Electronic and magnetic properties of the RuX₃ (X = Cl, Br, I) family: two siblings—and a cousin?

The RuX₃ (X = Cl, Br, I) family are layered Ru-based trihalides with honeycomb layers, attracting attention due to potential realization of Kitaev physics. α-RuCl₃ is a spin–orbit-assisted Mott insulator whose low-energy magnetism can be described by Jeff = 1/2 moments with strongly anisotropic exchange; it orders into a zigzag antiferromagnetic state at TN ≈ 7 K, yet exhibits signatures associated with Kitaev physics at finite temperature and fields. RuBr₃, recently synthesized, is insulating with zigzag order at TN = 34 K and exhibits differences in Weiss constant and ordered moment direction relative to RuCl₃, suggesting varying proximity to the Kitaev limit. RuI₃, synthesized by two groups, shows quasi-metallic behavior and lack of clear magnetic order, challenging a localized Jeff = 1/2 description. The research question is whether RuI₃ is intrinsically metallic or, like its siblings, a spin–orbit-assisted Mott insulator, and how electronic correlations and spin–orbit coupling across the series control magnetic interactions and ground states. The study aims to reconcile experimental transport, thermodynamics, and susceptibility with ab initio electronic structure and derived low-energy spin models, and to clarify the role of disorder and ligand SOC.

Extensive prior work established α-RuCl₃ as a proximate Kitaev material with anisotropic exchanges and zigzag order, supported by neutron scattering, thermodynamics, and transport studies. Standard Curie–Weiss analyses can be misleading in strong SOC systems due to van Vleck contributions; modified analyses reveal predominantly ferromagnetic interactions in RuCl₃. RuBr₃ was reported to be an insulator with zigzag order and higher TN than RuCl₃; Curie–Weiss fits suggested AFM interactions, but SOC-aware analyses can reverse this. RuI₃ measurements reported large, nearly temperature-independent resistivity and varying susceptibility behavior (temperature-independent or Curie-like upturn), indicating possible sample-quality dependence. Theoretical studies highlighted the fragility of magnetic orders in Ru trihalides, sensitivity to structural details, correlation strength, and SOC, and emphasized the need to include ligand SOC beyond the magnetic-ion atomic SOC picture.

- Experimental data analysis: Compiled and compared reported resistivity, specific heat, and magnetic susceptibility for RuCl₃, RuBr₃, and RuI₃ from the literature. - Modified Curie–Weiss fitting: Employed a SOC-aware Curie–Weiss formalism incorporating temperature-dependent effective moment μeff(T, Δ) capturing van Vleck-like contributions due to crystal-field splitting Δ and SOC λ. Fitted powder-averaged susceptibility over 150–300 K with parameters χ0, χ∞, θ||, θ⊥, fixing Δ from first-principles/quantum chemistry constraints (g-tensor anisotropy). Example: Δ = 0.018 eV, λ = 0.15 eV. - DFT calculations: Used GGA and GGA+U with SOC in VASP (PAW method; Ru_pv pseudopotential with Ru 4p as valence; Γ-centered 8×8×8 k-mesh; 350 eV cutoff; 1×10⁻⁸ eV convergence) and Wien2k (LAPW; RKmax = 8; k-meshes 8×8×2 or 8×4×6; DOS with 12×12×3 or 12×6×9 meshes). Considered experimental C2/m and R3 structures (RuCl₃) and R3 for RuBr₃ and RuI₃. Surveyed magnetic configurations (FM, zigzag, Néel, etc.). Hubbard Ueff selected based on CRPA trends across the series. - CRPA: Computed orbitally averaged on-site Hubbard Uavg, Hund’s Javg, and intersite Vavg using FHI-gap on top of Wien2k electronic structure, projecting to five Ru 4d orbitals and excluding screening within the target window. Ensured k-point and energy cutoff convergence. - Derivation of magnetic models (projED): Built effective electronic models using full-relativistic FPLO-derived complex hopping (projective Wannier) and CRPA interactions, then obtained bilinear pseudospin Jeff = 1/2 Hamiltonians Heff via exact diagonalization and projection onto the low-energy subspace. Included ligand SOC through complex hoppings; considered all five 4d orbitals (with two-site clusters). - g-tensor calculations: Performed quantum chemistry (ORCA 3.0.3, TPSSh functional, def2-TZVP basis, CAS(5,5)) on [RuX₃] molecules to obtain anisotropic g-tensors (g||, g⊥). - Exact diagonalization (ED): Solved Jeff = 1/2 models on a 24-site periodic honeycomb cluster. Identified magnetic order from the static spin structure factor S(k). Determined ordered moment directions from the leading eigenvector of the correlation matrix at ordering vector Q and mapped to magnetic moments via the g-tensor. Also evaluated Kitaev plaquette operator expectation values to assess proximity to Kitaev spin liquid behavior. - Parameter choices: Ueff guided by CRPA: RuCl₃ 2.7 eV, RuBr₃ 2.1 eV, RuI₃ 1.4 eV (with tests down to 1.0 eV for RuI₃); SOC λ = 0.15 eV; Δ ≈ 0.018 eV for susceptibility fits.

- Transport and thermodynamics: - RuI₃ shows large resistivity with weak or no temperature dependence (ρ ≈ 40 mΩ·cm and ≈ 4 mΩ·cm in different reports), exceeding typical bad-metal regimes and the Ioffe–Regel limit, inconsistent with a clean metal. - Specific heat in RuI₃ shows a T-linear coefficient γ ≈ 15–30 mJ K⁻² mol⁻¹; band-structure DOS implies γ₀ ≈ 3 mJ K⁻² mol⁻¹, suggesting large mass renormalization if intrinsic; β (T³ term) scales reasonably with mass across RuX₃. - Susceptibility: Modified Curie–Weiss analysis incorporating SOC-induced μeff(T) yields positive Weiss constants for RuBr₃ (Θ|| ≈ 5 K, Θ⊥ ≈ 17 K, Θavg ≈ 9 K) and for RuCl₃ (Θ|| ≈ +55 K, Θ⊥ ≈ +33 K, Θavg ≈ +48 K), indicating predominantly ferromagnetic interactions, in contrast to standard CW fits. - Electronic structure (GGA+SOC+U): - Ueff trend (CRPA-guided): RuCl₃ 2.7 eV, RuBr₃ 2.1 eV, RuI₃ 1.4 eV; effective interactions decrease from Cl→Br→I. - Band gaps: RuCl₃ ≈ 1.0 eV (fundamental and direct, matching optical data), RuBr₃ ≈ 0.56 eV, RuI₃ ≈ 0.1 eV; RuI₃ gap closes near Ueff ≲ 1.0 eV, placing pristine RuI₃ near a Mott transition. - Total-energy differences: RuCl₃ EZZ − EFM ≈ 2 meV/Ru (FM metastable), RuBr₃ favors zigzag, RuI₃ shows near-degeneracy (ENéel − EZZ ≈ 1 meV/Ru) and small, varying Ru moments, indicating frustration. - Magnetic interactions (projED): - Across RuX₃, dominant nearest-neighbor ferromagnetic Kitaev K₁; subdominant ferromagnetic Heisenberg J₁ that nearly vanishes in RuI₃; Γ₁ comparable in magnitude to J₁ and changes sign for I; Γ₁′ becomes important, especially in RuI₃. - Further-neighbor exchanges (2NN, 3NN) generally increase in importance from Cl→I due to enhanced p–d hybridization and larger further-neighbor hoppings; nearest-neighbor hoppings decrease with heavier ligands. - SOC from ligands competes with magnetic-ion SOC, breaking simple atomic SOC-limit expectations and leading to non-monotonic anisotropic exchanges. - g-tensor: g|| > g⊥ across the family, implying stronger Zeeman coupling for in-plane fields. - Ground states (ED on 24-site cluster): - RuCl₃ and RuBr₃: zigzag AFM order, with computed ordered-moment tilt angles θM ≈ 34° (RuCl₃) and ≈ 32° (RuBr₃). RuCl₃ matches experiment (θM = 32 ± 3°). RuBr₃’s reported θM ≈ 64° cannot be obtained by simply increasing K₁ proximity; even near the Kitaev QSL transition θs < 46°, and θM is smaller due to g anisotropy. Achieving 64° would require sizable negative Γ₁, incompatible with ab initio results. - RuI₃: spin structure factor lacks a dominant ordering vector; plaquette operator ⟨Wp⟩ ≈ 0.29 (vs ≤0.04 in classical states, 1 in pure Kitaev), suggesting a magnetically disordered state potentially distinct from the Z₂ Kitaev spin liquid, stabilized by further-neighbor interactions. - Predicted Weiss temperatures from models (powder-averaged): RuCl₃ +39.1 K, RuBr₃ +35.6 K, RuI₃ +15.4 K. - Interpretation of RuI₃ experiments: Many observations reconcile with insulating grains surrounded by metallic grain boundaries (and/or disorder pushing towards the Mott transition), explaining large, weakly T-dependent resistivity and susceptibility backgrounds.

The comparative ab initio and model analysis shows that all three RuX₃ compounds are best understood as spin–orbit-assisted Mott insulators, with quantitative evolution of correlation strength and exchange couplings from Cl to I. The reported quasi-metallic behavior of RuI₃ conflicts with its calculated proximity to, yet still on the insulating side of, a Mott transition; a consistent resolution is that sample disorder and metallic grain boundaries dominate transport and susceptibility backgrounds. The derived magnetic Hamiltonians clarify that ferromagnetic Kitaev interactions dominate across the family, while the conventional Heisenberg term is small, nearly vanishing in RuI₃, rendering interactions highly anisotropic and increasing the role of further neighbors. This explains zigzag order in RuCl₃ and RuBr₃ and suggests strong frustration and a possible quantum spin liquid (non-Kitaev type) or incommensurate order in RuI₃. Incorporating ligand SOC is essential to capture the non-monotonic anisotropy and to avoid misleading conclusions from standard Curie–Weiss fits; the modified fits reconcile RuBr₃ data with predominantly ferromagnetic interactions. The discrepancy between predicted and reported RuBr₃ moment tilt angles indicates either experimental complexities (e.g., structural distortions) or the need for conditions not captured by the experimental structures used, reinforcing the sensitivity of these systems to details.

- All three ideal RuCl₃, RuBr₃, and RuI₃ compounds are spin–orbit-assisted Mott insulators; the gap decreases from Cl to I, with RuI₃ near the metal–insulator transition. - Pristine RuBr₃ favors zigzag order even more than RuCl₃, while RuI₃ exhibits strong magnetic frustration; the ab initio models predict RuI₃ to host either incommensurate order or a quantum spin liquid distinct from the Z₂ Kitaev state. - Many experimental peculiarities of RuI₃ (large, weakly T-dependent resistivity; susceptibility backgrounds) can be explained by sample-quality issues such as metallic grain boundaries and disorder promoting metallic pathways. - Dominant nearest-neighbor ferromagnetic Kitaev K₁ exists in all three materials; the subdominant ferromagnetic J₁ nearly vanishes in RuI₃, enhancing anisotropy. Ligand and metal SOC compete, yielding non-monotonic anisotropic exchanges across the series. - RuBr₃’s interactions are predominantly ferromagnetic when SOC-induced van Vleck contributions are accounted for in Curie–Weiss analysis. Theoretical models reproduce zigzag order and moment tilt similar to RuCl₃ but cannot explain the reported ≈64° tilt without substantial, ab initio-incompatible parameter changes (e.g., large negative Γ₁). Future directions include synthesizing higher-quality RuI₃ crystals to isolate intrinsic properties; detailed structural studies of RuBr₃ to assess distortions affecting exchange anisotropy; larger-scale or alternative numerical methods to reduce finite-size effects in RuI₃; and extended theories including explicit ligand states to refine exchange derivations in systems with strong ligand SOC.

- Magnetic orders and metallicity in Ru trihalides are fragile and sensitive to structural details, correlation parameters, and computational choices; small variations can change energy hierarchies. - Curie–Weiss analyses depend on assumed crystal-field splitting Δ and g-tensor anisotropy; only powder-averaged data were available for RuBr₃, limiting precise parameter extraction. - The derivation of exchange interactions uses two-site clusters and projected models; while comprehensive, it may miss longer-range multi-spin effects. - Exact diagonalization is limited to a 24-site cluster and can suffer from finite-size effects, potentially obscuring incommensurate orders or subtle QSL signatures in RuI₃. - Reported experimental discrepancies (e.g., RuBr₃ tilt angle) suggest possible unaccounted structural distortions or sample-dependent effects not captured by the used experimental structures. - Interpretation of RuI₃ transport and susceptibility relies on sample-quality hypotheses (e.g., grain boundaries), which need direct microstructural confirmation.