Theory of magnetism in the van der Waals magnet ${\mathrm{CrI}}_3$

We study the microscopical origin of anisotropic ferromagnetism in the van der Waals magnet ${\text{CrI}}_3$. We conclude that the nearest-neighbor exchange is well described by the Heisenberg-Kitaev-$\mathrm{\Gamma }$ ($\mathrm{HK}\mathrm{\Gamma }$) model, and we also find a nonzero Dzyaloshinskii-Moriya interaction (DMI) on next-nearest neighbors. Both Kitaev and DMI are known to generate a nontrivial topology of the magnons in the honeycomb lattice and have been used separately to describe the low-energy regime of this material. We discuss how including one or the other leads to different signs of the Chern number. Furthermore, the topological gap at the $\mathit{K}$ point seems to be mainly produced by the DMI, despite its being one order of magnitude smaller than Kitaev. Finally, we show that, by applying an external electric field perpendicular to the crystal plane, it is possible to induce DMI on nearest neighbors, and this could have consequences in noncollinear spin textures, such as domain walls and skyrmions.

Figure 1

(upper panel) Top view of the ${\text{CrI}}_3$ layer, with different plaquettes ${\text{Cr}}_2{\text{I}}_2$ highlighted in color. Three kind of NN-links are enumerated from 1 to 3. (lower-left panel) Plaquette 1 in the coordinate system $\{x^{\prime },y^{\prime },z^{\prime }\}$. Each plaquette is composed by two Cr sites $[A$ (red) and $B$ (blue)], and two iodines (green) ${\text{I}}_{\mathrm{top}}$ and ${\text{I}}_{\mathrm{bottom}}$. Axis $z^{\prime }$ is normal to the plaquette. Plaquettes 2 and 3 can be obtained by a (111) threefold rotation, or equivalently by permuting the axes $x^{\prime }, y^{\prime }, z^{\prime }$. (lower-right panel) Comparison of the two coordinate systems $\{x,y,z\}$ (blue arrows) and $\{x^{\prime },y^{\prime },z^{\prime }\}$ (pink arrows), black spheres represent Cr sites.

Figure 2

(a) Electronic gap. (b) Magnetic moment on each Cr site. Both quantities are shown as function of the Hubbard parameter $U$, and several ratios $J_H/U$ are included.

Figure 3

(a) Magnetic constants for different values of the Hubbard parameter $U$, using a fixed value of $\lambda =0.6\text{eV}$. (b) Magnetic constants as functions of the SOC parameter $\lambda$, using $U=2.9\text{eV}$. In both plots the Hund's coupling is fixed in $J_H=0.25U$.

Figure 4

Magnon bands for $\lambda =0.6\text{eV}$. $U=2.9\text{eV}$ and $J_H=0.25U$.

Figure 5

Gaps in the magnonic spectrum at $\mathit{\Gamma }$ and $\mathit{K}$ points as a function of iodine's SOC parameter $\lambda$.

Figure 6

Topological phase diagram with respect to the parameters $K$ and $d_{nnn}^z$. Purple zone has Chern number $-1$, and green zone has Chern number $+1$. The sky blue market at the origin is the only point with $\mathcal{C}=0$. The parameters that we use to describe ${\text{CrI}}_3$ in this article are highlighted with a red triangle (i). Black triangles (ii) and (iii) represent the scenarios with $K=0$ and $d_{nnn}^z=0$, respectively, which are described in the main text.

Figure 7

Components of the DM vector, as a function of the electric field, using $\lambda =0.6\text{eV}$. The inset shows the behavior of $c_{xy}\left(\lambda \right)=d_{xy}/E_0$, and $c_z\left(\lambda \right)=d_z/E_0$ as a function of the iodine's SOC parameter $\lambda$. In this calculation we used $U=2.9\text{eV}$ and $J_H/U=0.25$.