The nature of the gap observed at the zone border in the spin excitation spectrum of ${\mathrm{CrI}}_3$ quasi-two-dimensional single crystals is still controversial. We perform first-principles calculations based on time-dependent density functional perturbation theory, which indicate that the observed gap results from a combination of spin-orbit and interlayer interaction effects. The former give rise to the anisotropic spin-spin interactions that are responsible for its very existence, while the latter determine both its displacement from the $K$ point of the Brillouin zone, due to the in-plane lattice distortions induced by them, and an enhancement of its magnitude, in agreement with experiments and previous theoretical work based on a lattice model.

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• Spin dynamics
• Spin-orbit coupling
• 2-dimensional systems
• Time-dependent DFT