Theory of magnetism in the van der Waals magnet CrI3

We study the microscopical origin of anisotropic ferromagnetism in the van der Waals magnet CrI3. We conclude that the nearest-neighbor exchange is well described by the Heisenberg-Kitaev-F (HKF) 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 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.