Proximity exchange effects in ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ heterostructures with ${\mathrm{CrI}}_3$: Twist angle, layer, and gate dependence

Proximity effects in two-dimensional (2D) van der Waals heterostructures offer controllable ways to tailor the electronic band structure of adjacent materials. Proximity exchange in particular is important for making materials magnetic without hosting magnetic ions. Such synthetic magnets could be used for studying magnetotransport in high-mobility 2D materials, or magneto-optics in highly absorptive nominally nonmagnetic semiconductors. Using first-principles calculations, we show that the proximity exchange in monolayer ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ due to ferromagnetic monolayer ${\mathrm{CrI}}_3$ can be tuned (even qualitatively) by twisting and gating. Remarkably, the proximity exchange remains the same when using antiferromagnetic ${\mathrm{CrI}}_3$ bilayer, paving the way for optical and electrical detection of layered antiferromagnets. Interestingly, the proximity exchange is opposite to the exchange of the adjacent antiferromagnetic layer. Finally, we show that the proximity exchange is confined to the layer adjacent to ${\mathrm{CrI}}_3$, and that adding a separating hBN barrier drastically reduces the proximity effect. We complement our ab initio results with tight-binding modeling and solve the Bethe-Salpeter equation to provide experimentally verifiable optical signatures (in the exciton spectra) of the proximity exchange effects.

Figure 1

Calculated band structures without SOC, geometries, and absorption spectra of ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$. [(a) and (b)] Side and top view of the small supercell geometry ($0^{\circ }$ twist angle) with labels for the different atoms and the definition of the interlayer distance $d_{\mathrm{TMDC}/{\mathrm{CrI}}_3}$. One unit cell of ${\mathrm{CrI}}_3$ is highlighted, by the red dashed line, in the top view. (c) Band structure along high-symmetry lines, with the energy gap of the heterostructure $\mathrm{\Delta }E$. The bands corresponding to ${\mathrm{MoSe}}_2$ are emphasized by red (spin up) and blue (spin down) spheres. [(d) and (e)] Zoom to the CB and VB edge corresponding to ${\mathrm{MoSe}}_2$. Symbols are DFT data and solid lines are the fitted model Hamiltonian. [(f)–(i)] The top view, calculated band structure, and the zoom to the low-energy bands for the larger supercell geometry (${30}^{\circ }$ twist angle). [(j) and (k)] Calculated first absorption peak of intralayer excitons for small (b) and large (f) geometry (including SOC) with the vertical solid (dashed) arrows indicating the peak position for the absorption at $K$ ($K^{\prime }$) point.

Figure 2

Calculated band structure of ${\mathrm{MoSe}}_2/{\mathrm{CrI}}_3$ with SOC for the $0^{\circ }$ twist angle structure. (a) Band structure along high-symmetry lines. Color corresponds to the $s_z$-expectation value. (b,c) Zoom to the CB edge corresponding to ${\mathrm{MoSe}}_2$ at $K$ and $K^{\prime }$. Symbols are DFT data and solid lines are the fitted model Hamiltonian. [(d) and (e)] Same as (b) and (c) but for VB edge.

Figure 3

(a) Band structure along high-symmetry lines for bilayer ${\mathrm{MoSe}}_2$ on top of monolayer ${\mathrm{CrI}}_3$. In (a)–(h), spin up (down) bands are plotted in red (blue), while symbols emphasize the character of the bands. In (a), the bands corresponding to bilayer ${\mathrm{MoSe}}_2$ are emphasized by thicker spheres. (b) Zoom to the CB edge around the $K$ point. [(c) and (d)] Same as (b), but for VB edge around $K$ and $\mathrm{\Gamma }$ point. In (b)–(d), the bands corresponding to the first (second) layer of the bilayer ${\mathrm{MoSe}}_2$ are emphasized by filled spheres (open triangles). (e) Side view of the geometry with definition of first and second layer of 2H bilayer ${\mathrm{MoSe}}_2$. (f) Band structure along high-symmetry lines for monolayer ${\mathrm{MoSe}}_2$ on top of bilayer ${\mathrm{CrI}}_3$ in the AFM ($\uparrow \downarrow$) configuration. (g) Zoom to the CB edge around the K point (symbols), with a fit to the model Hamiltonian (solid line). (h) Same as (g), but for VB edge. (i) Side view of the geometry of monolayer ${\mathrm{MoSe}}_2$ on bilayer ${\mathrm{CrI}}_3$.

Figure 4

Fit parameters as a function of transverse electric field for $\mathrm{TMDC}/{\mathrm{CrI}}_3$ heterostructures for calculations without SOC. (a) The orbital gap $\mathrm{\Delta }$ of the spectrum, (b) the Fermi velocity $v_{\mathrm{F}}$, (c) the dipole of the heterostructure, [(d) and (e)] the proximity exchange parameters $B_{\mathrm{c}}$ and $B_{\mathrm{v}}$, and (f) the band gap $\mathrm{\Delta }E$, as defined in Fig. 1. (g) Schematic representation of the exchange interaction in the band structure of ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$ at $K$ and $K^{\prime }$ points depicting the lowest energy optically allowed transitions (vertical dashed arrows). (h) Single-particle and intralayer exciton valley splitting for the first active optical transition between the top VB and the first (second) CB in ${\mathrm{MoSe}}_2$ (${\mathrm{WSe}}_2$) [indicated by the circle in (g)] as function of the external electric field. For the excitons, the valley splitting is taken as the energy difference between the first absorption peaks (see Fig. 1 and Ref. [71]).