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

We investigate the twist-angle and gate dependence of the proximity-induced exchange coupling in the monolayer transition-metal dichalcogenides (TMDCs) MoSe$_2$ and WSe$_2$ due to the vdW coupling to the ferromagnetic semiconductor CrI$_3$, from first-principles calculations. A model Hamiltonian, that 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° to 30°, we find a transition of the TMDC valence band (VB) edge exchange splitting from about $-2$ to $2$ meV, while the conduction band (CB) edge exchange splitting remains nearly unchanged at around $-3$ meV. For the VB of WSe$_2$ (MoSe$_2$) on CrI$_3$, the exchange coupling changes sign at around 8° (16°). 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 non-uniform, 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. We complement our \textit{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 WSe$_2$/CrI$_3$, which is equivalent to external magnetic fields of up to about 60 Tesla. Our results highlight the important impact of the twist angle and gating when employing magnetic vdW heterostructures in experimental geometries.