Interlayer superexchange in bilayer chromium trihalides

We construct a microscopic model based on superexchange theory for moir\'e bilayer in chromium trihalides (Cr$X_3$, $X=$Br, I). In particular, we derive analytically the interlayer Heisenberg exchange and Dzyaloshinskii-Moriya interaction with arbitrary distances (x) between spins. Importantly, our takes into account sliding twisting geometries $X$-$X$ hopping processes. Our approach can directly access $x$-dependent without large unit-cell calculations. argue that deducing by various bilayers may lead to an incomplete result bilayer. Using \textit{ab initio} tight-binding Hamiltonian, numerically evaluate interactions CrI$_3$. find analytical agrees previous comprehensive density functional studies. Furthermore, findings reveal important role of correlation effects $X$'s $p$ orbitals, which gives rise rich magnetic remarkable tunability.