Revealing the Underlying Mechanisms of Stacking Order and Interlayer Magnetism in Bilayer CrBr$_3$

Aiming to clarify the mechanisms governing the interlayer magnetic coupling, we have investigated the stacking energy and interlayer magnetism of bilayer CrBr$_3$ systemically. The magnetic ground states of bilayer CrBr$_3$ with different R-type and H-type stacking orders are established, which is found to be in good agreement with recent experiment (Science $\mathbf{366}$,983(2019)).Further analyses indicate that the stacking energy is mainly determined by the Coulomb interaction between the interlayer nearest-neighbor Br-Br atoms. While interlayer magnetism can be understood by a competition between super-super-exchange interactions involving $t_{2g}$-$t_{2g}$ and $t_{2g}$-$e_g$ orbitals and semi-covalent exchange interactions of $e_g$-$e_g$ orbitals. Our studies give an insightful understanding for stacking order and interlayer magnetism of bilayer CrBr$_3$, which should be useful to understand quantum confinement effect of other layered magnets in two-dimensional limit.