Difference in magnetic anisotropy of the ferromagnetic monolayers ${\mathrm{VI}}_3$ and ${\mathrm{CrI}}_3$

Concerning the magnetic anisotropy and magnetic moments of the $M^{3+}$ $(M=\mathrm{V}, \mathrm{Cr})$ ions in ferromagnetic $\left(\mathrm{FM}\right){M\mathrm{I}}_3$ 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 ${\mathrm{V}}^{3+}\left(d^2,S=1\right)$ 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., ∼8 vs ∼0.6 meV). The ${\mathrm{V}}^{3+}$ ion exhibits uniaxial magnetism because its orbital quantum number L is not zero $\left(L=1\right)$, in contrast to the ${\mathrm{Cr}}^{3+}$ ion $\left(L=0\right)$.

Figure 1

(a) Projection view of ${M\mathrm{I}}_3$ ($M=\mathrm{V}$, Cr in blue; I in violet) monolayer along the $c$ direction (i.e., perpendicular to the layer plane). (b) Local coordinate chosen for the ${M\mathrm{I}}_6$ octahedron in the ${M\mathrm{I}}_3$ monolayer, where the $z$ axis taken along the threefold rotational axis of ${M\mathrm{I}}_6$ is parallel to the $c$ direction. The $t_{2g}$ and $e_g$ states of an ideal ${M\mathrm{I}}_6$ octahedron are also shown.

Figure 2

The MAE vs ${\theta }_M$ in the polar coordinate system for (a) ${\mathrm{VI}}_3$ monolayer and (b) ${\mathrm{CrI}}_3$ monolayer. Red and blue are for the angles in the $xz$ plane, and black is for the angles in the $xy$ plane. (c) A sketch showing the definition of ${\theta }_M$ in the $xz$ plane. (d) The MAE vs ${\theta }_M$ in the Cartesian coordinate system for the $xz$ plane, with ${\theta }_M=0^{\circ }$ representing the moment lying in the $x$ axis. The dashed lines represent the fitting curves defined as a function of $\mathrm{si}{\mathrm{n}}^2{\theta }_M$ [see Eq. (1)].

Figure 3

(a) Relationship between the spin and orbital moment angles, ${\theta }_{MS}$ and ${\theta }_{ML}$, respectively, calculated for the ${\mathrm{VI}}_3$ monolayer. (b) Relative orientations of the ${\theta }_{ML}$ and ${\theta }_{MS}$ of the ${\mathrm{VI}}_3$ monolayer in the $xz$ plane.

Figure 4

The electron configurations for the ${\mathrm{V}}^{3+}$ $\left(d^2,S=1\right)$ $\left(d^2,S=1\right)$ ion of ${\mathrm{VI}}_6$ octahedra: (a) ${\mathrm{\Psi }}_{\mathrm{V},1}={(1a\uparrow )}^1(1e_x\uparrow ,1e_y{\uparrow )}^1$, $L=1$ state; (b) ${\mathrm{\Psi }}_{\mathrm{V},1}={(1a\uparrow )}^0(1e_x\uparrow ,1e_y{\uparrow )}^2$, $L=0$ state. Here the ${\mathrm{VI}}_6$ octahedron was assumed to be regular in shape so the 1 $a$ and 1 $e$ states are degenerate. (c) The electron configuration ${\mathrm{\Psi }}_{\mathrm{Cr}}={(1a\uparrow )}^1(1e_x\uparrow ,1e_y{\uparrow )}^2$ for the ${\mathrm{Cr}}^{3+}$ ($d^3$, $S=3/2$) ion of ${\mathrm{CrI}}_6$ octahedra in the ${\mathrm{CrI}}_3$ monolayer, where the HOMO and LUMO are given by the $t_{2g}\uparrow$ and $e_g\uparrow$ states, respectively. The ${\mathrm{CrI}}_6$ octahedron is assumed to be regular in shape, so the 1 $a$ and 1 $e$ states are degenerate. (d) The energy lowering $\mathrm{\Delta }{E}_1$ associated with the interaction between the HOMO and LUMO in the ${\mathrm{V}}^{3+}$ ion in ${\mathrm{VI}}_3$ monolayer. (e) The energy lowering $\mathrm{\Delta }{E}_2$ associated with the interaction between the HOMO and LUMO in the ${\mathrm{Cr}}^{3+}$ ion of ${\mathrm{CrI}}_3$ monolayer.