Environmental screening and ligand-field effects to magnetism in CrI3 monolayer

We present a detailed study on the microscopic origin of magnetism in suspended and dielectrically embedded CrI$_3$ monolayer. To this end, we down-fold two distinct minimal generalized Hubbard models with different orbital basis sets from \emph{ab initio} calculations using constrained random phase approximation. Within mean-field approximation, show that these are capable describing formation localized magnetic moments reproducing electronic properties full calculations. utilize force theorem to exchange channels between manifolds. find multi-orbital super-exchange mechanism as resulting interplay effective ferro- anti-ferromagnetic Cr-Cr $d$ coupling channels, which is decisively affected by ligand (I) $p$ orbitals. how environmental screening such encapsulation hexagonal boron nitride (hBN) monolayer affects Coulomb interaction film successively controls its properties. Driven non-monotonic nearest next-nearest neighbour interactions magnon dispersion Curie temperature be non-trivially dielectric screening.