  The nature of the gap observed at the zone border in the spin-excitation
spectrum of CrI$_3$ quasi-2D 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 inter-layer 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.
