Electric-field tuning of the magnetic properties of bilayer ${\mathrm{VI}}_3$: A first-principles study

The magnetic properties of the two-dimensional ${\mathrm{VI}}_3$ bilayer are the focus of our first-principles analysis, highlighting the role of $t_{2g}$ orbital splitting and carried out in comparison with the ${\mathrm{CrI}}_3$ prototypical case, where the splitting is negligible. In ${\mathrm{VI}}_3$ bilayers, the empty $a_{1g}$ state is found to play a crucial role in both stabilizing the insulating state and in determining the interlayer magnetic interaction. Indeed, an analysis based on maximally localized Wannier functions allows one to evaluate the interlayer exchange interactions in two different ${\mathrm{VI}}_3$ stackings (labeled AB and ${\mathrm{AB}}^{\prime }$), to interpret the results in terms of the virtual-hopping mechanism, and to highlight the strongest hopping channels underlying the magnetic interlayer coupling. Upon application of electric fields perpendicular to the slab, we find that the magnetic ground state in the ${\mathrm{AB}}^{\prime }$ stacking can be switched from antiferromagnetic to ferromagnetic, suggesting the ${\mathrm{VI}}_3$ bilayer as an appealing candidate for electric-field-driven miniaturized spintronic devices.

Figure 1

(a) ${\mathrm{VI}}_6$ (${\mathrm{CrI}}_6$) edge-sharing octahedra in the honeycomb layer structure. (b) and (c) The orbital splitting of $d$ levels in V $3d^2$ and Cr $3d^3$, respectively. Five $3d$ wannier functions reflect the $D_{3d}$ orbital basis set in monolayer ${\mathrm{VI}}_3$ (d) and the $O_h$ orbital basis set in monolayer ${\mathrm{CrI}}_3$ (e). The isosurface levels of the wannier functions were set at $1.5a_0^{-3/2}$ (yellow) and $-1.5a_0^{-3/2}$ (blue), where $a_0$ is the Bohr radius.

Figure 2

The partial DOS (within $U$ = 2.0 eV) projected onto (a) V-$3d\text{−}{D}_{3d}$ and (b) Cr-$3d\text{−}{O}_h$ orbital basis set, respectively, via wannier functions. Blue (red) color represents for majority spin (minority spin). The Fermi level is set as the energy origin.

Figure 3

(a) and (b) Top views and (c) and (d) side views of atomic structure in bilayer ${\mathrm{VI}}_3$ in AB and ${\mathrm{AB}}^{\prime }$ stacking. The red and orange hexagons represent honeycomb structure of V atoms in top and bottom layers, and gray balls represent I atoms. The black arrow indicates the vector which connects equivalent atoms located in two layers and shows how the top layer is sliding with respect to the bottom layer. Interlayer exchange coupling $J_{ij}$ in bilayer ${\mathrm{VI}}_3$ for (e) AB and (f) ${\mathrm{AB}}^{\prime }$ stacking.

Figure 4

MLWFs relevant to interlayer exchange coupling in bilayers (a)–(c) ${\mathrm{VI}}_3$ and (d)–(f) ${\mathrm{CrI}}_3$. $\uparrow$ and $\downarrow$ denote the majority and minority spin states, respectively. The arrows show the electron hopping from an occupied orbital state to an unoccupied orbital state; the dashed and solid lines denote the parallel-spin and anti-parallel-spin configurations, respectively. Values of the hopping integrals (meV) are also shown near the arrows. Isosurface levels were set at $\pm 0.45a_0^{-3/2}$ for (a)–(c) and $\pm 0.3a_0^{-3/2}$ for (d)–(f).

Figure 5

The energy difference between the FM and AFM ordering for (a) AB and (b) ${\mathrm{AB}}^{\prime }$ stacking as a function of an external electric field. The positive value of $\mathrm{\Delta }\mathrm{E}$ means FM is favored and negative means AMF is favored. $\mathit{d}$-orbital projected density of state for top (solid filled) and bottom (solid line) layer of bilayer ${\mathrm{VI}}_3$ with AFM ${\mathrm{AB}}^{\prime }$ stacking in AFM ordering (c) without and (d) with external electric field. Black arrow presents for the energy shifted by applied electric field. Vertical dash line denotes the Fermi energy.