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The observation of ferromagnetic order in a monolayer of CrI$_3$ has beenrecently reported, with a Curie temperature of 45 Kelvin and off-plane easyaxis. Here we study the origin of magnetic anisotropy, a necessary ingredientto have magnetic order in two dimensions, combining two levels of modeling,density functional calculations and spin model Hamiltonians. We find twodifferent contributions to the magnetic anisotropy of the material, bothfavoring off-plane magnetization and contributing to open a gap in the spinwave spectrum. First, ferromagnetic super-exchange across the $\simeq $ 90degree Cr-I-Cr bonds, are anisotropic, due to the spin orbit interaction of theligand I atoms. Second, a much smaller contribution that comes from the singleion anisotropy of the $S=3/2$ Cr atom. Our results permit to establish the XXZHamiltonian, with a very small single ion anisotropy, as the adequate spinmodel for this system. Using spin wave theory we estimate the Curie temperatureand we highlight the essential role played by the gap that magnetic anisotropyinduces on the magnon spectrum.

On the origin of magnetic anisotropy in two dimensional CrI$_3$