Abstract
Motivated by reports of metallic behavior in the recently synthesized RuI_{3}, in contrast to the Mottinsulating nature of the actively discussed αRuCl_{3}, as well as RuBr_{3}, we present a detailed comparative analysis of the electronic and magnetic properties of this family of trihalides. Using a combination of firstprinciples calculations and effectivemodel considerations, we conclude that RuI_{3}, 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 RuI_{3} to have a nearly vanishing conventional nearestneighbor 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 RuI_{3}. Furthermore, for the Mott insulator RuBr_{3,} we obtain a magnetic Hamiltonian of a similar form to that in the muchdiscussed αRuCl_{3} and show that this Hamiltonian is in agreement with experimental evidence in RuBr_{3}.
Introduction
RuI_{3} and RuBr_{3} are recent additions to the RuX_{3} family (X = Cl, Br, I) of layered Rubased trihalides (Fig. 1a). The first member, αRuCl_{3} (in the following ‘RuCl_{3}’) has attracted considerable attention in recent years as a candidate material for the Kitaev honeycomb model^{1}. RuCl_{3} is a spin–orbitassisted Mott insulator^{2,3,4,5} whose magnetic lowenergy 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 socalled 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}.
Recently, a sister compound with a heavier halogen, X = Br, was synthesized^{25}. Analogous to RuCl_{3}, it is insulating and shows zigzag magnetic order, albeit with higher Néel temperature T_{N} = 34 K^{25}. In contrast to RuCl_{3}, the authors of ref. ^{25} reported a Weiss constant with dominant antiferromagnetic interactions and a direction of the zigzagordered moment different from RuCl_{3} and argued that this deviation suggests closer proximity to the pure Kitaev model.
To complete the RuX_{3} family, two independent groups have now synthesized RuI_{3} with the even heavier halogen iodine^{26,27}. In contrast to the two ‘sibling’ compounds, a quasimetallic behavior was observed in RuI_{3}, questioning the description in terms of localized j_{eff} = 1/2 moments. Even though the dc resistivities measured in RuI_{3} are orders of magnitude smaller than those of RuCl_{3} or RuBr_{3}, the reported values of 10^{−3}–10^{−2}Ω cm^{26} are uncharacteristically large for metals or even typical bad metals^{28}, and practically temperatureindependent. 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 temperatureindependent^{26} or with a strong upturn at low temperatures^{27}, 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 firstprinciples calculations and extracted lowenergy models. We find that: (i) The behavior of RuI_{3} is not that far from RuCl_{3} and RuBr_{3} and the variations across the series are more quantitative than qualitative. (ii) Pristine samples of RuI_{3} should be insulating with highly anisotropic magnetic exchange and nearly vanishing conventional Heisenberg interaction. We argue that the reported metallic behavior in RuI_{3} could have its origin in sample quality. (iii) The magnetism in the Mott insulator RuBr_{3} 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 spinorbit coupling effects in the Curie–Weiss behavior.
Our study derives model parameters and magnetic Hamiltonians for the whole RuX_{3} family from abinitio, 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 Rutheniumonly SOC picture in the case of RuBr_{3} and RuI_{3}.
Results and discussion
Comparative analysis of experiments
In the following, we analyze the reported electrical resistivity, specific heat, and magnetic susceptibility data for RuX_{3} (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 RuI_{3}, the resistivity has a weak^{27} or almost no^{26} 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^{−5}Ω 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 RuI_{3} (ρ ~ 40 mΩ cm^{27} and ρ ~ 4 mΩ cm^{26}) are high even for bad metals, surpassing the Ioffe–Regel limit by more than an order of magnitude. Even more relevant, the lowerresistivity set of data^{26} 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 RuI_{3} 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 ingrain 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 Mottinsulating one, as has been shown for example for the Mott insulator κ(BEDTTTF)_{2}Cu[N(CN)_{2}]Cl^{30}. Indeed, in our firstprinciples calculations discussed below, we find the ideal RuI_{3} to already be quite close to a Mott–metal transition.
Turning to the sibling compounds RuCl_{3} and RuBr_{3}, 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 blueshift of the gap with increasing temperatures.
Considering specific heat data in the compounds, the specific heat for RuCl_{3} displays a welldefined peak at T_{N} ≈ 7 K denoting the onset of the zigzag order, while the onset of longrange magnetic order in RuBr_{3} is observed by a kink at T_{N} = 34 K^{25}. None of this is observed for RuI_{3}^{26,27}. In Table 1 we summarize specific heat parameters reported experimentally^{25,26,27,31}, where γ (β) is the Tlinear (T^{3}) contribution to C(T).
In RuI_{3}, the Tlinear 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 RuI_{3} shown below, we find that the unrenormalized metallic (i.e., nonmagnetic, not Ucorrected) density of states corresponds to γ_{0} ≈ 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 T^{3} 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 powderaveraged measured magnetic susceptibilities χ(T) as reported in refs. ^{25,26,27,32}. At low temperatures, the RuCl_{3} data^{32} shows a clear signature of a transition to the ordered magnetic phase at 7 K. For RuBr_{3}, 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 measurements^{25}. The experimental report on powder samples of RuBr_{3} utilized a standard Curie–Weiss (CW) fit, yielding an average Curie–Weiss temperature \({{{\Theta }}}_{{{{\rm{std}}}}}^{{{{\rm{avg}}}}}=58\) K^{25}, indicating predominantly AFM interactions. However, as we have recently shown^{33}, 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 Rubased trihalides. With SOC, temperaturedependent vanVleck contributions can arise, which can be effectively captured in a temperaturedependent 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 RuCl_{3}, a standard CW fit would lead to \({{{\Theta }}}_{{{{\rm{std}}}}}^{{{{\rm{avg}}}}}=20\) K, whereas an improved CW fit taking into account such vanVlecklike contributions^{33} provides CW constants Θ^{∥} = + 55 K for the magnetic field in the honeycomb plane and Θ^{⊥} = + 33 K for the outofplane 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 RuCl_{3}^{34,35,36,37}.
Considering a similar strategy (see the “Methods” section), we fit the average susceptibility χ^{avg} of RuBr_{3}^{25}. However, since the crystalfield parameter Δ primarily controls the inplane vs. outofplane anisotropy, and for RuBr_{3} only powderaveraged data are available, we do not aim at extracting Δ by fitting. Instead, we first fix Δ using our firstprinciples 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 RuBr_{3}, as seen before in RuCl_{3}. In Fig. 2d we further analyze how the bestfit Weiss constants evolve for other choices of Δ. Indeed, for a wide range of reasonable Δ around the firstprinciples 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 (RuBr_{3} or RuCl_{3}) 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 RuI_{3} (~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 RuI_{3}. Actually, the two available susceptibility measurements on RuI_{3} display different behaviors, one nearly temperatureindependent^{26}, and the other^{27} showing a Curielike rise at low temperatures, where a standard CW fit yields μ_{eff} = 0.53μ_{B} and \({{{\Theta }}}_{{{{\rm{CW}}}}}^{{{{\rm{avg}}}}}=3\) K^{27}. 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 data^{27} hint at larger insulating grains, hence less metallic boundaries are present, leading to the lowtemperature Curielike 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 RuX_{3} family obtained from a combination of density functional theory (DFT) and exact diagonalization of abinitioderived lowenergy models. Details of the calculations are given in the “Methods" section.
Past experience with firstprinciples calculations for the Rubased 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 RuCl_{3}, and the suggested R\(\bar{3}\) structures for RuBr_{3}^{25} and RuI_{3}^{27}. Structural details of the four models are summarized in the Supplementary Information. For RuCl_{3}, 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 orbitallyaveraged Hubbard onsite (U_{avg}) and Hund’s coupling (J_{avg}) as obtained from constrained randomphase approximation (cRPA) calculations (see the “Methods" section for calculation details). In contrast to previous cRPA estimates for RuCl_{3}^{42}, our estimates incorporate all five d orbitals and extend to the complete Rubased 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 RuI_{3} compared to RuCl_{3} when hybridizing with I instead of Cl.
For RuCl_{3} 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 meV^{43}. We systematically reduced U_{eff} to 2.1 eV for RuBr_{3} and 1.4 eV for RuI_{3} following the trend given by the cRPA results. With these values, RuBr_{3} shows a gap of 0.56 eV (Fig. 3b), while RuI_{3} shows a small gap of 0.1 eV (Fig. 3c). The gap closes in RuI_{3} when U_{eff} is further reduced to 1 eV. These results indicate a spin–orbit assisted Mott insulating state in disorderfree RuI_{3} 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 insulators^{30}. Note that these results hold regardless of the assumed magnetic pattern in the calculations.
In order to analyze the magnetic structure of the RuX_{3} compounds, we first consider spinpolarized 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 RuCl_{3}, 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 RuCl_{3}^{37,44}. Correspondingly, in our effective pseudospin model of RuCl_{3} 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 RuBr_{3} we find an energy minimum for the zigzag ordering in agreement with the experiment. Interestingly, for RuI_{3} 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 RuBr_{3} has been suggested to be closer to the pure Kitaev limit than in RuCl_{3}^{25}, and, secondly, with our proposed scenario of a Mott insulating state for RuI_{3}, the question of its magnetic properties is open. To investigate these issues from the first principles, we derive via the abinitio projED method^{45} the pseudospin models \({\mathcal{H}}_{\rm{eff}}={\sum}_{ij}{\bf{S}}_{i}\cdot {\mathbb{J}}_{ij}\cdot {\bf{S}}_{j}\) of the three RuX_{3} compounds. Here, S stands for the relativistic pseudospin j_{eff} = 1/2 moment^{2}.
In the conventional parametrization of Kitaev materials, the exchange matrix \({{\mathbb{J}}}_{ij}\) in R\(\bar{3}\) symmetry on a nearestneighbor Z_{1}bond (defined in Fig. 1) follows the form
with the isotropic Heisenberg exchange J_{1}, the bonddependent anisotropic Kitaev exchange K_{1}, the bonddependent offdiagonal exchange terms Γ_{1} and \({{{\Gamma }}}_{1}^{\prime}\) and correction terms η_{1} and ν_{1}. 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 Ybonds follow by respective C_{3} rotations about the outofplane 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 fullrelativistic DFT (magnitudes shown in Fig. 4b–d) and the projED method, we extracted the exchange constants shown in Fig. 5b–d.
Evaluating the magnetic interactions of the complete RuX_{3} family, we find a nearestneighbor ferromagnetic Kitaev interaction K_{1} to be the dominant in all three compounds. Additionally, a subdominant ferromagnetic nearestneighbor Heisenberg exchange J_{1} is present, which is, however, almost vanishing for the iodine case. The symmetric offdiagonal Γ_{1} interaction is of similar magnitude as J_{1}, changing sign going from Cl and Br to I. \({{{\Gamma }}}_{1}^{\prime}\), often neglected in the RuCl_{3} analysis, may become rather important, particularly for RuI_{3}. Furtherneighbor interactions are generally smaller than their nearestneighbor counterparts for all three systems, but increase for larger ligand atomic number and may play, especially in RuI_{3}, an important role.
That the anisotropic interactions do not monotonically increase with stronger spinorbit coupling of halogen elements can be related to the SOC source. In the original JackeliKhaliullin mechanism^{2}, the heavy magnetic ions are solely responsible for SOC effects, which can be well described within the SOC atomic limit. In the case of RuBr_{3} and RuI_{3}, 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 RuX_{3} 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. RuBr_{3} and RuI_{3} 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 CrI_{3}^{46}.
Along the halogen series Cl–Br–I we observe a decrease for nearestneighbor 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 nearestneighbor 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 nearestneighbor parameters. In contrast, the second and third neighbors show a very different dependence on the halogen element. From the dominant furtherneighbor hoppings (Fig. 4c, d), the hoppings show an overall tendency to increase, with few exceptions. Certain furtherneighbor magnetic exchange parameters, depending on their relation to the individual hopping parameters, become therefore increasingly important for RuBr_{3} and especially for the magnetic properties of RuI_{3}.
Finally, we also computed the gyromagnetic gtensor for the RuX_{3} 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 gtensor 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 inplane fields.
We now discuss the ramifications of the derived magnetic models for the magnetism in these materials. For RuCl_{3}, 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 wellbenchmarked recent models^{19,37,38}, and is therefore expected to also describe the material quite well. As we apply the same abinitio setup for the new members of the RuX_{3} family, we expect our models to be reliable for them too.
The directionaveraged Weiss constant (\({{{\Theta }}}_{{{{\rm{CW}}}}}^{{{{\rm{avg}}}}}\) in Table 2) is predicted to be positive across the RuX_{3} family, characteristic of ferromagnetic exchange interactions. This is in line with our analysis of the experimental magnetic susceptibilities of RuCl_{3} and RuBr_{3} above (Fig. 2). While in RuCl_{3} and RuBr_{3} a large FM contribution to the Weiss constant comes from a significant FM nearestneighbor Heisenberg interaction J_{1}, this interaction nearly vanishes for RuI_{3} (Fig. 5b), leading to a smaller Weiss constant. Furthermore, the small J_{1} in RuI_{3} renders the nearestneighbor interactions to be extremely anisotropic, with a dominant Kitaev interaction K_{1}. While at first glance this might suggest a spinliquid ground state in RuI_{3}, the increased strength of the furtherneighbor interactions in RuI_{3} (see, e.g., J_{2}, K_{2} in Fig. 5c) also needs to be considered^{47}.
To find the magnetic ground state properties, we perform exact diagonalization (ED) calculations of the derived j_{eff} = 1/2 models on the 24site cluster shown in Fig. 6a. In Table 2 and Fig. 6b we summarize the encountered ground states, i.e. zigzag for RuCl_{3} and RuBr_{3}, and possibly a quantum spin liquid (QSL) in RuI_{3}. This is discussed in detail below.
For RuCl_{3}, 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 RuBr_{3}. The static spin structure factor for the RuBr_{3} model of Fig. 5 is shown in Fig. 6b, indicating also a zigzag AFM order (k = M and C_{6}rotated vectors), in agreement with experiment^{25}. However, the calculated tilt angle of the magnetic moment, θ_{M} = 32^{∘}, is more in line with RuCl_{3} than with the reported measured θ_{M} = 64° of RuBr_{3}^{25}. The authors of ref. ^{25} argued that this anomalously large tilt angle indicates an exceptionally strong relative Kitaev coupling, i.e., larger ∣K_{1}/J_{1}∣ and ∣K_{1}/Γ_{1}∣ compared to RuCl_{3}. To investigate to what extent closer proximity to the pure Kitaev model could produce such high tilt angles, we take our RuBr_{3} Hamiltonian of Fig. 5 as a starting point and tune it towards the pure Kitaev model, where K_{1} is the only nonzero coupling. This is done by multiplying every exchange coupling except K_{1} 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 gtensor (Fig. 5a). A reconciliation with the reported θ_{M} = 64^{∘} would therefore require quite drastic changes to the gtensor anisotropy and/or the exchange parameters. While in the whole J_{1}−K_{1}−Γ_{1} parameter space with Γ_{1} > 0, no angles of θ_{S} beyond ~ 40^{∘} are expected in the zigzag phase^{48}, significant negative Γ_{1} < 0 can in principle lead to θ_{S} beyond 60°^{49}. However such terms seem incompatible with the abinitio results and would likely need strong distortions from the present considered RuBr_{3} crystal structure to be realized.
More distinct from the other two compounds are our results for RuI_{3}. 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}〉 = 1^{1}. 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 furtherneighbor interactions, as we find a clear ferromagnetic ground state when omitting the second and thirdneighbor interactions in the present model. While a QSL scenario for our full RuI_{3} model is compelling, we note that finitesize effects in our calculation could play a role. In particular, the finitesize 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 Rubased trihalide family, including the recently synthesized RuBr_{3} and RuI_{3}, by combining stateoftheart 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–orbitassisted Mott insulators, but their fundamental gap decreases with higher ligand atomic number, Cl → Br → I, with RuI_{3} coming rather close to a metal–insulator transition.

2.
From DFT totalenergy calculations, in ideal, pristine crystals the zigzag magnetic order is even more stable in RuBr_{3} than in RuCl_{3}, while RuI_{3} shows significant magnetic frustration. Our abinitio extracted lowenergy models predict RuI_{3} 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 RuI_{3} 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 nearestneighbor interaction is FM Kitaev K_{1}, with a subdominant FM Heisenberg Interaction J_{1}, that nearly vanishes for RuI_{3}. We observe a nonmonotonous 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.
RuBr_{3} has predominantly ferromagnetic interactions, in contrast to what is suggested by standard Curie–Weiss analysis^{25}. Such interactions are consistent with the experimental susceptibility when taking into account hightemperature SOC effects. Our abinitio magnetic model predicts zigzag order in agreement with the experiment, with a tilting angle of θ_{M} = 32^{∘} for the magnetic moments, similar to RuCl_{3}, 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 Γ_{1} < 0. Those would necessitate strong distortions on the reported RuBr_{3} crystal structures.
Answering the question posed in the title, our results and analysis strongly suggest that the ideal RuCl_{3}, RuBr_{3}, and RuI_{3} compounds constitute a family of three Mottinsulating siblings. The challenging task of getting better samples will hopefully help resolve the open issues.
Methods
Modified Curie–Weiss fit of RuBr_{3}
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
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 outofhoneycombplane 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 planewave (LAPW) basis as implemented in Wien2k^{54}. Throughout the paper, we have used the generalized gradient approximation (GGA^{55}) for the exchangecorrelation functional. Hubbard correlation effects were included on a meanfield level in the rotationally invariant implementation of the GGA + U method^{56}. 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 Γcentered 8 × 8 × 8 mesh in the nonmagnetic rhombohedral Brillouin zone was used, or the correspondingly scaled meshes for other structures. The energy cutoff 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 lowestenergy result was selected as the ground state. Individual results can be found in the Supplementary Information. For Wien2k we chose the planewave 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 RuCl_{3} while a 1 × 2 × 1 supercell of the R\(\bar{3}\) structures for RuBr_{3} and RuCl_{3}.
cRPA calculations
In order to obtain abinitio estimates for the effective Coulomb interaction for the Rutrihalide family, we employed the constrained randomphase approximation (cRPA)^{57,58}, as implemented in the FHIgap code^{59}, based on the Wien2K electronic structure. The lowenergy 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.
DFTbased derivation of magnetic models
To derive bilinear exchange parameters for each material, we employed the projED method^{45}, which consists of two steps. First, complex abinitio hopping parameters between the ruthenium ions are estimated with projective Wannier functions^{60} applied on full relativistic FPLO^{61} 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 abinitio 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 twoparticle 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 lowenergy subspace, mapped onto pseudospin 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 RuCl_{3} 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 twosite clusters, (iii) SOC effects from both Ru^{3+} 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 gtensor, we considered [RuX_{6}]^{3−} molecules within the quantum chemistry ORCA 3.03 package^{62,63} with the functional TPSSh, basis set def2TZVP 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 24site 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 S^{49}, which relates to the magnetic moment direction \({\bf{M}}\propto {\mathbb{G}}\cdot {\bf{S}}\), as measured by neutron diffraction, via the anisotropic gtensor \({\mathbb{G}}\).
Data availability
The datasets generated during the current study are available from the corresponding authors upon reasonable request.
Code availability
The custom codes implementing the calculations of this study are available from the authors upon reasonable request.
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Acknowledgements
We thank Stephen M. Winter, Robert J. Cava, Yoshinori Imai, and Elena Gati for discussions and Yoshinori Imai for sharing the structural information of RuBr_{3} with us before publication. R.V., A.R., K.R. and D.A.S.K. acknowlegde support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding through Project No. 411289067 (VA117/151) and TRR 288—422213477 (project A05). Y.L. acknowledges support from the National Natural Science Foundation of China (Grant No. 12004296) and the China Postdoctoral Science Foundation (Grant No. 2019M660249). I.I.M. acknowledges support from the U.S. Department of Energy through grant #DESC0021089. R.V. and I.I.M. thank the Wilhelm und Else Heraeus Stiftung for financial support.
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R.V. conceived and supervised the project. Density functional theory calculations were performed by K.R., A.R., Y.L., I.I.M., cRPA calculations by S.B., projED calculations by K.R., and calculations on magnetic models by D.A.S.K. All authors contributed to the manuscript.
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Kaib, D.A.S., Riedl, K., Razpopov, A. et al. Electronic and magnetic properties of the RuX_{3} (X = Cl, Br, I) family: two siblings—and a cousin?. npj Quantum Mater. 7, 75 (2022). https://doi.org/10.1038/s41535022004813
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DOI: https://doi.org/10.1038/s41535022004813