Superexchange and spin-orbit coupling in monolayer and bilayer chromium trihalides

We build a microscopic model to study the intralayer and interlayer superexchange due to electrons hopping in chromium trihalides (CrX₃, $X=$ Cl, Br, and I). In evaluating the superexchange, we identify the relevant intermediate excitations in the hopping. In our study, we find that the intermediate hole-pair excitations in the p orbitals on X ion play a crucial role in mediating various types of exchange interactions. In particular, the interlayer antiferromagnetic exchange may be realized by the hole-pair-mediated superexchange. Interestingly, we also find that these virtual hopping processes compete with each other, leading to weak intralayer ferromagnetic exchange. In addition, we also study the spin-orbit coupling effects on the superexchange and investigate the Dzyaloshinskii-Moriya interaction. Finally, we extract the microscopic model parameters from density functional theory for analyzing the exchange interactions in a monolayer CrI₃.