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

Abstract

Motivated by reports of metallic behavior in the recently synthesized RuI₃, in contrast to the Mott-insulating nature of the actively discussed α-RuCl₃, as well as RuBr₃, we present a detailed comparative analysis of the electronic and magnetic properties of this family of trihalides. Using a combination of first-principles calculations and effective-model considerations, we conclude that RuI₃, similarly to the other two members, is most probably on the verge of a Mott insulator, but with much smaller magnetic moments and strong magnetic frustration. We predict the ideal pristine crystal of RuI₃ to have a nearly vanishing conventional nearest-neighbor Heisenberg interaction and to be a quantum spin liquid candidate of a possibly different kind than the Kitaev spin liquid. In order to understand the apparent contradiction to the reported resistivity ρ, we analyze the experimental evidence for all three compounds and propose a scenario for the observed metallicity in existing samples of RuI₃. Furthermore, for the Mott insulator RuBr_3, we obtain a magnetic Hamiltonian of a similar form to that in the much-discussed α-RuCl₃ and show that this Hamiltonian is in agreement with experimental evidence in RuBr₃.

Introduction

RuI₃ and RuBr₃ are recent additions to the RuX₃ family (X = Cl, Br, I) of layered Ru-based trihalides (Fig. 1a). The first member, α-RuCl₃ (in the following ‘RuCl₃’) has attracted considerable attention in recent years as a candidate material for the Kitaev honeycomb model¹. RuCl₃ is a spin–orbit-assisted Mott insulator^2,3,4,5 whose magnetic low-energy degrees of freedom can be described in terms of j_eff = 1/2 moments that interact through strongly anisotropic exchange^2,6,7,8. While the material enters a so-called zigzag antiferromagnetic order (Fig. 1b) at low temperatures T_N ≈ 7 K^4,9,10, various experiments at finite temperature^11,12,13,14 or at finite magnetic field^{4,15,16,17,18} have been interpreted as hallmarks of Kitaev physics, a subject which is presently under intensive debate^{19,20,21,22,23,24}.

Fig. 1: RuX₃ (X = Cl, Br, I) crystal structure, magnetic structure, and resistivity.

a Honeycomb layer in the RuX₃ (X = Cl, Br, I) trihalides with bond definitions, cubic axes (xyz) and crystallographic axes (abc) in the R$\bar{3}$ structure, b Zigzag magnetic order in a honeycomb layer from two perspectives, with definitions of in-plane-angle ϕ and out-of-plane-angle θ. c Comparison of experimental dc resistivities as a function of temperature. Data was extracted from plots in the following references and labeled by respective first-author names: RuCl₃¹⁰, RuBr₃²⁵, and RuI₃^26,27. The shaded background depicts a typical range of resistivity for bad metals²⁸.

Recently, a sister compound with a heavier halogen, X = Br, was synthesized²⁵. Analogous to RuCl₃, it is insulating and shows zigzag magnetic order, albeit with higher Néel temperature T_N = 34 K²⁵. In contrast to RuCl₃, the authors of ref. ²⁵ reported a Weiss constant with dominant antiferromagnetic interactions and a direction of the zigzag-ordered moment different from RuCl₃ and argued that this deviation suggests closer proximity to the pure Kitaev model.

To complete the RuX₃ family, two independent groups have now synthesized RuI₃ with the even heavier halogen iodine^26,27. In contrast to the two ‘sibling’ compounds, a quasi-metallic behavior was observed in RuI₃, questioning the description in terms of localized j_eff = 1/2 moments. Even though the dc resistivities measured in RuI₃ are orders of magnitude smaller than those of RuCl₃ or RuBr₃, the reported values of 10⁻³–10⁻²Ω cm²⁶ are uncharacteristically large for metals or even typical bad metals²⁸, and practically temperature-independent. While neither of the groups found clear signatures of magnetic ordering^26,27, they reported different behaviors of the magnetic susceptibility, which is either found to be temperature-independent²⁶ or with a strong upturn at low temperatures²⁷, suggesting that sample quality plays a crucial role.

In order to understand the apparently distinct behavior of this family of trihalide materials, in this work, we analyze the available experimental data and perform a detailed comparative study of the electronic and magnetic properties of the systems via first-principles calculations and extracted low-energy models. We find that: (i) The behavior of RuI₃ is not that far from RuCl₃ and RuBr₃ and the variations across the series are more quantitative than qualitative. (ii) Pristine samples of RuI₃ should be insulating with highly anisotropic magnetic exchange and nearly vanishing conventional Heisenberg interaction. We argue that the reported metallic behavior in RuI₃ could have its origin in sample quality. (iii) The magnetism in the Mott insulator RuBr₃ has predominantly ferromagnetic interactions, in contrast to what is suggested by the Curie–Weiss analysis of ref. ²⁵. We show that such interactions are consistent with experiment when taking into account spin-orbit coupling effects in the Curie–Weiss behavior.

Our study derives model parameters and magnetic Hamiltonians for the whole RuX₃ family from ab-initio, which will be useful for future theoretical studies of these systems. In contrast to the usual model derivations that only include local spin–orbit coupling (SOC) on the magnetic ion^2,6,8, our approach includes all SOC effects in the crystal. In fact, we show that SOC from the ligands leads to significant deviations from the Ruthenium-only SOC picture in the case of RuBr₃ and RuI₃.

Results and discussion

Comparative analysis of experiments

In the following, we analyze the reported electrical resistivity, specific heat, and magnetic susceptibility data for RuX₃ (X = Cl, Br, I)^{10,25,26,27,29}.

In Fig. 1c we summarize the temperature dependence of the experimental resistivity data^10,25,26,27 in all three compounds. In RuI₃, the resistivity has a weak²⁷ or almost no²⁶ temperature dependence (Fig. 1c). Traditionally, metals are classified as materials where the resistivity ρ increases with temperature, distinguishing conventional metals (e.g., Cu) as those where in clean samples at temperatures roughly 300–600 K, ρ ~ 10⁻⁶ to ~10⁻⁵Ω cm, and bad metals as those with resistivities of ~1−10 mΩ cm. This range is shown as a background shading in Fig. 1c. The reported resistivities for RuI₃ (ρ ~ 40 mΩ cm²⁷ and ρ ~ 4 mΩ cm²⁶) are high even for bad metals, surpassing the Ioffe–Regel limit by more than an order of magnitude. Even more relevant, the lower-resistivity set of data²⁶ shows no discernible temperature dependence at all, while the data in ref. ²⁷ show a very weak positive derivative dρ/dT, but the absolute value is above anything traditionally considered metallic.

Seemingly, as also pointed out in ref. ²⁷, electron transport in existing RuI₃ samples may be contaminated by grain boundaries. One possibility to interpret the measurements is that the pristine material is metallic, but insulating grain boundaries prevent percolation. Then, the in-grain resistivity can be neglected and what is measured is the resistivity of the insulating grain boundaries. In that case, however, thermal activation of carriers in the boundaries should give a positive temperature gradient of the resistivity, which is not observed. The opposite scenario is that of an insulating behavior in the bulk and (possibly bad) metallic one between the grains. In that case, the large resistivity reflects the small relative volume of metallic boundaries, where the transport is dominated by the residual resistivity. This scenario is compatible with the observations. Morphology of the grain boundaries can vary wildly depending on the growth conditions, including but not limited to vacancies, twins, dislocation, and plain chemical dirt. Grain boundaries in semiconductors are often observed to be metallic. Apart from grain boundaries contaminating resistivity measurements, disorder (in form of vacancies, stacking faults, etc.^26,27) could promote the bulk metallic phase over the Mott-insulating one, as has been shown for example for the Mott insulator κ-(BEDT-TTF)₂Cu[N(CN)₂]Cl³⁰. Indeed, in our first-principles calculations discussed below, we find the ideal RuI₃ to already be quite close to a Mott–metal transition.

Turning to the sibling compounds RuCl₃ and RuBr₃, the resistivity (Fig. 1c) decreases with temperature, as expected for Mott insulators, and both systems show an approximate exponential activation gap behavior, ${E}_{{\rm {g}},{{{\rm{eff}}}}}(T)=-{k}_{{{{\rm{B}}}}}{T}^{2}({{{\rm{d}}}}\ln \rho /{{{\rm{d}}}}T)$, although with a significant blue-shift of the gap with increasing temperatures.

Considering specific heat data in the compounds, the specific heat for RuCl₃ displays a well-defined peak at T_N ≈ 7 K denoting the onset of the zigzag order, while the onset of long-range magnetic order in RuBr₃ is observed by a kink at T_N = 34 K²⁵. None of this is observed for RuI₃^26,27. In Table 1 we summarize specific heat parameters reported experimentally^25,26,27,31, where γ (β) is the T-linear (T³) contribution to C(T).

Table 1 Overview of reported specific heat parameters.

In RuI₃, the T-linear contribution, even though contaminated by an extrinsic raise at small temperatures in ref. ²⁷ attributed to the nuclear quadrupole moment of Ru, yields γ ~ 15−30 mJ K⁻² mol^{−1 26,27}. From our electronic structure calculations of RuI₃ shown below, we find that the unrenormalized metallic (i.e., nonmagnetic, not U-corrected) density of states corresponds to γ₀ ≈ 3 mJ K⁻³ mol⁻¹, suggesting a mass renormalization (if this γ is intrinsic) of a factor of 7–12. In the scenario where the metallic grain boundaries take up a sizeable fraction of the sample volume, this renormalization shall be even stronger, encroaching into the heavy fermions domain. This suggests that the origin of the anomalously large residual heat capacity may not be related to intrinsic metallicity. It is worth noting that the T³ term β of C(T), on the other hand, is rather reasonable for the three systems and scales roughly as the harmonic average ${M}_{{{{{\rm{RuX}}}}}_{3}}$ of the atomic masses (last column in Table 1).

We now turn our attention to magnetic susceptibility measurements. Figure 2a summarizes the powder-averaged measured magnetic susceptibilities χ(T) as reported in refs. ^25,26,27,32. At low temperatures, the RuCl₃ data³² shows a clear signature of a transition to the ordered magnetic phase at 7 K. For RuBr₃, the Néel transition T_N ≈ 34 K is less apparent from the susceptibility, but the maximum in dχ/dT is consistent with the distinct transition seen in NMR relaxation measurements²⁵. The experimental report on powder samples of RuBr₃ utilized a standard Curie–Weiss (CW) fit, yielding an average Curie–Weiss temperature ${{{\Theta }}}_{{{{\rm{std}}}}}^{{{{\rm{avg}}}}}=-58$ K²⁵, indicating predominantly AFM interactions. However, as we have recently shown³³, the Weiss constants obtained with such a standard CW fit may not anymore reflect the intrinsic exchange couplings in the case of significant SOC in the material, as is the case for the Ru-based trihalides. With SOC, temperature-dependent van-Vleck contributions can arise, which can be effectively captured in a temperature-dependent magnetic moment μ_eff(T, Δ)³³, as shown for Δ = 0.018 eV in Fig. 2b, where Δ can be directly associated to the crystal field splitting resulting from the distorted octahedral environment of Ru. In fact, for the sister compound RuCl₃, a standard CW fit would lead to ${{{\Theta }}}_{{{{\rm{std}}}}}^{{{{\rm{avg}}}}}=-20$ K, whereas an improved CW fit taking into account such van-Vleck-like contributions³³ provides CW constants Θ^∥ = + 55 K for the magnetic field in the honeycomb plane and Θ^⊥ = + 33 K for the out-of-plane field, revealing an average CW constant, ${{{\Theta }}}^{{{{\rm{avg}}}}}=\frac{2{{{\Theta }}}^{\parallel }+{{{\Theta }}}^{\perp }}{3}$, of ≈48K. This indicates predominant ferromagnetic (FM) interactions, as they have become established for the magnetic Hamiltonian in RuCl₃^34,35,36,37.

Fig. 2: Magnetic susceptibility and modified Curie–Weiss fit.

a Experimental direction-averaged RuX₃ susceptibility data, extracted from plots in the following references and labeled by respective first-author names: RuCl₃³², RuBr₃²⁵, RuI₃²⁷, RuI₃²⁶. Note that the RuCl₃ curve is scaled by $\frac{1}{2}$. b Calculated temperature-dependent effective moment μ_eff(T) for Δ = 0.018 eV and SOC λ = 0.15 eV. c Modified Curie–Weiss fit of RuBr₃ data, taking into account such μ_eff(T). d Dependence of best-fit Weiss constants on assumed Δ. The vertical dashed line indicates Δ = 0.018 eV.

Considering a similar strategy (see the “Methods” section), we fit the average susceptibility χ^avg of RuBr₃²⁵. However, since the crystal-field parameter Δ primarily controls the in-plane vs. out-of-plane anisotropy, and for RuBr₃ only powder-averaged data are available, we do not aim at extracting Δ by fitting. Instead, we first fix Δ using our first-principles calculations, enforcing ${\mu }_{{{{\rm{eff}}}}}^{\parallel }/{\mu }_{{{{\rm{eff}}}}}^{\perp }(T=0\,{\rm {K}})$ ∝ g_∥/g_⊥, where g_∥/g_⊥ are taken from quantum chemistry calculations (see Fig. 5a, discussed below). This leads to Δ = 0.018 eV. The best CW fit accounting for the implied μ_eff(T, Δ = 0.018 eV) (shown in Fig. 2c) yields Weiss constants Θ^∥ ≈ 5 K, Θ^⊥ ≈ 17 K and Θ^avg ≈ 9 K, which are positive, indicating predominately ferromagnetic interactions for RuBr₃, as seen before in RuCl₃. In Fig. 2d we further analyze how the best-fit Weiss constants evolve for other choices of Δ. Indeed, for a wide range of reasonable Δ around the first-principles value (indicated by the dashed vertical line), the average Weiss constant Θ^avg remains positive.

Importantly, for both materials, a standard residual ‘background’ term has to be included in the fitting, which in our case, depending on the material (RuBr₃ or RuCl₃) ranges from ~−3.5 × 10^−4 emu/mol to 1.5 × 10⁻⁴. This is of the same order of magnitude as the corresponding term in RuI₃ (~3 to 8 × 10⁻⁴ emu/mol)^26,27. Since in the former cases an intrinsic Pauli origin can be excluded, this observation also casts doubts on a metallic interpretation of this term in RuI₃. Actually, the two available susceptibility measurements on RuI₃ display different behaviors, one nearly temperature-independent²⁶, and the other²⁷ showing a Curie-like rise at low temperatures, where a standard CW fit yields μ_eff = 0.53μ_B and ${{{\Theta }}}_{{{{\rm{CW}}}}}^{{{{\rm{avg}}}}}=-3$ K²⁷. These differences are consistent with our hypothesis that the measured samples consist of magnetic insulating grains surrounded by metallic boundaries. Then, the samples with larger resistivity data²⁷ hint at larger insulating grains, hence less metallic boundaries are present, leading to the low-temperature Curie-like upturn in the susceptibility, compared to the samples in ref. ²⁶.

Electronic and magnetic calculations

In the following, we present a comparison of the electronic and magnetic properties of the trihalide RuX₃ family obtained from a combination of density functional theory (DFT) and exact diagonalization of ab-initio-derived low-energy models. Details of the calculations are given in the “Methods" section.

Past experience with first-principles calculations for the Ru-based trihalides^4,7,38,39,40 indicates that the magnetic order and, to a considerably lesser extent, metallicity is very fragile, with several closely competing different magnetic phases. The ground states may vary depending on small changes in the crystal structure, on the way in which strong correlations are accounted for, and even on tiny details of the computational protocol. With this in mind, it is imperative to compare the calculated properties across the series, using the exact same computational setup.

For the electronic structure calculations, we consider the experimentally reported C2/m^4,9 and R$\bar{3}$⁴¹ structures for RuCl₃, and the suggested R$\bar{3}$ structures for RuBr₃²⁵ and RuI₃²⁷. Structural details of the four models are summarized in the Supplementary Information. For RuCl₃, the R$\bar{3}$ results are shown in Supplementary Information due to very similar results to the C2/m ones.

Figure 3 shows the relativistic density of states (DOS) obtained within GGA + SOC + U as implemented in Wien2k, where a zigzag magnetic configuration with magnetic moments polarized perpendicular to the ab plane was considered. For the choice of U_eff = U−J we take as a reference the ab initio estimates for the orbitally-averaged Hubbard on-site (U_avg) and Hund’s coupling (J_avg) as obtained from constrained random-phase approximation (cRPA) calculations (see the “Methods" section for calculation details). In contrast to previous cRPA estimates for RuCl₃⁴², our estimates incorporate all five d orbitals and extend to the complete Ru-based trihalide family. As shown in Fig. 4a, the effective Hubbard interaction parameters decrease with increasing ligand atomic number from Cl to I, which can be attributed to the more delocalized nature of the Ru d orbitals in RuI₃ compared to RuCl₃ when hybridizing with I instead of Cl.

Fig. 3: Density of states for RuX₃ (X = Cl, Br, I).

Density of states (DOS) for the experimental structures of RuCl₃, RuBr_3, and RuI₃, obtained from GGA + SO + U calculations with Wien2k, considering antiferromagnetic zigzag magnetic configurations. For RuCl₃ we employed U_eff = 2.7 eV, for RuBr₃ U_eff = 2.1 eV and for RuI₃ U_eff = 1.4 eV. Shown is also the contribution of Ru and halogen states to the DOS.

Fig. 4: Ab-initio-computed multi-orbital Hubbard model parameters across the RuX₃ family.

a cRPA results for the orbitally averaged on-site Hubbard interaction (U_avg), Hund's coupling (J_avg), and the nearest-neighbor V_avg coupling. b Absolute magnitude of hopping parameters t₁ = t_(yz, yz), t₂ = t_(xz, yz), t₃ = t_(xy, xy), and $\tilde{t}={t}_{(xy,{z}^{2})}$ on nearest-neighbor (NN), second-neighbor (2NN), and third-neighbor (3NN) Z-bonds.

For RuCl₃ a U_eff = 2.7 eV yields both the fundamental and direct gap to be ≈1 eV (Fig. 3a) in agreement with the reported optical gap, apart from the presence of multiplets at 200 meV⁴³. We systematically reduced U_eff to 2.1 eV for RuBr₃ and 1.4 eV for RuI₃ following the trend given by the cRPA results. With these values, RuBr₃ shows a gap of 0.56 eV (Fig. 3b), while RuI₃ shows a small gap of 0.1 eV (Fig. 3c). The gap closes in RuI₃ when U_eff is further reduced to 1 eV. These results indicate a spin–orbit assisted Mott insulating state in disorder-free RuI₃ samples, which is on the verge of a metal–insulator transition. Possibly, as discussed above, a disorder in the experimental samples could act as effective pressure, and bring the samples closer to or over the Mott transition as seen in other Mott insulators³⁰. Note that these results hold regardless of the assumed magnetic pattern in the calculations.

In order to analyze the magnetic structure of the RuX₃ compounds, we first consider spin-polarized total energy calculations with VASP in the GGA + SOC + U approximation (see also the “Methods” section). Detailed results are listed in the Supplementary Information. For RuCl₃, the calculated energy of the ferromagnetic state E_FM is very competitive with the energy of the experimentally observed zigzag ordered state E_ZZ: E_ZZ−E_FM ≈ 2 meV/Ru. This observation is consistent with the evidence for a metastable ferromagnetic state in RuCl₃^37,44. Correspondingly, in our effective pseudospin model of RuCl₃ discussed below, classically, the energy of the ferromagnet is below that of zigzag, and only by including quantum fluctuations the zigzag ground state is recovered (as in, e.g., ref. ³⁷). For RuBr₃ we find an energy minimum for the zigzag ordering in agreement with the experiment. Interestingly, for RuI₃ Néel and zigzag orders are energetically almost degenerate E_Néel−E_ZZ ≈ 1 meV/Ru, with the rest of magnetic orders we scanned being energetically rather close. All orders show very small and varying magnetic moments for Ru. These results hint at a magnetic frustration.

We proceed with the derivation of magnetic exchange models. In the first place, the magnetic Hamiltonian of RuBr₃ has been suggested to be closer to the pure Kitaev limit than in RuCl₃²⁵, and, secondly, with our proposed scenario of a Mott insulating state for RuI₃, the question of its magnetic properties is open. To investigate these issues from the first principles, we derive via the ab-initio projED method⁴⁵ the pseudospin models ${\mathcal{H}}_{\rm{eff}}={\sum}_{ij}{\bf{S}}_{i}\cdot {\mathbb{J}}_{ij}\cdot {\bf{S}}_{j}$ of the three RuX₃ compounds. Here, S stands for the relativistic pseudospin j_eff = 1/2 moment².

In the conventional parametrization of Kitaev materials, the exchange matrix ${{\mathbb{J}}}_{ij}$ in R$\bar{3}$ symmetry on a nearest-neighbor Z₁-bond (defined in Fig. 1) follows the form

$${{\mathbb{J}}}_{ij}=\left(\begin{array}{lll}{J}_{1}+{\nu }_{1}&{{{\Gamma }}}_{1}&{{{\Gamma }}}_{1}^{\prime}+{\eta }_{1}\\ {{{\Gamma }}}_{1}&{J}_{1}-{\nu }_{1}&{{{\Gamma }}}_{1}^{\prime}-{\eta }_{1}\\ {{{\Gamma }}}_{1}^{\prime}+{\eta }_{1}&{{{\Gamma }}}_{1}^{\prime}-{\eta }_{1}&{J}_{1}+{K}_{1}\end{array}\right),$$

(1)

with the isotropic Heisenberg exchange J₁, the bond-dependent anisotropic Kitaev exchange K₁, the bond-dependent off-diagonal exchange terms Γ₁ and ${{{\Gamma }}}_{1}^{\prime}$ and correction terms η₁ and ν₁. The latter correction terms are found to be small in our calculated Hamiltonians, and are neglected in what follows. The exchange matrices on X- and Y-bonds follow by respective C₃ rotations about the out-of-plane axis ([111] in pseudospin coordinates). Analogously follow the definitions for second and third neighbor exchange terms (or see, e.g., ref. ⁸).

Using U_avg and J_avg from cRPA (Fig. 4a), the complex hopping parameters extracted from full-relativistic DFT (magnitudes shown in Fig. 4b–d) and the projED method, we extracted the exchange constants shown in Fig. 5b–d.

Fig. 5: Ab-initio computed pseudospin models across the RuX₃ family.

a Quantum chemistry results for local gyromagnetic g-tensor components g_∥ (in-plane) and g_⊥ (out-of-plane). b–d projED results for the magnetic exchange couplings on nearest-neighbor (NN), second-neighbor (2NN), and third-neighbor (3NN) bonds. Tabular form of all values is given in the Supplementary Information.

Evaluating the magnetic interactions of the complete RuX₃ family, we find a nearest-neighbor ferromagnetic Kitaev interaction K₁ to be the dominant in all three compounds. Additionally, a subdominant ferromagnetic nearest-neighbor Heisenberg exchange J₁ is present, which is, however, almost vanishing for the iodine case. The symmetric off-diagonal Γ₁ interaction is of similar magnitude as J₁, changing sign going from Cl and Br to I. ${{{\Gamma }}}_{1}^{\prime}$, often neglected in the RuCl₃ analysis, may become rather important, particularly for RuI₃. Further-neighbor interactions are generally smaller than their nearest-neighbor counterparts for all three systems, but increase for larger ligand atomic number and may play, especially in RuI₃, an important role.

That the anisotropic interactions do not monotonically increase with stronger spin-orbit coupling of halogen elements can be related to the SOC source. In the original Jackeli-Khaliullin mechanism², the heavy magnetic ions are solely responsible for SOC effects, which can be well described within the SOC atomic limit. In the case of RuBr₃ and RuI₃, however, ligand SOC starts to play an important role. To evaluate the interplay of these two SOC sources, we extracted ab initio values for the RuX₃ materials and compared them to the SOC atomic limit (see Supplementary Information). We find that in these compounds SOC effects from magnetic ions and ligands do not enhance each other, but do compete. This leads to the observed inhomogeneous behavior of the magnetic anisotropic terms in Fig. 5 as a function of ligand atomic number. Another consequence of this breakdown of the SOC atomic limit is that the established analytic perturbation theory expressions^2,6,8 become unjustified in Kitaev materials where SOC arises from both the metal and the ligand elements. RuBr₃ and RuI₃ are therefore cases where more general approaches, like ours, are indispensable. Another approach would be perturbation theory taking into account ligand orbitals, as recently derived for the S = 3/2 material CrI₃⁴⁶.

Along the halogen series Cl–Br–I we observe a decrease for nearest-neighbor couplings (Fig. 5b) and an overall increase in magnitude for second and third neighbors (Fig. 5c, d). This can be understood by consideration of the ligand–metal (p–d) hybridization. We quantify the hybridization strength by integrating the DFT(GGA) density of states (DOS) with Ru 4d orbital character in the energy window dominated by the ligand p orbitals (between −7 and −1.05 eV). In spite of respective larger Ru–Ru distances, this can be related to the magnetic exchange by consideration of the ab initio hopping parameters between Wannier d orbitals. As also pointed out in ref. ³⁹, in spite of the stronger hybridization the nearest-neighbor hopping parameters are reduced for heavier ligands, illustrated in Fig. 4b. This is reflected in the magnetic exchange parameters (Fig. 5b) in an overall reduced magnitude in the nearest-neighbor parameters. In contrast, the second and third neighbors show a very different dependence on the halogen element. From the dominant further-neighbor hoppings (Fig. 4c, d), the hoppings show an overall tendency to increase, with few exceptions. Certain further-neighbor magnetic exchange parameters, depending on their relation to the individual hopping parameters, become therefore increasingly important for RuBr₃ and especially for the magnetic properties of RuI₃.

Finally, we also computed the gyromagnetic g-tensor for the RuX₃ family from first principles, in order to relate the pseudospin S of the effective Hamiltonian to the magnetic moment ${\bf{M}}={\mu }_{\rm{B}}{\mathbb{G}}\cdot {\bf{S}}$. The g-tensor can be approximately characterized by two components, the value parallel to the honeycomb plane, g_∥, and the one perpendicular to it, g_⊥, which are shown in Fig. 5a. We consistently find g_∥ > g_⊥ for the whole family, promoting a stronger Zeeman term for in-plane fields.

We now discuss the ramifications of the derived magnetic models for the magnetism in these materials. For RuCl₃, we can compare our result to a vast available literature of models that have been shown to reproduce various experimental observations. Indeed, the model presented here in Fig. 5, derived completely from first principles without adjustments or external parameters, is remarkably close to some well-benchmarked recent models^19,37,38, and is therefore expected to also describe the material quite well. As we apply the same ab-initio setup for the new members of the RuX₃ family, we expect our models to be reliable for them too.

The direction-averaged Weiss constant (${{{\Theta }}}_{{{{\rm{CW}}}}}^{{{{\rm{avg}}}}}$ in Table 2) is predicted to be positive across the RuX₃ family, characteristic of ferromagnetic exchange interactions. This is in line with our analysis of the experimental magnetic susceptibilities of RuCl₃ and RuBr₃ above (Fig. 2). While in RuCl₃ and RuBr₃ a large FM contribution to the Weiss constant comes from a significant FM nearest-neighbor Heisenberg interaction J₁, this interaction nearly vanishes for RuI₃ (Fig. 5b), leading to a smaller Weiss constant. Furthermore, the small J₁ in RuI₃ renders the nearest-neighbor interactions to be extremely anisotropic, with a dominant Kitaev interaction K₁. While at first glance this might suggest a spin-liquid ground state in RuI₃, the increased strength of the further-neighbor interactions in RuI₃ (see, e.g., J₂, K₂ in Fig. 5c) also needs to be considered⁴⁷.

Table 2 Properties of derived pseudospin models.

To find the magnetic ground state properties, we perform exact diagonalization (ED) calculations of the derived j_eff = 1/2 models on the 24-site cluster shown in Fig. 6a. In Table 2 and Fig. 6b we summarize the encountered ground states, i.e. zigzag for RuCl₃ and RuBr₃, and possibly a quantum spin liquid (QSL) in RuI₃. This is discussed in detail below.

Fig. 6: Exact diagonalization of RuX₃ pseudospin models.

a Employed periodic cluster. Labeled sites 1, …, 6 define the Kitaev plaquette operator ${W}_{p}={2}^{6}{S}_{1}^{x}{S}_{2}^{y}{S}_{3}^{z}{S}_{4}^{x}{S}_{5}^{y}{S}_{6}^{z}$. b Static spin structure factor in reciprocal space. Inner (outer) hexagon mark the edge of the first (third) Brilluoin Zone. High-symmetry k-points ${{\Gamma }},{{{\rm{M}}}},{{\Gamma }}^{\prime}$ are labeled. Color scale is the same for all three plots. c Out-of-plane angle θ_S (θ_M) of the pseudospin (magnetic moment) within the zigzag phase when tuning from the RuBr₃ model (f = 0) towards the pure Kitaev model (f = 1). Dashed vertical line indicates phase transition to the Kitaev spin liquid, identified by a peak in −∂²E/∂f².

For RuCl₃, the model in Fig. 5 (as well as the R$\bar{3}$ model discussed in the Supplementary Information) yields zigzag AFM order, identifiable by a maximum at k = M in the static spin structure factor, shown in Fig. 6b. The computed ordered magnetic moment direction (see the “Methods” section), parametrized by θ and ϕ in Table 2 (compare Fig. 1b), is found to be tilted by θ_M ≈ 34° out of the plane, in excellent agreement with the recent experiment, where θ_M = 32 ± 3°³⁶ was reported. Interestingly, on the classical level, the ferromagnetic state is lower in energy than the zigzag state, meaning that the latter only becomes the ground state through quantum fluctuations, as discussed also in ref. ³⁷.

We will now focus on the recently synthesized compounds, starting with RuBr₃. The static spin structure factor for the RuBr₃ model of Fig. 5 is shown in Fig. 6b, indicating also a zigzag AFM order (k = M and C₆-rotated vectors), in agreement with experiment²⁵. However, the calculated tilt angle of the magnetic moment, θ_M = 32^∘, is more in line with RuCl₃ than with the reported measured θ_M = 64° of RuBr₃²⁵. The authors of ref. ²⁵ argued that this anomalously large tilt angle indicates an exceptionally strong relative Kitaev coupling, i.e., larger ∣K₁/J₁∣ and ∣K₁/Γ₁∣ compared to RuCl₃. To investigate to what extent closer proximity to the pure Kitaev model could produce such high tilt angles, we take our RuBr₃ Hamiltonian of Fig. 5 as a starting point and tune it towards the pure Kitaev model, where K₁ is the only non-zero coupling. This is done by multiplying every exchange coupling except K₁ by (1−f) and sweeping f from 0 to 1. As shown in Fig. 6c, the moment indeed rotates further away from the honeycomb plane upon moving towards the pure Kitaev model, however even right before the transition to the Kitaev spin liquid (indicated by the vertical dashed line), θ_S does not exceed 46^∘. ${\theta }_{{{{\bf{M}}}}}=\arccos \left(\frac{\cos {\theta }_{{{{\bf{S}}}}}}{\sqrt{{{g}_{\parallel }}^{2}{\cos }^{2}{\theta }_{{{{\bf{S}}}}}+{{g}_{\perp }}^{2}{\sin }^{2}{\theta }_{{{{\bf{S}}}}}}}\right)$, which is to be compared to the neutron diffraction experiment, is even smaller due to the anisotropy g_∥ > g_⊥ in our calculated g-tensor (Fig. 5a). A reconciliation with the reported θ_M = 64^∘ would therefore require quite drastic changes to the g-tensor anisotropy and/or the exchange parameters. While in the whole J₁−K₁−Γ₁ parameter space with Γ₁ > 0, no angles of θ_S beyond ~ 40^∘ are expected in the zigzag phase⁴⁸, significant negative Γ₁ < 0 can in principle lead to θ_S beyond 60°⁴⁹. However such terms seem incompatible with the ab-initio results and would likely need strong distortions from the present considered RuBr₃ crystal structure to be realized.

More distinct from the other two compounds are our results for RuI₃. As discussed above, in our GGA + SOC + U calculations we find a very flat energy landscape of competitive magnetic configurations, indicative of strong magnetic frustration. Fittingly, the ground state from the exact diagonalization of the present exchange model does not show a dominant ordering wave vector in the spin structure factor (see Fig. 6b). Although this is a signature generally associated with quantum spin liquid (QSL) states, we note that in the present model, the Kitaev ${{\mathbb{Z}}}_{2}$ flux operator yields $\langle {W}_{{\rm {p}}}\rangle ={2}^{6}\langle {S}_{1}^{x}{S}_{2}^{y}{S}_{3}^{z}{S}_{4}^{x}{S}_{5}^{y}{S}_{6}^{z}\rangle \approx 0.29$ (site indices refer to Fig. 6a). While this is clearly elevated compared to classical collinear states, where 〈W_p〉 is restricted to $| \langle {W}_{{\rm {p}}}\rangle | \,\le \,\frac{1}{27}\, < \,0.04$, it is still significantly below the value of the pure unperturbed Kitaev spin liquid, where 〈W_p〉 = 1¹. Hence, if the ground state constitutes a QSL state, it is presumably not the ${{\mathbb{Z}}}_{2}$ Kitaev spin liquid. The precise nature of the encountered magnetically disordered state might be interesting for future studies. It appears to be stabilized by the further-neighbor interactions, as we find a clear ferromagnetic ground state when omitting the second- and third-neighbor interactions in the present model. While a QSL scenario for our full RuI₃ model is compelling, we note that finite-size effects in our calculation could play a role. In particular, the finite-size cluster could be incompatible with the supposed correct ordering wave vector of the model, e.g. in the case of an incommensurate ordering vector.

To summarize, we have presented a comparative analysis of the electronic and magnetic properties of the Ru-based trihalide family, including the recently synthesized RuBr₃ and RuI₃, by combining state-of-the-art ab initio microscopic modeling with analysis of reported resistivity, specific heat, and magnetic susceptibility data. The evolution of the magnetic order and Mott–Hubbard correlations along the halogen series, as well as the possible role of disorder, have been a central part of our study. We conclude that:

• 1. All three ideal compounds are spin–orbit-assisted Mott insulators, but their fundamental gap decreases with higher ligand atomic number, Cl → Br → I, with RuI₃ coming rather close to a metal–insulator transition.

• 2. From DFT total-energy calculations, in ideal, pristine crystals the zigzag magnetic order is even more stable in RuBr₃ than in RuCl₃, while RuI₃ shows significant magnetic frustration. Our ab-initio extracted low-energy models predict RuI₃ to feature either an incommensurate magnetic ordered state or a quantum spin liquid, which, interestingly, is possibly of a different kind to the ${{\mathbb{Z}}}_{2}$ Kitaev spin liquid.

• 3. A number of reported experimental observations seem to be adversely affected by the sample quality, in particular by dirty grain boundaries. In fact, most of the observations in RuI₃ can be reconciled with theory by assuming insulating grains surrounded by (bad) metallic boundaries. The experimental evidence is consistent with a ‘dirty’ insulator, or a bad metal. Disorder would favor either of these.

• 4. In all three systems the dominant nearest-neighbor interaction is FM Kitaev K₁, with a subdominant FM Heisenberg Interaction J₁, that nearly vanishes for RuI₃. We observe a non-monotonous behavior of the magnetic anisotropic terms as a function of ligand atomic number that we trace back to a competition of the SOC effects from magnetic ions and ligands.

• 5. RuBr₃ has predominantly ferromagnetic interactions, in contrast to what is suggested by standard Curie–Weiss analysis²⁵. Such interactions are consistent with the experimental susceptibility when taking into account high-temperature SOC effects. Our ab-initio magnetic model predicts zigzag order in agreement with the experiment, with a tilting angle of θ_M = 32^∘ for the magnetic moments, similar to RuCl₃, but in contradiction to the reported θ_M = 64^∘25. We showed that such a large angle cannot be simply explained by proximity to the pure Kitaev model, but would require quite drastic changes to the exchange parameters, such as sizeable negative Γ₁ < 0. Those would necessitate strong distortions on the reported RuBr₃ crystal structures.

Answering the question posed in the title, our results and analysis strongly suggest that the ideal RuCl₃, RuBr₃, and RuI₃ compounds constitute a family of three Mott-insulating siblings. The challenging task of getting better samples will hopefully help resolve the open issues.

Methods

Modified Curie–Weiss fit of RuBr₃

We fit the experimental average susceptibility of ref. ²⁵ with four fitting parameters ${\chi }_{0}^{\perp },{\chi }_{0}^{\parallel },{{{\Theta }}}^{\perp },{{{\Theta }}}^{\parallel }$ using the modified Curie–Weiss formula

$${\chi }^{{{{\rm{avg}}}}}(T)\approx \frac{2}{3}\left({\chi }_{0}^{\parallel }+\,\frac{{C}^{\parallel }(T)}{T-{{{\Theta }}}^{\parallel }}\right)+\frac{1}{3}\left({\chi }_{0}^{\perp }+\,\frac{{C}^{\perp }(T)}{T-{{{\Theta }}}^{\perp }}\right),$$

(2)

where Θ^∥, Θ^⊥ are the Weiss constants and ${C}^{\alpha }(T)\propto {[{\mu }_{{{{\rm{eff}}}}}^{\alpha }(T,{{\Delta }})]}^{2}$ is determined through Δ as described in ref. ³³. Superscripts ∥ and ⊥ indicate the in- and out-of-honeycomb-plane direction, respectively. The susceptibility is fitted over the temperature range 150–300 K and SOC strength λ = 0.15 eV is taken.

DFT calculations

To make sure that the calculated features within density functional theory are robust with respect to the choice of the basis set, we have tested the results using two different methods: the projector augmented wave method^50,51 as implemented in the VASP code^52,53, and the full potential linearized augmented plane-wave (LAPW) basis as implemented in Wien2k⁵⁴. Throughout the paper, we have used the generalized gradient approximation (GGA⁵⁵) for the exchange-correlation functional. Hubbard correlation effects were included on a mean-field level in the rotationally invariant implementation of the GGA + U method⁵⁶. All calculations included spin–orbit coupling (SOC) effects. For VASP we used the Ru_pv pseudopotential, treating Ru p states as valence, and the standard pseudopotentials for the halogens. The Γ-centered 8 × 8 × 8 mesh in the nonmagnetic rhombohedral Brillouin zone was used, or the correspondingly scaled meshes for other structures. The energy cut-off was 350 eV, and the energy convergence criterion 1 × 10⁻⁰⁸ eV. For each type of magnetic order, a number of collinear starting configurations with randomly selected Néel vectors were used, and the lowest-energy result was selected as the ground state. Individual results can be found in the Supplementary Information. For Wien2k we chose the plane-wave cutoff ${K}_{\max }$ corresponding to RK${}_{\max }=8$ and a k mesh of 8 × 8 × 2 for the R$\bar{3}$ structure in the hexagonal Brillouin zone and 8 × 4 × 6 in the first Brillouin zone of the conventional unit cell for the C/2m structure. The density of states are calculated using a k mesh of 12 × 12 × 3 for the $\mathrm{R}\bar{3}$ structure and 12 × 6 × 9 for the C/2m structure. The zigzag configurations are constructed using a conventional cell of the C/2m structure for RuCl₃ while a 1 × 2 × 1 supercell of the R$\bar{3}$ structures for RuBr₃ and RuCl₃.

cRPA calculations

In order to obtain ab-initio estimates for the effective Coulomb interaction for the Ru-trihalide family, we employed the constrained random-phase approximation (cRPA)^57,58, as implemented in the FHI-gap code⁵⁹, based on the Wien2K electronic structure. The low-energy limit of the screened interaction was projected on the five Ru d orbitals, where screening processes in the same window were excluded. Convergence with respect to the discretization of the Brillouin zone and energy cutoff was ensured.

DFT-based derivation of magnetic models

To derive bilinear exchange parameters for each material, we employed the projED method⁴⁵, which consists of two steps. First, complex ab-initio hopping parameters between the ruthenium ions are estimated with projective Wannier functions⁶⁰ applied on full relativistic FPLO⁶¹ calculations on a 12 × 12 × 12k mesh. This allows constructing an effective electronic model ${{{{\mathcal{H}}}}}_{{{{\rm{tot}}}}}={{{{\mathcal{H}}}}}_{{{{\rm{hop}}}}}+{{{{\mathcal{H}}}}}_{{{{\rm{U}}}}}$, where the complex ab-initio hopping parameters enter the kinetic term ${{{{\mathcal{H}}}}}_{{{{\rm{hop}}}}}={\sum }_{ij\alpha \beta }{\sum }_{\sigma {\sigma }^{\prime}}{t}_{i\alpha ,j\beta }^{\sigma {\sigma }^{\prime}}\,{c}_{i\alpha \sigma }^{{\dagger} }{c}_{j\beta {\sigma }^{\prime}}$ and the cRPA effective Coulomb interaction parameters enter the two-particle term ${{{{\mathcal{H}}}}}_{{{{\rm{U}}}}}={\sum }_{i\alpha \beta \gamma \delta }{\sum }_{\sigma {\sigma }^{\prime}}{U}_{i\alpha \beta \gamma \delta }^{\sigma {\sigma }^{\prime}}\,{c}_{i\alpha \sigma }^{{\dagger} }{c}_{i\beta {\sigma }^{\prime}}^{{\dagger} }{c}_{i\delta {\sigma }^{\prime}}{c}_{i\gamma \sigma }$. Second, the effective spin Hamiltonian ${{{{\mathcal{H}}}}}_{{{{\rm{eff}}}}}$ is extracted from the electronic model via exact diagonalization (ED) and projection of the resulting energy spectrum onto the low-energy subspace, mapped onto pseudo-spin operator representation in the j_eff picture with the projection operator ${\mathbb{P}}$: ${\mathcal{H}}_{\rm{eff}}={\mathbb{P}}{\mathcal{H}}_{\rm{tot}}{\mathbb{P}}={\sum}_{ij}{\bf{S}}_{i}\,{{\mathbb{J}}}_{ij}\,{\bf{S}}_{j}$.

Note that for RuCl₃ the exchange constants slightly differ from previously calculated values by some of the authors^8,38. The reason for this lies in the following details of the calculation setup: (i) first principles input parameters U_avg and J_avg from cRPA in contrast to previous choices, (ii) consideration of all five 4d ruthenium orbitals with the cost of restriction onto two-site clusters, (iii) SOC effects from both Ru³⁺ and ligands considered through complex hopping parameters in contrast to the atomic limit, and (iv) consideration of the experimental crystal structure in contrast to relaxed ambient pressure structure as it was done in ref. ³⁸.

For the calculation of the gyromagnetic g-tensor, we considered [RuX₆]³⁻ molecules within the quantum chemistry ORCA 3.03 package^62,63 with the functional TPSSh, basis set def2-TZVP and complete active space for the d orbitals CAS(5,5).

Exact diagonalization

Exact diagonalization calculations of the j_eff = 1/2 models were performed on the 24-site cluster shown in Fig. 6a. To identify possible magnetic ordering, we analyze the static spin structure factor ${\sum }_{\mu = x,y,z}\langle {S}_{-{{{\bf{k}}}}}^{\mu }{S}_{{{{\bf{k}}}}}^{\mu }\rangle$. For the ordered moment direction, we compute the eigenvector with the maximal eigenvalue of the correlation matrix ${(\langle {S}_{-{{{\bf{k}}}}}^{\mu }{S}_{{{{\bf{k}}}}}^{\nu }\rangle )}_{\mu ,\nu }$ (μ, ν ∈ {x, y, z}) at the ordering wave vector k = Q. This eigenvector then represents the ordered pseudospin direction S⁴⁹, which relates to the magnetic moment direction ${\bf{M}}\propto {\mathbb{G}}\cdot {\bf{S}}$, as measured by neutron diffraction, via the anisotropic g-tensor ${\mathbb{G}}$.