Two-dimensional (2D) ferromagnetic materials are highly promising for spintronic nanodevices and magnon-based spintronics. Chromium trihalides (CrX₃, X = I, Br, Cl) have shown potential in this area, with experimental observations of magnons. However, the impact of the environment on the magnetic properties of these materials remains a subject of debate. Theoretical predictions suggest a strong dependence on film thickness and spin-orbit coupling, while experimental studies indicate tunability using electrostatic gating. The nearest-neighbor magnetic exchange couplings in CrX₃ are reported to be between 1 and 3.2 meV, highlighting the need for rigorous theoretical descriptions, particularly concerning Coulomb interactions, which are enhanced in layered materials due to reduced polarization. Environmental screening can significantly modify these Coulomb interactions, influencing properties like exciton and plasmon formation. This study aims to understand how environmental screening affects the magnetic ground state and magnonic excitations of CrI₃, described by effective spin Hamiltonians representing exchange interactions between Cr atoms. Conventional DFT methods struggle to accurately capture environmentally induced modifications to Coulomb interactions; therefore, this research employs a multi-step approach combining ab initio calculations, constrained random phase approximation (CRPA), and mean-field Hartree-Fock (HF) theory to overcome these limitations.

Existing literature on chromium trihalides shows a range of theoretical predictions for nearest-neighbor magnetic exchange couplings (1-3.2 meV), depending on the calculation method. Studies highlight the importance of considering Coulomb interactions, enhanced in layered materials due to reduced polarization. Environmental effects, including dielectric screening and electrostatic gating, are shown to tune magnetic properties. However, the precise mechanism by which the environment affects exchange couplings remains unclear, and conventional DFT methods are insufficient for accurate modeling. The Kugel-Khomskii (KK) formalism provides a theoretical framework for understanding magnetic exchange in transition metal compounds, but its applicability to CrI3 and the influence of environmental screening requires further investigation. Previous work has used various methods, including LSDA(+U), but a comprehensive understanding involving the full multi-orbital nature of interactions and environmental effects was lacking.

This study uses a multi-step computational approach to investigate the impact of environmental screening on the magnetic properties of CrI₃ monolayer. First, spin-unpolarized ab initio band structure calculations are performed using DFT to obtain the electronic structure. These results are then used to construct two minimal models: one using only Cr d states, and another incorporating both Cr d and I p states. The Coulomb tensors for these models are calculated using CRPA, accounting for screening effects. The resulting multi-orbital Hubbard models are solved within mean-field Hartree-Fock (HF) theory to obtain interacting and spin-resolved quasi-particle band structures. The magnet force theorem is applied to these solutions to determine orbitally resolved magnetic exchange interactions. The CRPA calculations explicitly exclude Kohn-Sham states within a specific energy range to avoid unrealistically strong metallic screening. The analysis focuses on static local density-density elements and local Hund's exchange elements, neglecting non-local interactions. Renormalization of hopping parameters due to retarded Coulomb interactions is also considered, but found to be negligible in this system. The d-only model and the extended (d+p) model are compared to assess the importance of ligand p orbitals. The effects of environmental screening are modeled by modifying the CRPA Coulomb interaction tensor according to a Wannier function continuum electrostatics (WFCE) approach, which accounts for dielectric screening. The impact of screening on the band structure, exchange interactions (nearest and next-nearest neighbor), magnon dispersion, and Curie temperature is analyzed. The spin Hamiltonian is used to calculate the magnon dispersion and Curie temperature using Tyablikov's decoupling approximation.

The d-only model, while qualitatively reproducing the Cr d band structure, incorrectly predicts antiferromagnetic ordering. This highlights the crucial role of ligand p orbitals in mediating the super-exchange mechanism. The (d+p) model, including both Cr d and I p states, correctly predicts ferromagnetic ordering, with exchange couplings consistent with previous LSDA+U calculations. The CRPA screening significantly reduces the Coulomb interactions, with an orbital-dependent effect. Environmental screening, modeled through the WFCE approach, leads to non-monotonic changes in the magnetic exchange interactions. Increasing dielectric screening reduces the nearest-neighbor exchange interaction while slightly enhancing the next-nearest-neighbor interaction. This non-monotonic interplay between nearest and next-nearest neighbor interactions leads to non-trivial effects on the magnon dispersion and Curie temperature. The magnon dispersion shows a non-monotonic dependence on screening, with the energy at the K point initially increasing before decreasing at higher screening levels. Similarly, the Curie temperature initially increases with screening, reaches a maximum, and then decreases, mirroring the behavior of the optical magnon branch at the M point. The study concludes that a reliable description of CrI₃ magnetism requires considering the full multi-orbital structure of Cr atoms and the ligand p orbitals. Environmental screening exerts a significant influence on the magnetic properties, particularly through its effects on Coulomb interactions.

This study provides a microscopic understanding of magnetism in CrI₃ monolayer, showing the necessity of including ligand p orbitals in the super-exchange mechanism. The non-monotonic behavior of the Curie temperature and magnon dispersion as a function of environmental screening highlights the complex interplay between Coulomb interactions and magnetic exchange. These findings are relevant to experimental setups involving supported or encapsulated CrI₃ films. The results suggest that magnetism in multilayer CrI₃ may involve layer-dependent changes in Coulomb interactions, and anisotropic magnetic interactions warrant further investigation. The tunability of magnetic properties through environmental screening opens avenues for designing CrI₃-based heterostructures with spatially structured environments, enabling non-invasive control of magnetic properties.

This work demonstrates that a multi-orbital super-exchange mechanism involving both Cr d and I p orbitals is crucial for accurately describing magnetism in CrI₃ monolayer. Environmental screening significantly modifies the magnetic properties through its impact on Coulomb interactions, resulting in non-monotonic changes in exchange interactions, magnon dispersion, and Curie temperature. This understanding is crucial for designing and engineering CrI₃-based spintronic devices. Future research could focus on exploring dynamical correlation effects using DMFT, investigating magnon-phonon couplings, and further examining the effects of anisotropic magnetic interactions and Dzyaloshinskii-Moriya interactions.

The study uses a mean-field Hartree-Fock approximation to treat Coulomb interactions, which might not fully capture all correlation effects. The extended (d+p) model, while significantly improving the accuracy, is still a minimal model and may not completely capture all the complexities of the real material. The WFCE approach for modeling environmental screening relies on certain approximations, such as the assumption of a constant eigenbasis for the Coulomb tensor.