Relativistic exchange interactions in CrX3 (X=Cl, Br, I) monolayers

It has been predicted theoretically and indirectly confirmed experimentally that single-layer $\mathrm{Cr}{X}_{3}$ ($X$ = Cl, Br, I) might be the prototypes of topological magnetic insulators (TMI). In this work, by using first-principles calculations combined with atomistic spin dynamics, we provide a complete picture of the magnetic interactions and magnetic excitations in $\mathrm{Cr}{X}_{3}$. The focus is here on the two most important aspects for the actual realization of TMI, namely the relativistic magnetic interactions and the finite-size (edge) effects. We compute the full interaction tensor, which includes both Kitaev and Dzyaloshinskii-Moriya (DM) terms, which are considered as the most likely mechanisms for stabilizing topological magnons. First, we instigate the properties of bulk ${\mathrm{CrI}}_{3}$ and compare the simulated magnon spectrum with the experimental data [Phys. Rev. X 8, 041028 (2018)]. Our results suggest that a large size of topological gap, seen in experiment ($\ensuremath{\approx}4$ meV), cannot be explained by considering pair-wise spin interactions only. We identify several possible reasons for this disagreement. The magnetic interactions in the monolayers of $\mathrm{Cr}{X}_{3}$ are also investigated. The strength of the anisotropic interactions is shown to scale with the position of halide atom in the periodic table, the heavier the element the larger is the anisotropy, in agreement with prior studies. Comparing the magnons for the bulk and single-layer ${\mathrm{CrI}}_{3}$, we find that the size of the topological gap becomes smaller in the latter case. The obtained next nearest-neighbor DM vector is oriented primarily in-plane of the monolayer and has relatively small $z$ component, which results in a small value of the topological gap. Finally, we investigate finite-size effects in monolayers and demonstrate that the anisotropic couplings between Cr atoms close to the edges are much stronger than those in ideal periodic structure. This should have impact on the dynamics of the magnon edge modes in Cr halides.