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

The RuX₃ (X = Cl, Br, I) family of layered Ru-based trihalides has garnered significant attention due to their diverse electronic and magnetic properties. α-RuCl₃, the first member, is a well-studied spin-orbit-assisted Mott insulator and a potential candidate for the Kitaev honeycomb model, exhibiting zigzag antiferromagnetic order below its Néel temperature (TN ≈ 7 K). The synthesis of RuBr₃, a sister compound with a heavier halogen, revealed similar insulating behavior and zigzag magnetic order but with a higher TN (34 K), suggesting closer proximity to the pure Kitaev model. The recent synthesis of RuI₃, however, has introduced a significant puzzle: it displays quasi-metallic behavior in contrast to its insulating siblings. The reported resistivity values, while high for a metal, are practically temperature-independent, raising questions about the nature of electron transport in this compound. Magnetic susceptibility measurements also show variability across different RuI₃ samples, indicating the potential impact of sample quality on the observed properties. To understand these discrepancies and the overall behavior of the RuX₃ family, this study uses first-principles calculations and effective model considerations to analyze existing experimental data and gain a deeper insight into the electronic and magnetic properties of RuCl₃, RuBr₃, and RuI₃.

The literature review focuses on previous studies of α-RuCl₃, RuBr₃, and the newly synthesized RuI₃. α-RuCl₃ has been extensively investigated as a potential Kitaev material, with numerous studies exploring its spin-orbit coupling, anisotropic exchange interactions, and low-temperature magnetic ordering. These studies have employed a variety of experimental techniques, including neutron scattering, angle-resolved photoemission spectroscopy, and thermodynamic measurements. The recent synthesis and characterization of RuBr₃ provided a comparative point, highlighting similarities and differences with α-RuCl₃, especially in the magnetic ordering and the Weiss constant. The contrasting metallic behavior observed in RuI₃, reported by two independent groups, prompted further investigation into the role of defects and sample quality in these compounds. This contrasts with the insulating behavior of RuCl₃ and RuBr₃, making it a key area for investigation.

The authors employed a multi-pronged approach combining experimental data analysis with theoretical calculations. For the experimental analysis, they collated and analyzed reported electrical resistivity, specific heat, and magnetic susceptibility data for all three RuX₃ compounds. This involved extracting data from published plots and analyzing the temperature dependence of these properties. In particular, the authors focused on understanding the unusual resistivity in RuI₃ and the implications of varying sample quality. The theoretical part relied heavily on density functional theory (DFT) calculations, using two different methods (VASP and Wien2k) with the generalized gradient approximation (GGA) for exchange-correlation, Hubbard U correction, and spin-orbit coupling (SOC). This approach aimed to determine the electronic structure and the energy of different magnetic configurations for each compound. The constrained random-phase approximation (CRPA) was used to estimate effective Coulomb interaction parameters. The authors utilized the projED method to derive low-energy pseudospin models from ab-initio hopping parameters, providing information on the magnetic exchange couplings between Ru ions. Finally, they used exact diagonalization of these models on a 24-site cluster to determine ground state properties and magnetic order. A modified Curie-Weiss fit was also employed to analyze the magnetic susceptibility data, accounting for the temperature-dependent van-Vleck contributions arising from spin-orbit coupling.

The key findings of this study include: 1. All three RuX₃ compounds (RuCl₃, RuBr₃, and RuI₃) are predicted to be spin-orbit-assisted Mott insulators. However, the band gap decreases significantly from RuCl₃ to RuI₃, with RuI₃ being close to a metal-insulator transition. 2. DFT total-energy calculations reveal that zigzag magnetic order is highly stable in RuBr₃, while RuI₃ exhibits substantial magnetic frustration. Ab-initio derived models suggest RuI₃ may be an incommensurate magnetically ordered state or a quantum spin liquid, possibly different from the Z₂ Kitaev spin liquid. 3. Experimental observations on RuI₃, particularly the resistivity, are likely significantly affected by sample quality, possibly through the presence of metallic grain boundaries or disorder. The authors suggest that a more accurate description of RuI₃ might be a ‘dirty’ insulator or a bad metal. 4. Across all three compounds, the dominant nearest-neighbor interaction is ferromagnetic Kitaev (K₁), with a subdominant ferromagnetic Heisenberg (J₁) interaction which almost vanishes in RuI₃. The behavior of magnetic anisotropic terms with the halogen ligand atomic number shows a non-monotonic relationship attributed to the competition between SOC effects from Ru and ligands. 5. RuBr₃ exhibits predominantly ferromagnetic interactions, contradicting initial Curie-Weiss analysis, but consistent with susceptibility data when accounting for high-temperature SOC effects. The ab-initio magnetic model correctly predicts zigzag ordering in RuBr₃, although the predicted tilting angle (32°) slightly differs from the experimental value (64°). The large experimental angle in RuBr3 is unlikely due to proximity to the pure Kitaev model; rather it may require significant negative Γ1 < 0 and strong structural distortions.

The findings address the initial research question of understanding the contrasting electronic and magnetic behavior across the RuX₃ family. The theoretical predictions of Mott insulating behavior for all three compounds, despite varying band gaps, provide a unified framework for the family. The explanation of RuI₃'s apparent metallicity as a consequence of sample quality is crucial, highlighting the challenges in material synthesis and characterization. The derived magnetic Hamiltonians offer valuable insights into the microscopic interactions responsible for the magnetic properties, revealing the complex interplay of Kitaev and Heisenberg exchange interactions. The non-monotonic trend observed in magnetic anisotropic terms is discussed in terms of the competing spin-orbit coupling effects from Ru and the ligands, highlighting the importance of going beyond simplified models. While the agreement with experimental observations is largely positive, there remain discrepancies such as the tilting angle of RuBr₃, prompting further investigation into the possible role of structural distortions or more complex interactions.

The paper successfully demonstrates that RuCl₃, RuBr₃, and RuI₃ form a family of spin-orbit-assisted Mott insulators. The varying band gaps, magnetic ordering, and the apparent metallicity of RuI₃ are explained by considering the effects of ligand atomic number, magnetic frustration, and sample quality. This comprehensive study provides a clear picture of the electronic and magnetic properties of these materials, suggesting future research directions could focus on improving sample quality and verifying the predicted quantum spin liquid behavior in RuI₃. Further refinement of the magnetic models incorporating additional complex interactions may be needed to achieve better agreement with experimental observations in specific compounds.

The study primarily relies on theoretical calculations, and the interpretations of experimental data are influenced by the assumption of sample quality affecting the measured properties. The exact diagonalization calculations are limited by the finite size of the cluster, potentially impacting the identification of magnetic ordering, especially for complex phases like incommensurate ordering. Furthermore, the discrepancies between theoretical and experimental tilting angles in RuBr₃ warrant further investigation.