Coexisting Z-type charge and bond order in metallic NaRu₂O₄

Abstract

How particular bonds form in quantum materials has been a long-standing puzzle. Two key concepts dealing with charge degrees of freedom are dimerization (forming metal-metal bonds) and charge ordering. Since the 1930s, these two concepts have been frequently invoked to explain numerous exciting quantum materials, typically insulators. Here we report dimerization and charge ordering within the dimers coexisting in metallic NaRu₂O₄. By combining high-resolution x-ray diffraction studies and theoretical calculations, we demonstrate that this unique phenomenon occurs through a new type of bonding, which we call Z-type ordering. The low-temperature superstructure has strong dimerization in legs of zigzag ladders, with short dimers in legs connected by short zigzag bonds, forming Z-shape clusters: simultaneously, site-centered charge ordering also appears. Our results demonstrate the yet unknown flexibility of quantum materials with the intricate interplay among orbital, charge, and lattice degrees of freedom.

Introduction

Chemical bonding lies at the heart of all-natural sciences like condensed matter physics, chemistry, and even biology. As such, the understanding of chemical bonding was a great puzzle for a long time. In the long and rich history of condensed matter physics, one cannot emphasize strongly enough the importance of the two basic concepts: dimerization¹ and charge ordering², both born in the 1930s. However, these two phenomena have been rarely found to coexist in one compound, much less in metallic systems.

Many studies in the late 1990s and early 2000s demonstrated, one example after another, how the new patterns emerge out of simple structures: hexamers in ZnCr₂O₄³, octamers in CuIr₂S₄⁴, spontaneous dimensionality reduction in Tl₂Ru₂O₇⁵, and the formation of trimeron in Fe₃O₄⁶, to name only a few. Despite the various properties, an ever occurring theme is that these phenomena typically happen near the metal–insulator boundary, with the ordered states usually insulating. This characteristic feature is naturally captured by Mott–Hubbard physics⁷, and it is particularly prominent among frustrated magnets^8,9,10,11. It is relatively less well-known how this rich physics plays out in metallic systems; it remains unchartered territory to date. One can imagine that this question may become more important in the newly emerging field of quantum materials that increasingly include metallic systems¹².

Ru has a special place in this extensive search because of its two characteristic features: the modest spin–orbit coupling (SOC), not too small nor too large compared to the Coulomb energy and the bandwidth; another is relatively stronger importance of orbital physics⁷. This combination of the two has been behind the active researches on Ru oxides; for instance, superconducting Sr₂RuO₄ and ferromagnetic SrRuO₃ are two prominent examples^13,14. In comparison, there have been few studies on Na–Ru–O systems, including NaRu₂O₄. Using neutron diffraction, the authors of ref. ¹⁵ determined that its room-temperature structure has the space group of Pnma, the typical orthorhombic structure among oxides: to our best knowledge, it was initially reported in 1975¹⁶ before being re-examined in 2006. It was reported to have a paramagnetic susceptibility down to the lowest temperature. While studying another Na compound, Na_2.7Ru₄O₉¹⁷ which was also investigated in the same ref. ¹⁵, we realized that the high-temperature phase of these Na–Ru–O systems could exhibit exciting new unreported properties and hence deserve careful studies.

In this report, we made extensive studies of both structure and physical properties of NaRu₂O₄ over a wide temperature range up to 600 K above which it becomes chemically unstable. We made the high-resolution XRD study above room temperature to find a surprising first-order phase transition during this study. We found that NaRu₂O₄ undergoes a metal–metal phase transition below 535 K with a distinct structure change, from a CaFe₂O₄-type structure with the orthorhombic Pnma symmetry at a higher temperature to a monoclinic structure P112₁/a below T_c. By combining experimental and theoretical studies, we found that at this transition we have simultaneously a site-centered charge ordering coexisting with dimerization.

Results and discussion

Bulk properties

Electrical resistivity (ρ) measurements were carried out using a homemade system equipped with a furnace (300–685 K) and a pulsed-tube cryostat with the base temperature of 3 K. The electrical resistance was measured in a four-point geometry, where contacts were made using silver paint and 25 µm gold wire as shown in the inset of Fig. 1a. The current was applied perpendicular to the single crystal length, which is the crystallographic b-axis. As the single crystals are all in a thin needle form, we could not measure the resistivity along with the other crystallographic directions. We tested several long rod-shaped as-grown crystals and discovered the same transition (see also Supplementary Fig. 1). The resistivity data shows that a phase transition occurs in NaRu₂O₄ at T_c = 535 K with small hysteresis in the electrical resistivity, a sign of a weak first-order phase transition at T_c. We will extensively discuss this transition below; it is the main feature of NaRu₂O₄.

Fig. 1: Bulk properties.

a The electrical resistivity data as a function of temperature is taken over single-crystal with the solid red line for the Bloch–Grüneisen model. Inset shows the single crystal used for the four-probe resistivity measurements. b Magnetic susceptibility (χ) measured at an applied magnetic field B = 0.5 T. The inset shows inverse susceptibility with the modified Curie–Weiss law (solid red line). c Temperature dependence of the specific heat C_p measured at zero fields (B = 0 T). Solid black circles are the raw data with C_p/T shown in the inset of the figure while solid red line is the corresponding C_p/T = γ + βT² fitting.

The resistivity ρ(T) data measured down to 3.3 K indicates metallic behavior with a room temperature resistivity of ~40 μΩ cm. To have a quantitative understanding_, we analyzed the results theoretically using the Bloch–Grüneisen model. The following formula gives the expression for the temperature-dependent part of the metallic resistivity:

$$\rho\approx \left(\frac{3}{\hslash{e}^{2}{v}_{{{{{{\mathrm{F}}}}}}}^{2}}\right)\frac{{k}_{{{{{\mathrm{B}}}}}}T}{{{{{{\mathrm{Mv}}}}}}_{{{{{\mathrm{s}}}}}}^{2}}\int {\left|v(q)\right|}^{2} \left[\frac{{\left(\hslash \omega /{k}_{{\mathrm {B}}}T\right)}^{2}{q}^{3}{{{{\mathrm{d}}}}}q}{\left({{{{{\mathrm{exp}}}}}}\left(\hslash \omega /{k}_{{\mathrm {B}}}T\right)-1\right)\left(1-{{{{{\mathrm{exp}}}}} }\left(-\hslash \omega /{k}_{{\mathrm {B}}}T\right)\right)}\right],$$

(1)

where v(q) is the Fourier transform of the potential associated with one lattice site and v_s is the sound velocity. When Eq. (1) was considered for the acoustic phonon contribution, it leads to the Bloch–Grüneisen function as shown below:

$${\rho }_{e-{{\mbox{ph}}}}\left(T,{\theta }_{{\mathrm {D}}}\right)=4{A}_{{{\mbox{ac}}}}{\left(T/{\theta }_{{\mathrm {D}}}\right)}^{4}\times T{\int }_{0}^{{\theta }_{{\mathrm {D}}}/T}{x}^{5}{\left({e}^{x}-1\right)}^{-1}{\left(1-{e}^{-x}\right)}^{-1}{{\mbox{d}}}x,$$

(2)

where $x$ = ħω/k_BT, θ_D is Debye temperature, and A_ac is a proportionality constant. The total resistivity of a material can be written as

$$\rho (T)={\rho }_{0}+{\rho }_{{\mathrm {e}}-{{\mbox{ph}}}}\left(T,{\theta }_{{\mathrm {D}}}\right),$$

where ${\rho }_{0}$ is the temperature-independent residual resistivity and the second term represents the electron–phonon scattering. The Bloch–Grüneisen model is found to work well for our study, and the theoretically estimated ρ is quite consistent with the experimental data for NaRu₂O₄. The characteristic Debye temperature (θ_ρ) and residual resistivity (ρ₀) is found to be 452.2 K and 35.9 μΩ cm, respectively.

We also measured magnetic susceptibility and specific heat of NaRu₂O₄ in the accessible temperature range of 2–300 K. The susceptibility (χ) for NaRu₂O₄ was measured in the temperature range between 1.9 ≤ T ≤ 300 K with a 0.5 T of the external field applied perpendicular to the length of the as-grown crystal as in Fig. 1b. No magnetic ordering was observed in the measured temperature range. It is noticeable that the susceptibility is almost temperature-independent over the wide temperature range, showing a sizeable paramagnetic behavior consistent with the reported susceptibility data on the polycrystalline sample¹⁵. The low-temperature upturn is apparently due to some magnetic impurities, and it can be explained by using a modified Curie–Weiss formula as discussed in Supplementary Note 2 (see the red line in Fig. 1b). The fit yields the following values: Curie–Weiss temperature θ_CW = −1.8 K, observed moment μ_eff = 0.04 μ_B per Ru atoms and χ₀ = 1.15 × 10⁻⁷ (cm³/Ru mol)⁻¹, where the latter is the temperature-independent van Vleck contribution to the susceptibility. The Curie–Weiss fit gives a significantly smaller observed moment, indicating negligible localized magnetic moments in NaRu₂O_4, similar to the reported polycrystalline sample¹⁵.

The heat capacity was measured between 1.9 ≤ T ≤ 300 K and is displayed in Fig. 1c. The heat capacity did not show any λ-type anomaly over the entire temperature range, confirming no long-range magnetic order from 300 to 1.9 K. Low-temperature heat capacity data were analyzed using the usual formula of C_p = γT + βT³, where γ is the electronic specific-heat coefficient and β is the phonon contribution at low temperature (see the inset of Fig. 1c). The electronic contribution to the specific heat (γ) for NaRu₂O₄ was 3.93 mJ/mol K², consistent with the metallic behavior.

Single-crystal XRD data and crystal structures at high and low temperatures

To understand the high-temperature phase transition seen in the resistivity data, we conducted high-resolution single-crystal diffraction experiments to obtain the most informative data. We performed the temperature-dependent SC-XRD experiment from 300 to 575 K using a single crystal diffractometer (XtaLAB P200, Rigaku) (see Fig. 2). In our room-temperature data, we observed q = (0, ½, 0) superlattice peak (satellite reflections). It is important to note that this presence of the satellite reflection cannot possibly be explained by the Pnma space group proposed in ref. ¹⁵.

Fig. 2: Room-temperature SC-XRD refinement for LT-phase: Single-crystal XRD reciprocal-lattice images and lattice parameters across the structural transition.

a The SC-XRD refinement includes a modified structural model with a new space group (P112₁/a) determined from CBED experiments. The refinement is performed by including a monoclinic merohedral twin model under twin law as (-100) (010) (00-1). The solid red and blue circles are fundamental and superlattice reflections, respectively. The inset of the figure shows the optical image of the measured crystal. b Single-crystal x-ray diffraction data measured at various temperatures below and above the phase transition (T_c = 535 K) for NaRu₂O₄. c The cell parameters with error bars of NaRu₂O₄ at different temperatures extracted from single-crystal XRD data. The vertical line corresponds to T_c.

While searching battling for a possible true structure at room temperature, we decided to employ the convergence beam electron diffraction (CBED) as it is the most powerful technique in determining the space group and gives crucial information about the lattice’s symmetry¹⁸ (see Fig. 3). The weak superlattice reflections in the x-ray diffraction experiments already form strong evidence for broken symmetry (both translation and rotational) and lattice distortion in a crystal. In addition, the superlattice peak at q = (0, ½, 0) disappears upon heating just above T_c. We collected CBED diffraction patterns along the various [100], [130], [001], and [201] axes as shown in Fig. 3a–c at room temperature. The CBED experiment revealed P112₁/a symmetry (no. 14 with setting unique axis c¹⁹) at room temperature, a subgroup of the reported Pnma symmetry. Therefore, a modified structure model under group–subgroup relation is used to solve the room temperature SC-XRD data. We used the Shelx software²⁰ to address the twin problem in our refinement. In the twin refinement for NaRu₂O₄ SC-XRD data, we used a monoclinic merohedral twin model under twin law as (-100) (010) (00-1) to get better refinement parameters and statistics as well. Supplementary Note 3 provides details of monoclinic merohedral twin model under twin law. All the reflection patterns collected at 300 K could be indexed as monoclinic P112₁/a symmetry with a primitive lattice of a = 9.273 (6) Å, b = 5.643 (3) Å, and c = 11.17 (7) Å.

Fig. 3: The converge beam electron diffraction (CBED) patterns at room temperature in NaRu₂O₄ crystal.

The CBED patterns are taken with (a) [001], (b) [100], and (c) [130] incidences. Whole pattern symmetries of the CBED patterns are two-fold symmetry for [001] incidence, mirror symmetry perpendicular to the c-axis for [100] and [130] incidences. Dynamical extinction lines were observed at 0 0 l_odd and h_odd k 0 reflections at [130] incidence.

Table 1 summarizes the SC-XRD refinement results for the low-temperature (LT) phase (300 K). All the crystallographic sites were considered to be fully occupied and were kept fixed during the refinement. We carried out the structure analysis by including the twin structure of monoclinic cells. The structural parameters were finally determined from the least-squares refinement of the single-crystal XRD data. For the single-crystal refinement, we used a total of 5021 reflections, and all the reflections were well indexed by a monoclinic a₀ × 2b₀ × c₀ cell. We used P112₁/a (no. 14) with a table setting choice: c1. No constraints were placed on the atomic positions. The comparison of calculated and observed intensity (F²) after the structural refinement at room temperature is shown in Fig. 2a.

Table 1 The single-crystal x-ray diffraction measurement of NaRu₂O₄ at 300 K is used for the refinement.

As regards the high-temperature (HT) phase, our single-crystal data could not uniquely solve the structure above T_c (HT-phase) due to the sizeable diffuse scattering. Most probably, at such high temperatures, lighter Na ions located in 1D tunnels of NaRu₂O₄ start to move within the tunnels. In this situation, we decided to begin with positional parameters of distorted phase (Table 2, LT-phase with a₀ × 2b₀ × c₀ P112₁/a from Shelx refinement with twin), and generate undistorted positional parameters (undistorted P112₁/a a₀ × b₀ × c₀). The undistorted positional parameters of P112₁/a symmetry are used to get positional parameters for orthorhombic HT-phase of Pnma symmetry under cell-symmetry and group-subgroup relation¹⁹. Expected coordinates at HT Pnma phase based on LT P112₁/a structure (a₀ × b₀ × c₀) are shown in Table 2. Excellent agreement is seen between expected positions and reported one¹⁵ (by missing superlattice reflections: a₀ × b₀ × c₀), as shown in Table 3.

Table 2 Expected coordinates at HT Pnma phase based on LT P112₁/a structure (a₀ × b₀ × c₀) (atoms are at 4c site¹⁹).

Table 3 Reported LT Pnma structure (a₀ × b₀ × c₀) (atoms are at 4c site)¹⁹.

The crystal structure of NaRu₂O₄ is close to that of the prototype compound CaFe₂O₄ with similar coordination environments. In our revised crystal structure for NaRu₂O₄, the most characteristic feature we found is the edge-sharing octahedra double RuO₆ chain, which runs along the crystallographic b-axis. These double chains form a two-leg zigzag ladder along the crystallographic b-axis. The RuO₆ octahedra are edge-sharing within each chain, tied to the neighboring chain through the corner oxygen. This leads to an interesting lattice geometry with pseudo-triangular tunnels, in which a single Na atom can be accommodated, as shown in Fig. 4a, which is also found in various CaFe₂O₄-type lattices^{21,22,23,24,25,26}. Our detailed single-crystal x-ray diffraction analysis shows a transition from the HT (high-temperature) Pnma structure to the low-temperature (LT) P112₁/a structure at T_c = 535 K, as shown in Fig. 4b, c: which is consistent with the results of resistivity measurements shown in Fig. 1a.

Fig. 4: High-temperature and low-temperature crystal structures.

a The HT-phase in NaRu₂O₄ has an orthorhombic structure with a Pnma space group, similar to the CaFe₂O₄-type structure¹. The octahedra and Ru atoms are represented in light brown and O atoms in red color. b The HT-phase crystal structure (Pnma) of NaRu₂O₄ in bc-plane. c The LT-phase crystal structure of NaRu₂O₄ has monoclinic symmetry P112₁/a.

According to our diffraction experiments done between 300 ≤ T ≤ 575 K, the q = (0, ½, 0) superlattice peak (satellite reflection) persist right up to 535 K as shown in Fig. 5. The line cut from the temperature-dependent reciprocal image analysis for peak q = (0, ½, 0) shows clearly the suppression of intensity as a function of temperature as in Fig. 5b, c. The q = (0, ½, 0) superlattice peak is substantially suppressed just above the first-order transition (T_c = 535 K).

Fig. 5: Single-crystal x-ray diffraction (SC-XRD).

a Reciprocal lattice maps of the SC-XRD data for NaRu₂O₄ obtained at 470 and 575 K with the first-order transition at T_c = 535 K. b The line cut for various temperatures below and above the first-order phase transition as extracted from the reciprocal space analysis. c Temperature dependence of the (4 3 0) super-lattice peak.

Charge ordering and bond valence sum (BVS)

The structure of NaRu₂O₄ can be visualized as the double chain of edge-sharing RuO₆ octahedra running in the b-direction and forming 2-leg zigzag ladders (see Fig. 6a). The corner-sharing oxygen atoms connect these ladders, also having a zigzag pattern. In effect, there appear in this system, not one, but three different types of zigzag ladders: those in edge-sharing double chains (ladders-2 marked by red in Fig. 6b. For shortness, we will call them “red” ladders), and two types of corner-sharing zigzag ladders, marked in Fig. 6b, c by blue and green. When describing this class of materials, one often pays primary attention to the phenomena occurring in the “red” ladders (see ref. ²⁷). But the corner-sharing ladders are sometimes found to play a more critical role in systems like hollandite K₂Cr₈O₁₆²⁸. As we will see below, these edge-sharing ladders also play a crucial role in NaRu₂O₄.

Fig. 6: Crystal structure of the tunnel compound NaRu₂O₄.

a Top view of the low-temperature structure. b Illustration of different types of zigzag ladders: “red” ladders with edge-sharing RuO₆ octahedra; “green” and “blue” ladders with corner-sharing octahedra. c The detailed structure of corner-sharing and edge-sharing ladders.

Here it is worthwhile to think about possibility of certain effect due to off-stoichiometry. In particular, one can ask whether the observed metallic behavior is related to a sample quality issue. For instance, it is known that such stoichiometry plays an important role in determining the low-temperature properties of RuP²⁹, whose high-quality powder samples exhibit a metal–insulator transition. On the other hand, single crystal of RuP, presumably off-stoichiometric, has a metallic ground state. For NaRu₂O₄, we can rule out this issue as our powder sample also shows a metallic behavior (see also Supplementary Fig. 2). Another point is that our heat capacity measurements taken on two different samples consistently show a finite gamma value at low temperature, which is consistent with the data reported in ref. ¹⁵.

An exciting picture now emerges from the detailed structural study. The HT-phase has two independent Ru sites (Ru₁ and Ru₂), equally spaced (2.82 Å) with the lattice constant b₀ along two Ru chains. The most prominent feature of the structural changes is the appearance of strong dimerization in Ru chains (legs of zigzag ladders), leading to cell-doubling in the b-direction (see Fig. 7a). Ru–Ru distance in short Ru dimers is 2.60 Å—even shorter than the Ru–Ru distance in Ru metal (2.65 Å). It is a clear sign of direct metal–metal bonding in such dimers—which quite unexpectedly coexist with metallic conductivity. Simultaneously with the dimerization in 1D Ru chains—legs of zigzag ladders, Ru sites themselves become inequivalent, Ru(A) and Ru(B), a clear sign of a site-centered charge ordering in NaRu₂O₄. The coexistence of bond- and site-centered orderings is extremely rare in metallic systems, making NaRu₂O₄ especially interesting.

Fig. 7: Superstructure in the low-temperature phase and the results of the LSDA + U band structure calculations.

a The unit-cell of the low-temperature phase with two dimerized Ru bonds and the valence of Ru as determined from bond valence sum (BVS) calculations. b Structure of the blue corner-sharing ladder with different valence states of Ru and different bond lengths marked. c–f Orbital-resolved DOS for four nonequivalent Ru atoms calculated within LSDA + U with U = 3.7 eV and J = 0.7 eV, assuming the charge order shown in (a). Ru orbitals are defined in local frames with the axes directed to the nearest O ions.

Another essential feature of this superstructure is the difference of distances in the inter-chain bond length (interdimer), both in edge-sharing “red” and in corner-sharing “green” and “blue” ladders (Fig. 6b). The Ru₁(A)–Ru₁(A) and Ru₂(A)–Ru₂(A) distance in the edge-sharing ladder is 3.10 and 3.09 Å, while the Ru₁(A)–Ru₁(B) and Ru₂(A)–Ru₂(B) distance are 3.12 and 3.11 Å, respectively, with the remaining distance between dimers (Ru₁(B)–Ru₁(B) and Ru₂(B)–Ru₂(B)) being 3.16 Å (see Fig. 7a, b). From the viewpoint of the bond order, short bonds have a shape like a “letter-Z” (see Fig. 8). The same is true for two other types of ladders, edge-sharing “blue” and “green” ladders. We thus see that the dimers also form Z-type orbital clusters in corner-sharing ladders. The relative difference of interdimer bonds in these is even more prominent than in the “red” edge-sharing ladders: it is ~1.7% compared to ~0.7% in the “red” ladders. This already hints that what happens in corner-sharing ladders is probably more critical than in edge-sharing ones, as in hollandite K₂Cr₈O₁₆²⁸. According to our ab-initio calculations based on the charge valence of our choice [Ru₁(A)³⁺, Ru₁(B)⁴⁺, Ru₂(B)⁴⁺, Ru₂(A)³⁺), the nn hoppings in corner-sharing blue and green zigzags are: for blue corner-sharing bonds t = 0.40 eV, for green corner-sharing ones t = 0.29 eV, and for red edge-sharing bonds t = 0.35 eV. For other charge ordering the values are somewhat different but the blue corner-sharing bonds are always the strongest. It shows the dominance of the blue bond among all relevant bonds.

Fig. 8: Z-order in the two-leg corner- and edge-sharing RuO₆ zigzag ladders.

The sketch of self-organized Z-clusters in the low-temperature phase, formed within the edge-sharing red and corner-sharing blue ladders of NaRu₂O₄ lattice in the low-temperature phase.

For the determination of the oxidation states of cations (Na and Ru), we carried out calculations by using the BVS method with an inbuilt program in Fullprof software³⁰. The principle of BVS method in determining the valence V from different cation sites^31,32 can be given by an expression V = Σ_i[exp(d₀−d_i)/0.37]. Here, d_i is the experimental bond lengths to the surrounding ions, and d₀ is a tabulated empirical value characteristic for the cation-anion pair^31,32. The BVS result for NaRu₂O₄ was determined for the LT-phase. The charge at different Ru sites is illustrated by different colors (brown-Ru⁴⁺ and green-Ru³⁺) in Figs. 6 and 7. The BVSs for the eight oxygen sites were between 1.86 and 2.20. The BVSs for the four Ru cations sites for different charge configurations (we call them four different models) are shown below in Table 4. The respective reliability parameter as Global Instability Index (GII) is also listed for each BVS model for NaRu₂O₄.

Table 4 The four different models corresponding to four options for two dimers: counting from the left, (3+, 4+) in the upper dimer, (3+, 4+) in the lower dimer.

To be more technical, an important question is how different Ru valence states are distributed in the low-temperature phase. Specifically, one has to know which ions, in which charge state, are connected by these shorter zigzag rungs in orbital Z clusters. For that, we used the results from the BVS analysis at each Ru, an empirical measure of its charge or its valence state^33,34. In the HT-phase, all Ru are equivalent, and BVS gives the valence Ru^3.5+. In contrast, Ru ions in a short dimer are indeed inequivalent in the LT-phase, with the valence estimated from BVS being 3.536 and 3.106; that is, one can think of Ru⁴⁺ and Ru³⁺ in each dimer. However, from our structural data alone, although collected with high-resolution single-crystal x-ray diffraction and the total number of 5021 Bragg peaks, we cannot uniquely discriminate the four possible charge ordering patterns, mainly because an x-ray experiment fundamentally suffers from the weak scattering signals of oxygen. The reliability parameters (GII) for four different charge ordering patterns are comparable to one another, about 10–11% (see Table 4). Still, a slightly better index is obtained for charge distribution with Ru⁴⁺ ions at the ends of short diagonal bonds in blue and green ladders (reliability factor 10.43% for model 3 in Table 4). In contrast, these are Ru³⁺ in edge-sharing red ladders. Note that different ladders are connected in the structure of NaRu₂O₄ so that the charges at short zigzags in neighboring ladders are opposite (see Fig. 8).

Theoretical studies

As we commented above, there is unavoidable ambiguity about our determination of the charge valence using the BVS method although we used the total number of 5021 Bragg peaks from the high-resolution single-crystal x-ray diffraction experiment. Therefore we employed ab-initio LSDA and LSDA + U calculations to clarify the charge pattern favored by the LT crystal structure. We used different Coulomb repulsion U values in the range 2.7–4.2 eV and Hund’s coupling J_H = 0.7 eV as estimated from LSDA. All calculations were performed for the experimental HT and LT crystal structures. No attempts to optimize lattice geometry have been made. The effect of SOC on Ru t_2g bands was relatively weak. In the following, for simplicity, we discuss the results of scalar-relativistic calculations. Our theoretical studies show that different self-consistent spin and charge-ordered solutions can be stabilized for sufficiently large U (3.7 eV). These solutions have the following common features: Ru d_xy states (in local coordinates, with the axes directed from Ru to O) are strongly split into occupied bonding and unoccupied antibonding molecular orbital states (see Fig. 7c–f). Second, each dimer’s Ru ions become formally Ru³⁺, with its d_xz, d_yz orbitals being doubly occupied. Another Ru ion of the dimer, Ru⁴⁺, has one half-occupied orbital, approximated by a linear combination of d_xz and d_yz lying in the plane perpendicular to the b-axis. The Ru⁴⁺ ion acquires a spin moment of about 0.8 μ_B. The orbital-resolved density of Ru d states illustrates the charge disproportionation. With decreasing U, the hybridization becomes more robust between the unoccupied d_xz, d_yz orbital of Ru⁴⁺ and occupied t_2g orbitals of Ru³⁺, a gap separating the unoccupied d_xz, d_yz band from the top of the valence band closes, and for U ~ 2.7 eV the charge disproportionation between Ru³⁺ and Ru⁴⁺ ions disappears (see Fig. 7c–f). It should be noted that this is only a rough estimate for the critical U value which may change if optimized crystal structure were used in the calculations and is in general implementation specific.

Out of the four charge ordering patterns allowed for the LT structure, three are nearly degenerate, with the energy difference being <3 meV/f.u. This finding agrees with the BVS analysis results, which give comparable reliability factors for different charge ordering. Still, the lowest total energy is found for one charge ordering pattern: for which the lowest value of GII was obtained experimentally from BVS. The final charge pattern is that the short zigzag bonds in Z-cluster in corner-sharing “blue” ladders connect Ru⁴⁺ - Ru⁴⁺ ions (see Fig. 8).

Finally, we would like to note that the structure of NaRu₂O₄ is composed of frustrated 2-leg zigzag ladders with a spin-singlet ground state, which makes this material quite interesting. For more details see Supplementary Figs. 4 and 5 in Supplementary Note 4. As shown in Fig. 9a, b, the orthorhombic Pnma crystal structure of NaRu₂O₄ has two ruthenium atoms Ru₁ and Ru₂. The edge-sharing bond between the two RuO₆ octahedra for each dimer 1 (2.60 Å), dimer 2 (2.61 Å) and Z diagonal-bond (3.10 Å) give rise to two Ru–O–Ru bonds which are deviated from 90° for each and also all of the edge-sharing octahedral shows significant trigonal-distortion. The schematic representation shows the double chain-forming two-leg zigzag ladders (solid thick light-green, reddish-brown, and sky-blue lines) running along the crystallographic b-axis in crystal. The 2-leg zigzag ladders are connected in two different forms: the corner- (blue or green ladders) or the edge- (red ladders) sharing RuO₆ octahedra. The t_2g energy-level sketch with respective electron fillings is shown for a charge-ordered state, so-called Z-order (left panel) and Ru-dimer (right panel) in Fig. 9d.

Fig. 9: CaFe₂O₄-type tunnel structure and edge-sharing RuO₆ two-leg ladders in NaRu₂O₄ lattice.

a The orthorhombic Pnma crystal structure. b The bond between the two RuO₆ octahedra for each dimer 1 (2.60 Å), dimer 2 (2.61 Å), and Z diagonal-bond (3.10 Å). c Schematic representation of the double chain with two-leg zigzag ladders (solid thick light-green, reddish-brown, and sky-blue lines). d The two-leg zigzag ladders in two different forms, either corner- (blue or green ladders) or edge- (red ladders), sharing RuO₆ octahedra.

Conclusions

Thus the following final picture emerges from both the experimental and theoretical studies. NaRu₂O₄ has very strong dimerization below T_c = 535 K (in other words, strong metal–metal bonding with the use of xy-orbitals) in one-dimensional chains forming legs of zigzag ladders (this is a bond-centered CDW). With the average valence Ru^3.5+, one active electron (or one hole) remains per a dimer. And simultaneously with this dimerization (actually the orbitally driven Peierls dimerization^35,36,37,38), these extra electrons order in each dimer—there appears also a site-centered charge ordering or site-centered CDW. The remaining electrons from one stronger diagonal bond between dimers in neighboring legs, creating Z-clusters (Fig. 8). The detailed form of orbitals in forming this short zigzag bond in the “blue” corner-sharing ladders is shown in Fig. 9c. But this extra bonding is not strong enough to make the system insulating, and it remains metallic below T_c due to the multi-orbital effect. These Ru–Ru dimers probably survive above T_c, forming dimer liquid as, e.g., in Li₂RuO₃³⁹; this dimer liquid phase in Li₂RuO₃ was found stable upon doping too⁴⁰. This could be checked by probes like EXAFS or PDF (pair distribution function) analysis.

To put our results in a broader perspective, we would like to comment on why the dimerization and the site-centered charge ordering coexistence are favorable for metallic NaRu₂O_4, while it has been rarely seen in other transition metal compounds: another example is IrTe₂⁴¹. Most importantly, the Ru 4d bands of NaRu₂O₄ exhibit the delicate balance between the correlations, the Coulomb U, and the bandwidth W. Two other factors play a crucial role: orbital freedom, leading to orbitally-driven dimerization, and the mixed-valence of Ru, Ru^3.5+, promoting site-centered charge ordering. These features seem to be the key to realizing charge ordering and dimer coexistence in a metallic NaRu₂O₄. Going further, we anticipate that our observations will prompt renewed efforts towards the hitherto scarcely investigated Mott–Hubbard physics in a metallic regime.

Methods

Sample preparations

Polycrystalline NaRu₂O₄ samples were synthesized by solid-state reaction of preheated RuO₂ (99.999%, Aldrich) and Na₂CO₃ (99.999%, Aldrich) under an Ar-gas environment at 950 °C for 90 h with several intermediate grindings and pelletization. Subsequently, high-quality single crystals of NaRu₂O₄ were grown from this polycrystalline powder via a modified self-flux vapor transport reaction under flowing Ar-gas (ultra-pure 99.999%). Long needle-shaped high-quality single crystals (1 × 0.1 × 0.1 mm³) were obtained from the final products. The synthesized polycrystalline sample’s phase purity was checked using a Bruker D8 Discover diffractometer with a Cu-Kα source with no impurity peaks observed. Elemental analysis was subsequently done confirming the samples’ stoichiometry: we used a COXI EM-30 scanning electron microscope equipped with a Bruker QUANTAX 70 energy dispersive x-ray system.

Bulk properties

Electrical resistivity (ρ) measurements were carried out using a homemade system equipped with a furnace (300–685 K) and a pulsed-tube cryostat (down to 3 K, Oxford). The electrical resistance was measured in the four-point geometry on a static sample holder, where the contacts to the sample were made using silver paint and 25 µm gold wire. The current was applied perpendicular to the single crystal length, the crystallographic b-axis. Magnetic susceptibility χ(T) measurements were taken using an MPMS-SQUID magnetometer (Quantum Design). Heat capacity C_p(T) measurements were made using the commercial Physical Property Measurement System (PPMS, Quantum Design).

Structure analysis

The temperature-dependent single-crystal x-ray diffraction (SC-XRD) was performed from 300 to 575 K using a single crystal diffractometer (XtaLAB P200, Rigaku). The room-temperature SC-XRD data exhibit superstructures q = (0, ½, 0). The crystal structure was refined by using Fullprof suite software and the Shelx program with a modified monoclinic merohedral twin model under twin law as (-100) (010) (00-1) are presented in Supplementary Note 3. We also carried out temperature-dependent SC-XRD measurements on NaRu₂O₄ crystal to confirm the structural phase transition. Besides, we made a convergent beam electron diffraction (CBED) experiment, which is the most accurate in determining the correct symmetry of materials, especially regarding the loss of the inversion center. This CBED measurement allows us to determine the exact space-group of NaRu₂O₄. The CBED result for NaRu₂O₄ gives P112₁/a symmetry and works well for the observed commensurate superstructures with q = (0, ½, 0).

LDA band calculations

Band structure calculations were performed for the experimentally obtained high- and low-temperature crystal structures using the LMTO method as implemented in the PY LMTO computer code⁴² and the Perdew–Wang⁴³ parameterization of the exchange-correlation potential in the local spin density approximation (LSDA). To account for Coulomb repulsion within Ru d shell, we used a rotationally invariant formulation of the LSDA + U method⁴⁴ with the double-counting term in the fully localized limit. An LDA calculation for the HT crystal structure gave a nonmagnetic metallic solution with partially occupied Ru₁ and Ru₂ t_2g orbitals. The band structure’s characteristic feature is Ru t_2g bands with stable quasi-one-dimensional dispersion along the b direction. These bands originate from Ru d_xy orbitals and are evidence of strong direct Ru d_xy–d_xy hopping along the b chains. Here and in the following, Ru orbital is defined in local frames with the axes directed approximately to three nearest O ions in such a way that the local [110] direction always points along the crystallographic b-axis. The LDA band structure calculated for the LT structure is qualitatively different. Although it remains nonmagnetic and metallic, the shortening of the Ru-Ru distance within dimers leads to the splitting of Ru d_xy drives bands into completely filled bonding states 2 eV below E_F and unoccupied antibonding ones 1 eV above E_F. The bonding d_xy bands accommodate one electron per Ru while the remaining 3.5 d electrons fill bands are formed by Ru d_xz, d_yz states.