Difference in magnetic anisotropy of the ferromagnetic monolayers VI 3 and CrI 3

Concerning the magnetic anisotropy and magnetic moments of the ${M}^{3+}$ $(M=\mathrm{V}, \mathrm{Cr})$ ions in ferromagnetic $(\mathrm{FM}){M\mathrm{I}}_{3}$ monolayers, which have a honeycomb pattern of edge-sharing ${M\mathrm{I}}_{6}$ octahedra, conflicting observations have been reported in experimental and theoretical studies. We resolve these conflicts by determining the magnetic anisotropy energies for the ${M}^{3+}$ ions of ${M\mathrm{I}}_{3}$ monolayers, by analyzing their preferred spin orientations in terms of the selection rules based on the highest occupied molecular orbital--lowest unoccupied molecular orbital interactions of the ${M\mathrm{I}}_{6}$ octahedra, and by discussing whether or not the ${M}^{3+}$ ions are uniaxial. Here we show that the FM monolayer ${\mathrm{VI}}_{3}$ is uniaxial, but that of ${\mathrm{CrI}}_{3}$ is not. The magnetic anisotropy energy for the ${\mathrm{V}}^{3+}({d}^{2},S=1)$ ion of ${\mathrm{VI}}_{3}$ is greater than that for the ${\mathrm{Cr}}^{3+}$ $({d}^{3},$ $S=3/2)$ ion of ${\mathrm{CrI}}_{3}$ by more than an order of magnitude (i.e., \ensuremath{\sim}8 vs \ensuremath{\sim}0.6 meV). The ${\mathrm{V}}^{3+}$ ion exhibits uniaxial magnetism because its orbital quantum number L is not zero $(L=\phantom{\rule{0.16em}{0ex}}1)$, in contrast to the ${\mathrm{Cr}}^{3+}$ ion $(L=0)$.