  Proximity effects in two-dimensional (2D) van der Waals heterostructures
offer controllable ways to tailor the electronic band structure of adjacent
materials. Exchange proximity 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
MoSe$_2$ and WSe$_2$ due to ferromagnetic monolayer CrI$_3$ can be tuned (even
qualitatively) by twisting and gating. Remarkably, the proximity exchange
remains the same when using antiferromagnetic 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 exchange proximity
is confined to the layer adjacent to CrI$_3$, and that adding a separating hBN
barrier drastically reduces the proximity effect. We complement our it 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.
