Topological magnonics in the two-dimensional van der Waals magnet ${\mathrm{CrI}}_3$

We report on the magnon spectrum of Kitaev-Heisenberg magnets and extract the parameters to model a two-dimensional ${\text{CrI}}_3$. Our minimal spin Hamiltonian includes a contribution stemming from a Heisenberg, isotropic exchange, and a contribution arising from a Kitaev interaction, anisotropic and frustrated exchange. Our calculations reveal a gap that opens at the $K$ and $K^{\prime }$ points and the topological nature of the magnons which lead to the thermal Hall effect. Furthermore, we calculate the magnon spectrum of nanoribbons illustrating the corresponding edge states.

Figure 1

(a) Top view of ${\text{CrI}}_3$ monolayer. The atoms in the unit cell are highlighted in color: big purple spheres represent chromium sites and small green spheres represent iodine sites. Lattices vectors ${\mathbf{a}}_{\mathbf{1}}=a_0(\frac{\sqrt{3}}{2},\frac{1}{2},0)$ and ${\mathbf{a}}_{\mathbf{2}}=a_0(\frac{\sqrt{3}}{2},-\frac{1}{2},0)$ were drawn with red arrows, $a_0$ is the lattice constant. (b) First Brillouin zone, with the special symmetry points $\mathit{\Gamma }$, $\mathrm{K}$, and ${\mathrm{K}}^{\prime }$.(c) A view of the ${\text{CrI}}_3$ monolayer in perspective, with the plaquettes $C{r}_2{I}_2$ colored according to their normal vectors ${\widehat{\mathbf{\gamma }}}_{\mathit{1}}$, ${\widehat{\mathbf{\gamma }}}_{\mathit{2}}$, and ${\widehat{\mathbf{\gamma }}}_{\mathit{3}}$. Three plaquettes in the unit cell are highlighted, and normal vectors form an orthonormal basis.

Figure 2

Energy spectrum of magnons within the first Brillouin zone. The blue line corresponds to the case $J=0.53\mathrm{meV}$, $A=0.44\mathrm{meV}$, and $K=0$. It can be see that there is no gap at the $\mathbf{K}$ point. There is a gap at the $\mathrm{\Gamma }$ point that arises from the anisotropy contribution [12]. On the other hand, the inclusion of the Kitaev interaction, $K=2J$ displayed in the red line, displays a gap opening at the $\mathbf{K}$ point revealing a nontrivial topology. The circles next to each bands correspond to the associated Chern numbers. These are calculated according to [21]. The $\mathrm{K}$-point gap is calculated as a function of the Kitaev interaction strength in the inset.

Figure 3

Thermal Hall conductivity ${\kappa }^{xy}$ vs temperature $T$. It can be seen that the conductivity changes of sign at different temperatures for different values of $K$. We highlight that for $K=4.07\mathrm{meV}$, the thermal conductivity changes of sign for $T=7.50\mathrm{K}$, which is highlighted by the grey dotted line.

Figure 4

Magnonic energy bands with $N=10$ unit cells zigzag nanoribbon. (Left) Situation with $K=0$. We can see a clear resemblance with those of graphene nanoribbons [43]. The otherwise flat bands in the mid-gap area are distorted due to the on-site energy discrepancy at the edge. The plots below represent the magnitude of the probabilities (grey circles with increasing radius for larger probabilities) for the states with $E=3.25\mathrm{meV}$. There is no localization in this case. (Right) Situation with $K=4.07\mathrm{meV}$. The bulk bands preserve their basic shape but the band-width is amplified. The degeneracy of the mid-gap bands, highlighted with red, toward the edge of the Brillouin zone gets lifted. The lower panels describe the probability density of the states with $E=8.53\mathrm{meV}$. We can see clear evidence of the localization of these states at the geometrical edges of the ribbons.

Figure 5

Dispersion relation and states corresponding to the marked energies. (Left) Situation with $K=0$. As with Fig. 4, there is a resemblance with graphene, where two of the top bands lower their energies due to the on-site energy discrepancy at the edge. The plots below represent the magnitude of the probabilities for the two possible states with $E=3.23\mathrm{meV}$. The probability is proportional to the radii of the circles. (Right) Situation with $K=4.07\mathrm{meV}$. Bands preserve their basic structure and the two central bands are separated. The light-blue colored area is the gap of the bulk, which shows which band should have a topological behavior. The lower plots represent the probability of the states with $E=8.63\mathrm{meV}$. There is a clear localization at both edges of the ribbon.

Figure 6

Zigzag nanoribbon. The numbers beside the site correspond to the indices $j$ and $j^{\prime }$ that are used in Hamiltonian (14).

Figure 7

Armchair nanoribbon. The numbers beside the site correspond to the indices $j$ and $j^{\prime }$ that are used in Hamiltonian (14).