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

Ferromagnetic layered van der Waals materials are promising for spintronic nanodevices. Recent experiments (IETS and magneto-Raman) have observed magnons in thin chromium trihalides, enabling exploration of magnon-based spintronics in 2D. How magnetic properties are modified in heterostructures via proximity, gating, or screening remains under debate. Theoretical and experimental works indicate strong dependence of magneto-optical response on film thickness and spin-orbit coupling in CrI₃, and tunability of monolayer/bilayer CrI₃ magnetism via electrostatic gating. For CrX₃ (X = I, Br, Cl), nearest-neighbor exchange couplings have been reported between 1–3.2 meV depending on method, underscoring the need for rigorous theory. Reduced environmental polarization in 2D enhances Coulomb interactions, and environmental screening can significantly modify them, enabling Coulomb engineering of many-body properties. It is unknown how efficiently environmental screening can tailor magnetic ground states or magnons in layered magnets. For CrI₃, effective spin Hamiltonians describe exchange between Cr spins; thus, screening-induced changes to exchange couplings can alter T_c and magnon spectra. Conventional DFT poorly captures environmental screening effects on Coulomb interactions; higher-level approaches (e.g., GW/RPA) are needed, but full ab initio GW for heterostructures is computationally prohibitive. The authors therefore pursue a multi-step approach: DFT band structures, down-folded minimal Wannier models (d-only and d+p), CRPA Coulomb tensors, mean-field Hartree-Fock solutions, and magnet force theorem to extract orbitally resolved exchanges, extended with a continuum electrostatics framework (WFCE) to include environmental screening.

• Experimental context: Magnons have been observed in thin chromium trihalides via IETS and magneto-Raman spectroscopy, opening avenues for magnonic devices. Electrostatic gating can tune magnetism in monolayer and bilayer CrI₃.
• Prior theory/values: Reported nearest-neighbor exchange couplings for CrX₃ span 1–3.2 meV depending on material and computational approach. Magneto-optical responses depend on film thickness and spin–orbit coupling.
• Methodological gaps: DFT struggles to capture environmental screening effects on Coulomb interactions; GW/RPA-level treatments are needed, but full heterostructure GW is challenging due to large supercells and lattice mismatch.
• Coulomb engineering: Environmental screening can alter Coulomb interactions and thus excitons, plasmons, charge order, superconductivity; its role in tailoring magnetic exchanges and magnon spectra in CrI₃ was unclear prior to this work.

• Electronic structure and model construction: Start from spin-unpolarized DFT (GGA) band structures of monolayer CrI₃. Construct localized Wannier models for two target spaces: (i) a minimal Cr d-only basis and (ii) an extended (d+p) basis including I p states. The d-only model uses Cr d Wannier orbitals; the (d+p)-model includes ligand p contributions in the kinetic part.
• Coulomb interactions via CRPA: Compute partially screened Coulomb interaction tensors using constrained RPA (CRPA), excluding the correlated subspace (Cr d states) from the polarization. Use a VASP CRPA implementation; 128 bands with rest polarization excluding Kohn-Sham states between −0.5 and 3 eV to avoid metallic screening of half-occupied t₂g band. Extract full rank-4 Coulomb tensor; retain static local density-density (U) and local Hund’s exchange (J_H) terms on Cr sites. Nonlocal density-density and nonlocal Hund’s terms are neglected based on prior evidence of minor impact on magnon dispersions. Static U(ω=0) is used, neglecting small bandwidth renormalizations.
• Hartree–Fock solver and double counting: Solve the multi-orbital Hubbard Hamiltonian at mean-field level (HF) allowing spin symmetry breaking. Use an effective single-particle HF Hamiltonian with spin-conserving and spin-mixing terms self-consistent in local occupations. In the (d+p)-model, apply a fully localized limit (FLL) double-counting correction on Cr d states using nominal Cr³⁺ d-count N_d=3 to align correlated d bands relative to uncorrelated ligand p states.
• Magnetic exchange extraction (MFT): Apply the magnetic force theorem (MFT) to HF solutions to compute orbitally resolved exchange couplings between Cr sites from second variations of the total energy with respect to infinitesimal spin rotations. Positive J denotes FM, negative J AFM. Analyze contributions from t₂g–t₂g (AFM superexchange) and t₂g–e_g (FM via ligand-mediated paths) channels.
• Spin-wave spectrum and T_c: Map to a spin Hamiltonian with isotropic nearest-neighbor (J) and next-nearest-neighbor (J′) exchanges and small anisotropies. Compute magnon dispersion using linearized Holstein–Primakoff transformation on honeycomb lattice. Estimate T_c via Tyablikov’s RPA decoupling, using the magnon spectrum.
• Environmental screening (WFCE): Incorporate dielectric environment using a Wannier-function-based continuum electrostatics (WFCE) approach. Diagonalize the bare Coulomb tensor in momentum space; identify a leading monopole-like screening channel ε₁(q) that is sensitive to external dielectrics, modeled by a dielectric slab Poisson solution. Fit free-standing layer parameters (thickness h ≈ 5.2 Å, intrinsic ε ≈ 8.7 for d+p model), then modify the CRPA tensor for arbitrary top/bottom dielectric constants (ε_above, ε_below). This screening equally reduces U and U′ (monopole density-density terms) but does not affect local Hund’s exchange J_H. Benchmarking against explicit CRPA with hBN layers is provided in Supplementary Methods.

• Coulomb parameters (d-only model): Local CRPA-screened intra-orbital interactions U_t ≈ 3.6 eV (t₂g) and U_e ≈ 3.1 eV (e_g), much larger than the ~1 eV t₂g bandwidth, indicating significant correlations. Screened Hund’s exchange J_t ≈ 0.49 eV and J_e ≈ 0.44 eV (only ~10% reduction from bare). Nearest-neighbor screened density-density interaction U_01 ≈ 1.4 eV; nearest-neighbor Hund’s interactions are ≈1–7 meV.
• d-only model fails for magnetism: MFT on HF solutions yields AFM total exchange for nearest neighbors J ≈ −1.964 meV and next-nearest neighbors J′ ≈ −0.042 meV, driven by dominant AFM t₂g–t₂g channel. Kugel–Khomskii-type estimates with effective hoppings and averaged U, J also predict AFM, confirming the insufficiency of a Cr d-only model to capture FM in CrI₃.
• Extended (d+p) model succeeds: Including ligand p states localizes Cr d Wannier functions, increasing bare and screened local interactions (screened intra-orbital U on order of 4 eV; inter-orbital ~3 eV; screened J_H ~0.5–0.7 eV). HF density of states matches GGA+U characteristics (band ordering, spin splittings, gaps). MFT yields correct FM exchange: nearest-neighbor J ≈ +1.76 meV and next-nearest-neighbor J′ ≈ +0.35 meV, comparable to prior LSDA+U-based MFT.
• Sensitivity to interaction channels: Reducing U or U′ by 20% enhances the AFM t₂g–t₂g exchange (as per KK intuition), while reducing J_H diminishes the FM t₂g–e_g exchange. Some deviations from simple KK expectations for the e_g–e_g channel highlight the need for full multi-orbital MFT analysis.
• Environmental screening trends (WFCE+MFT, d+p): External dielectric screening reduces U and U′ equally while leaving J_H unchanged. With increasing dielectric constant ε (free-standing to encapsulated), the total nearest-neighbor J decreases, predominantly due to strengthening of AFM t₂g–t₂g, whereas J′ increases slightly as the FM channel grows somewhat faster. Representative values from Table 4: at ε=1, J_t₂g–t₂g=−0.864 meV, J_total≈+1.756 meV, J′≈+0.352 meV; at ε=8, J_total≈+1.426 meV, J′≈+0.646 meV; at ε=20, J_total≈+0.774 meV, J′≈+0.742 meV.
• Magnon dispersion and Curie temperature: Environmental screening induces non-trivial, non-monotonic changes in magnon energies across the Brillouin zone. The K-point energy E(K)=3S(J+3J′+A) is non-monotonic versus ε because J+3J′ peaks near ε≈6. The optical branch near Γ is continuously lowered with increased screening; van Hove features near M show complex behavior, including flattened dispersions at intermediate ε. The Curie temperature T_c shows a non-monotonic dependence on ε, initially increasing (driven by rising J′), peaking around ε≈6, and decreasing at larger ε as J diminishes.

The work addresses how the microscopic origin of ferromagnetism in monolayer CrI₃ emerges and how it can be tuned by the dielectric environment. A d-only description fails because AFM t₂g–t₂g superexchange dominates, contradicting experiments. Incorporating ligand p orbitals reveals a multi-orbital superexchange mechanism in which the FM t₂g–e_g pathway, mediated by iodine p states, overcomes the AFM channel, yielding the correct FM ground state and realistic exchange magnitudes. Environmental (monopole-like) screening reduces density-density Coulomb terms (U, U′) equally but leaves Hund’s exchange unaffected, reshaping spin splittings and inter-manifold gaps. This selectively enhances AFM t₂g–t₂g relative to FM t₂g–e_g, causing the nearest-neighbor FM exchange J to decrease with ε while the next-nearest-neighbor J′ slightly increases. The competing, ε-dependent trends in J and J′ produce non-monotonic modifications to magnon dispersions (notably at K and M) and to T_c, with a maximum at intermediate ε. These findings demonstrate explicit pathways for Coulomb engineering of magnetic interactions, magnons, and thermal stability of magnetism in van der Waals magnets, guiding substrate/encapsulation choices to tailor spintronic and magnonic functionalities.

This study develops ab initio down-folded multi-orbital Hubbard models with CRPA interactions and HF+MFT analysis to reveal the multi-orbital, ligand-mediated superexchange origin of ferromagnetism in monolayer CrI₃. A minimal d-only model predicts AFM and is inadequate; an extended (d+p) model reproduces realistic band structures, FM nearest-neighbor exchange (~1.76 meV), and next-nearest-neighbor exchange (~0.35 meV). Incorporating environmental screening via a WFCE framework shows that monopole-like screening reduces U and U′, leaving J_H intact, which drives a decrease in J and a mild increase in J′. This leads to non-monotonic magnon spectra and Curie temperatures versus dielectric environment, with optimal T_c near ε≈6. Future directions include: (i) incorporating dynamical correlations (e.g., DMFT) to capture strong-correlation effects beyond HF, (ii) exploring anisotropic exchange and Dzyaloshinskii–Moriya interactions and their dielectric tunability, (iii) extending to multilayer systems where interlayer hybridization and layer-dependent screening are relevant, and (iv) leveraging spatially structured environments for real-space engineering of magnetic exchange and magnon dispersions in device architectures.

• Mean-field approximation: HF treatment neglects dynamical correlations; given U comparable to or larger than bandwidths, DMFT or beyond may be necessary to capture all correlation effects.
• Interaction truncation: Only local (onsite) density-density and local Hund’s exchange are retained; nonlocal interactions and frequency dependence U(ω) are neglected (justified partially by large CRPA plasmon frequency but still an approximation).
• Basis and double counting: Results depend on Wannier basis construction and double-counting correction (FLL). Some discrepancies in valence-band orbital composition compared to GGA+U are noted.
• Spin–orbit and anisotropies: Anisotropic exchanges and SOC effects are treated minimally for spin-wave calculations; detailed anisotropic interactions (including DMI) are not fully explored here.
• Environmental modeling: WFCE assumes dominant monopole-like screening channel and fixed eigenbasis of the Coulomb tensor; while benchmarked, it remains an approximate continuum description.
• Numerical specifics: Full charge self-consistency within correlated frameworks and potential substrate-induced symmetry breaking or strain effects are not comprehensively treated.