Strong manipulation of the valley splitting upon twisting and gating in ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$ and ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ van der Waals heterostructures

Van der Waals (vdW) heterostructures provide a rich playground to engineer electronic, spin, and optical properties of individual two-dimensional materials. We investigate the twist-angle and gate dependence of the proximity-induced exchange coupling in the monolayer transition-metal dichalcogenides (TMDCs) ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ due to the vdW coupling to the ferromagnetic semiconductor ${\mathrm{CrI}}_3$, from first-principles calculations. A model Hamiltonian, which captures the relevant band edges at the $K/K^{\prime }$ valleys of the proximitized TMDCs, is employed to quantify the proximity-induced exchange. Upon twisting from $0^{\circ }$ to ${30}^{\circ }$, we find a transition of the TMDC valence band (VB) edge exchange splitting from about $-2$ to $2\mathrm{meV}$, while the conduction band (CB) edge exchange splitting remains nearly unchanged at around $-3\mathrm{meV}$. For the VB of ${\mathrm{WSe}}_2$ (${\mathrm{MoSe}}_2$) on ${\mathrm{CrI}}_3$, the exchange coupling changes sign at around $8^{\circ }$ (${16}^{\circ }$). We find that even at the angles with almost zero spin splittings of the VB, the real-space spin polarization profile of holes at the band edge is highly nonuniform, with alternating spin up-and spin-down orbitals. Furthermore, a giant tunability of the proximity-induced exchange coupling is provided by a transverse electric field of a few V/nm. Within our first-principles framework, we are limited to commensurate structures, and thus considered a maximum strain of 2.2% for the individual monolayers. By investigating different atomic stacking configurations of the strained supercells, we demonstrate that proximity exchange varies locally in space within experimentally realistic setups. We complement our ab initio results by calculating the excitonic valley splitting to provide experimentally verifiable optical signatures of the proximity exchange. Specifically, we predict that the valley splitting increases almost linearly as a function of the twist angle. Furthermore, the proximity exchange is highly tunable by gating, allowing to tailor the valley splitting in the range of 0 to 12 meV in ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$, which is equivalent to external magnetic fields of up to about 60 T. Our results highlight the important impact of the twist angle and gating when employing magnetic vdW heterostructures in experimental geometries.

Figure 1

3D view of a TMDC (${\mathrm{MoSe}}_2$ or ${\mathrm{WSe}}_2$) on ${\mathrm{CrI}}_3$, where we define the interlayer distance, ${\mathrm{d}}_{\mathrm{int}}$. We twist the TMDC by an angle $\vartheta$ around the $z$ axis, with respect to the magnetic semiconductor ${\mathrm{CrI}}_3$, with magnetization $\mathit{M}$ along the $z$ direction. The twist-angle evolution of the proximitized TMDC band edges are sketched. Red bands are polarized spin up, while blue bands are polarized spin down. By twisting from $0^{\circ }$ to ${30}^{\circ }$, the TMDC VB edge splitting, which is due to proximity exchange, first vanishes and then reverses sign.

Figure 2

(a) DFT-calculated band structure of the ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$ heterostructure for a twist angle of $0^{\circ }$ along the high-symmetry path $M\text{−}K\text{−}\mathrm{\Gamma }$. Bands in red are spin up, while bands in blue are spin down. We define the heterostructure band gap $\mathrm{\Delta }E$, between the TMDC VB edge at K, and the minimum of spin polarized ${\mathrm{CrI}}_3$ in-gap states. We also define the bandwidth $\mathrm{\Delta }W$ of the in-gap states. (b) Zoom to the ${\mathrm{MoSe}}_2$ CB edge near the $K$ point, showing proximity exchange split bands. Symbols are DFT data and solid lines are the fitted model Hamiltonian results. (c) Same as (b), but for the VB edge. [(d)–(f)] Same as [(a)–(c)], but for a twist angle of ${30}^{\circ }$. Comparing (c) and (f), the spin ordering of bands is reversed.

Figure 3

Calculated twist-angle dependence of the proximity exchange couplings $B_{\text{c}}$ and $B_{\text{v}}$ for the ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$ and ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ heterostructures.

Figure 4

(Top) An experimentally realistic $0^{\circ }{\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ heterostructure (2123 atoms), where we consider a $21\times 21$ supercell of ${\mathrm{WSe}}_2$ on a $10\times 10$ supercell of ${\mathrm{CrI}}_3$, corresponding to strains of 0% and 0.5%. (Middle) Different high-symmetry stackings of the $0^{\circ }$ strained heterostructure are locally indicated, with their corresponding proximity exchange parameters $B_{\text{c}}$ and $B_{\text{v}}$. (Bottom) Calculated proximity-induced local magnetic moments (colored spheres) of the interfacial Se and W atoms for the different stackings.

Figure 5

Calculated electric-field and twist-angle dependence of the proximity-induced exchange parameters $B_c$ (top) and $B_v$ (bottom) for the ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ heterostructure, interpolated from Table S2.

Figure 6

Calculated real-space TMDC band edge spin polarizations, $\mathrm{\Delta }\rho ={\rho }_{\uparrow }-{\rho }_{\downarrow }$, for the twisted ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ heterostructures. The background color is used to group the subfigures for the three twist angles (a) $0^{\circ }$, (b) $8.2^{\circ }$, and (c) ${30}^{\circ }$. (a) (Middle) Sketch of the proximitized TMDC band edges for $0^{\circ }$, where we indicate the energy windows from which we calculate the spin densities ${\rho }_{\uparrow /\downarrow }$.

(Top/bottom) Calculated band edge spin polarization, taking into account only CB/VB states. The color red (blue) corresponds to $\mathrm{\Delta }\rho >0$ ($\mathrm{\Delta }\rho <0$). The isosurfaces correspond to isovalues (units ${\text{Å}}^{-3}$) as indicated. (b) The same as (a), but for $8.2^{\circ }$. For the VB, we also show a top view, with removed ${\mathrm{CrI}}_3$ layer, to show the nonuniform spin polarization on ${\mathrm{WSe}}_2$. (c) The same as (a), but for ${30}^{\circ }$.

Figure 7

(a) Valley splitting for the ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$ and ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ bilayers calculated from the single-particle (abbreviated by s.p., short dashed lines), and from the excitonic absorption (abbreviated as exc.) considering the region above the TMDC to be vacuum (long dashed lines) or hexagonal boron nitride (solid lines). The single-particle values follow nicely the excitonic calculations. (b) Valley splitting for the ${\mathrm{WSe}}_2/{\mathrm{CrI}}_3$ bilayer as function of the twist angle ($x$ axis) and applied electric field ($y$ axis) using the single-particle values, calculated via Eq. (6).