Topological magnon insulator spin excitations in the two-dimensional ferromagnet ${\mathrm{CrBr}}_3$

Topological magnons are bosonic analogues of topological fermions in electronic systems. They have been studied extensively by theory but rarely realized by experiment. Here, by performing inelastic neutron scattering measurements on single crystals of a two-dimensional ferromagnet ${\mathrm{CrBr}}_3$, which was classified as Dirac magnon semimetal featured by the linear bands crossing at the Dirac points, we fully map out the magnetic excitation spectra, and reveal that there is an apparent gap of $\sim 3.5$ meV between the acoustic and optical branches of the magnons at the K point. By collaborative efforts between experiment and theoretical calculations using a five-orbital Hubbard model obtained from first-principles calculations to derive the exchange parameters, we find that a Hamiltonian with Heisenberg exchange interactions, next-nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, and single-ion anisotropy is more appropriate to describe the system. Calculations using the model show that the lower and upper magnon bands separated by the gap exhibit Chern numbers of $\pm 1$. These results indicate that ${\mathrm{CrBr}}_3$ is a topological magnon insulator, where the nontrivial gap is a result of the DM interaction.

Figure 1

(a) Schematic crystal structure of ${\mathrm{CrBr}}_3$. Arrows denote the spins. $J_1, J_2$. and $J_3$ represent NN, NNN, and NNNN exchange couplings, respectively. (b) Schematic Brillouin zone with high-symmetry points and directions on the ($h$, $k$, 0) plane. (c), (d), and (e) show INS spectra collected at 6 K along ($h$, $h$), ($h$–0.5, $h+0.5$), and (–$k$, $k$) directions, respectively. These directions are illustrated in (b). The high-energy part ($\ge 4$ meV, above the horizontal dashed line) was obtained with $E_{\mathrm{i}}=30$ meV, while the low-energy part was obtained with $E_{\mathrm{i}}=10.7$ meV. The high-energy data were integrated with the orthogonal direction over $[-0.2,0.2]$ r.l.u., while the low-energy data were over $[-0.15,0.15]$ r.l.u, and $l$ over [–5, 5] r.l.u. for both cases. Solid curves are the calculated dispersions with the DM model, as discussed in the main text. Vertical dashed lines in (c) and (d) illustrate constant $\mathit{Q}$ cuts at the K points, and the cut results of (c) and (d) are plotted in (f) and (g), respectively. Solid curves in (f) and (g) are guides to the eye, and the vertical dashed lines denote the peak positions, between which we define to be the gap energy. (h) Magnetic spectrum along the [001] direction obtained with $E_{\mathrm{i}}=10.7$ meV. Throughout the paper, errors represent one standard deviation.

Figure 2

Calculated dependence of the Heisenberg exchange interactions $J_1, J_2$, and $J_3$, NN Kitaev interaction $K$, off-diagonal interactions $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$, NNN DM interaction $A$, and single-ion anisotropy term $D_z$, as a function of Hund's coupling $J_{\mathrm{H}}$ over the Coulomb interaction $U$. The vertical line indicates the material parameters with $U=3.05$ eV and $J_{\mathrm{H}}=0.95$ eV. An SOC constant $\lambda =30$ meV was used in the calculations.

Figure 3

Comparison between the experimental (on the left of each panel) and calculated excitation spectra using a Hamiltonian consisted of $J_1, J_2, J_3, A$, and $D_z$ (on the right of each panel). The experimental spectra were obtained under the same conditions as those in Figs. 1–1. Calculated dispersions with $J_1=-1.48$ meV, $J_2=-0.08$ meV, $J_3=0.11$ meV, $A=0.22$ meV, and $D_z=-0.02$ meV are plotted as solid lines on top of the experimental spectra on the left side in each panel. Chern numbers of the acoustic and optical bands are also labeled.