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Magnetic anisotropy is crucially important for the stabilization oftwo-dimensional (2D) magnetism, which is rare in nature but highly desirable inspintronics and for advancing fundamental knowledge. Recent works on CrI$_3$and CrGeTe$_3$ monolayers not only led to observations of the long-time-sought2D ferromagnetism, but also revealed distinct magnetic anisotropy in the twosystems, namely Ising behavior for CrI$_3$ versus Heisenberg behavior forCrGeTe$_3$. Such magnetic difference strongly contrasts with structural andelectronic similarities of these two materials, and understanding it at amicroscopic scale should be of large benefits. Here, first-principlescalculations are performed and analyzed to develop a simple Hamiltonian, toinvestigate magnetic anisotropy of CrI$_3$ and CrGeTe$_3$ monolayers. Theanisotropic exchange coupling in both systems is surprisingly determined to beof Kitaev-type. Moreover, the interplay between this Kitaev interaction andsingle ion anisotropy (SIA) is found to naturally explain the differentmagnetic behaviors of CrI$_3$ and CrGeTe$_3$. Finally, both the Kitaevinteraction and SIA are further found to be induced by spin-orbit coupling ofthe heavy ligands (I of CrI$_3$ or Te of CrGeTe$_3$) rather than the commonlybelieved 3d magnetic Cr ions.

Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI$_3$ and CrGeTe$_3$ monolayers