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