Effect of Spin-lattice coupling in CrI$_3$ Monolayer*

The microscopic understanding of how spin and lattice degree of freedom interact is important to gain control over magnetic ordering in ultrafast experiments. The orbital dynamics of the electrons in CrI$_3$ monolayer is expected to have a significant impact on coupled spin-lattice dynamics (SLD) because of strong hybridization of Cr$-3d$ and I-$p$ orbitals. We present a first-principles assessment of magnetic exchange interactions from a full relativistic approach to understand the microscopic origin of the effect of SLD on magnetism. The magnetic exchange interactions are sensitive both to the in-plane motion of magnetic atoms and out-of-plane motion of ligand atoms in full atomistic vibration. Orbital decomposition of isotropic exchange interactions unfold the dependence of ferromagnetic-antiferromagnetic sign inversion on the competition between the antiferromagnetic $t_{2g}-t_{2g}$ and ferromagnetic $t_{2g}-e_{g}$ orbitals which rely on I-Cr-I bond angle and Cr-Cr bond distance. Our calculated spin-lattice coupling (SLC) constants are $\approx$ 10 times larger than bcc Fe which guarantee a larger SLD effect in CrI$_3$ monolayers. SLD reduces symmetry in CrI$_3$ monolayers and enhances the the Dzyaloshinskii-Moriya interaction over Heisenberg interactions. The estimated SLC constants, along with the microscopic orbital analysis can act as a guiding principle for further studies of the thermodynamic properties and combined magnon-phonon excitations in two-dimensional magnets.