectrometer, while (b)–(d) are cuts from the TOF data set, integrated within $\pm 0.1$ r.l.u. in momentum directions orthogonal to the figure plane. The magnon dispersion calculated with the fitted value of the DMI is shown with thin red lines. The straight feature in the bottom-right-hand corner of (a), which is not captured by the spin-wave model, is a spurious peak from nonmagnetic multiple scattering. The magnon dispersion along the paths which do not contain high-symmetry points is shown in (b) and (d). The last two cuts (e) and (f) are centered at the $\mathbf{R}\left(\frac{3}{2}\frac{3}{2}\frac{3}{2}\right)$ point, where degenerate magnon bands are predicted in the absence of DMI.

Figure 3

The surface magnon band structure of the 75-layer-thick slab of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$(001) along the paths indicated in the inset shown on the left without the DMI, and on the right including the DMI. The special points $\mathrm{\Gamma }, \mathbf{R}, {\mathrm{\Gamma }}^1, {\mathbf{R}}^1, {\mathrm{\Gamma }}^2, {\mathbf{R}}^2, {\mathbf{R}}^3$, and ${\mathbf{R}}^4$ are the projections of the Weyl points onto the (001) plane, which are indicated with red (positive charge) and blue (negative charge) solid circles. The color scale represents the weight of the magnonic wave function along the slab.

Figure 4

Components of the thermal Hall conductivity tensor in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$. (a) Energy-dependent and (b) cumulative thermal Hall conductivity computed at 60 K. (c) The temperature dependence of the thermal Hall conductivity.

Figure 5

The effect of the nearest-neighbor Dzyaloshinskii-Moriya coupling on the positions of the Weyl points. (a), (b) The DMI vector is rotated about the $x$ axis with the initial direction along [010], and the evolution of the corresponding Weyl points with angle $\theta$ near (a) $\mathbf{R}$ and (b) $\mathrm{\Gamma }$ points is shown. The projection of each Weyl point onto the $k_x-k_y, k_x-k_z$, and $k_y-k_z$ planes is shown with black symbols in the corresponding planes. (c), (d) The evolution of the Weyl points near (c) $\mathbf{R}$ and (d) $\mathrm{\Gamma }$ as a function of the DMI strength, with the DMI vector following $D=c(-0.491,2.0,-1.41)\mathrm{meV}$, where the coefficient $c$ represented by the color scale is chosen from 0 to 1.2. The red circle marks a new pair of Weyl points as $c$ increases, and the projections are circled by red dotted circles.