Interplay of Dzyaloshinskii-Moriya and Kitaev interactions for magnonic properties of Heisenberg-Kitaev honeycomb ferromagnets

The properties of Kitaev materials are attracting ever increasing attention owing to their exotic properties. In realistic two-dimensional materials, the Kitaev interaction is often accompanied by the Dzyaloshinskii-Moriya interaction, which poses a challenge for distinguishing their magnitudes separately. In this paper, we demonstrate that it can be done by accessing magnonic transport properties. By studying honeycomb ferromagnets exhibiting Dzyaloshinskii-Moriya and Kitaev interactions simultaneously, we reveal nontrivial magnonic topological properties accompanied by intricate magnonic transport characteristics as given by thermal Hall and magnon Nernst effects. We also investigate the effect of a magnetic field, showing that it does not only break the symmetry of the system but also brings drastic modifications to magnonic topological transport properties, which serve as hallmarks of the relative strength of anisotropic exchange interactions. Based on our findings, we suggest strategies to estimate the importance of Kitaev interactions in real materials.

Figure 1

(a) Sketch of the structure of a honeycomb ${\mathrm{CrI}}_3$ monolayer. The unit cell is outlined with a thin black line, where blue balls represent ${\mathrm{Cr}}^{3+}$ ions and pink balls are iodide ions. The Kitaev bonds $\mathbf{x}$ (red), $\mathbf{y}$ (dark blue), $\mathbf{z}$ (yellow) are indicated with thick colored lines. The arrows mark the second-nearest-neighbor bond orientations along the black dotted lines that share a common sign of the out-of-plane DM vector. (b) Schematic diagram of the influence of an in-plane magnetic field $\mathbf{B}$ on the spin direction $\mathbf{S}$ whose polar angle is defined as ${\theta }_s$. (c) The perspective view of the ${\mathrm{Cr}}_2{\mathrm{I}}_2$ plane, where its normal vector is marked as $\widehat{\gamma }$ and the Kitaev angle is defined as the polar angle of $\widehat{\gamma }$.

Figure 2

(a) The comparison of magnon dispersions along high-symmetry lines for different values of the Kitaev parameters. Gray, blue, and red lines correspond to the $K$ values of 0, 5.2, and 3 meV. The dashed and solid lines correspond to the Kitaev angles $\theta$ of ${45}^{\circ }$ and $54.{74}^{\circ }$, respectively. The corresponding energy-resolved Chern number is shown in (b). The Berry curvature distribution of the first magnon branch in the first Brillouin zone for different $K$ values with $\theta =54.{74}^{\circ }$ is shown in the inset of (b). The color map ranges from $-40$ to 40 in arbitrary (arb.) units, and the exceeding values are marked with black. The temperature dependence of thermal Hall conductivity ${\kappa }_{\mathrm{TH}}^{xy}$ and magnon Nernst conductivity ${\kappa }_{\mathrm{N}}^{xy}$ are shown in (c) and (d), in units of ${10}^{-11}$ W/K and $k_B/2\pi$, respectively. (e) The map of ${\kappa }_{\mathrm{TH}}^{xy}/T$ as a function of temperature $T$ and parameter $K$ at the Kitaev angle of $\theta =54.{74}^{\circ }$. (f) The map of ${\kappa }_{\mathrm{TH}}^{xy}/T$ as a function of $T$ and $\theta$ at a constant $K$ value of 5.2 meV. The units of the color maps in (e) and (f) are chosen as W/${\mathrm{K}}^2$. The magnon band gap ${▵}_K$ as a function of $K$ (at $\theta =54.{74}^{\circ }$) and $\theta$ (at $K=5.2$ meV) is shown in (g) and (h), respectively.

Figure 3

(a) The comparison of magnon dispersions with different Kitaev and DMI parameters specified in the legend (in the units of meV and degrees). The magnon dispersions represented with black, blue, green, and red lines have almost the same band gap ${▵}_K$ between the two branches. (b) Band gap ${▵}_K$ as a function of $D$ and $K$ at $\theta =54.{74}^{\circ }$. (c) Dependence of thermal Hall conductivity ${\kappa }_{\mathrm{TH}}^{xy}$ on $K$ and $D$ at $T=100$ K and $\theta =54.{74}^{\circ }$. The $T$ dependence of thermal Hall conductivity ${\kappa }_{\mathrm{TH}}^{xy}$ is shown in (d)–(g). (d) ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of $T$ and $D$, with $K$ and $\theta$ corresponding to the case of ${\mathrm{CrI}}_3$. (e), (f) The corresponding BZ distribution of the Berry curvature (in arb. units) for the first band with $D=+0.4$ meV (e) and $-0.4$ meV (f) and the Kitaev interaction is the same as (d). (g) ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of $T$ and $K$ for $\theta =54.{74}^{\circ }$ and $D=-0.2$ meV. (h) ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of $\theta$ and $D$ at $T=100$ K and $K=5.2$ meV. The corresponding color map is in units of W/K.

Figure 4

(a), (b) Evolution of the band gap ${▵}_K$ with the angles ${\theta }_s$ and ${\varphi }_s$ in the Heisenberg-DMI model ($K=0$) (a), and Heisenberg-Kitaev model ($D=0$) (b). (a) and (b) share the same color map and the units are meV. (c) Thermal Hall conductivity ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of ${\theta }_s$ and ${\varphi }_s$ at $T=100\mathrm{K}$ [same parameters as in (b)]. (d) ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of temperature and ${\varphi }_s$ at ${\theta }_s={60}^{\circ }$ and (e), (f) ${\kappa }_{\mathrm{TH}}^{xy}$ as a function of temperature and ${\theta }_s$ for different values of $D$ assuming ${\varphi }_s=0^{\circ }$ and $K=5.2$ meV. (c)–(f) share the same color map in units of W/T.