Introduction

Superconductivity (SC) is a peculiar property of materials characterized by no resistance to direct current and expulsion of magnetic fields, hence enabling many applications without energy loss. Yet, its practical implementation is limited by the low critical temperature Tc reaching at best 135 K in the famous cuprates at ambient conditions1. SC is usually explained by the presence of bound electrons into Cooper pairs2. However, the origin of the attractive interaction between electrons still has to be understood and unified for all compounds. Among all superconductors, materials based on the An+1BnO3n+1 Ruddlesden-Popper structure attract the largest interest of solid-state physicists and chemists (Fig. 1a for n = 1). These structures are based on corner-sharing O6 groups centered on B cations, forming a building ABO3 perovskite block. Then, n perovskite blocks are stacked along the c-axis and are separated by a spacing AO layer. This generic structure hosts most oxide superconductors, entailing cuprates1 and ruthenates3 (La2-xSrxCuO4 and Sr2RuO4, n = 1 member), antimonates4 and bismuthates5,6 (Ba1-xKxSbO3, Ba1-xKxBi1-yPbyO3, n = ∞ members) and nickelates7,8,9 under high pressures (La3Ni2O7, n = 2) or after a chemical reduction aiming at removing apical oxygen (Nd6Ni5O12, n = 5 member, or R1-xAxNiO2, n = ∞ with R=La, Pr or Nd and A = Ca or Sr).

Fig. 1: Structural and electronic features exhibited by ruthenates A2RuO4.
figure 1

a Crystallographic structure exhibited by n = 1 undistorted Ruddlesden-Popper phase. The undistorted cell adopts an I4/mmm symmetry based on ABO3 perovskite building blocks stacked along c and separated by one AO layer. b Jahn-Teller distortion for one ABO3 layer, corresponding to a Q2 mode, producing a contraction and elongation of two B-O bond lengths with alternating contraction/expansion on nearest neighbor B sites. c–e Ru d states splitting according to O6 octahedral deformations for totally undistorted O6 groups (c), compressed (d), or extended (e). The distortion is determined by the co and ao ratios defined in (a). ∆CFo and ∆ex are octahedral crystal fields and exchange splitting, respectively. Majority (minority) spin channel is represented in blue (red). The point group symmetry for the undistorted octahedral situation is Oh, that can be lowered to D2h by a co/ao ≠ 1 and produce an additional ∆CF’ splitting.

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Interestingly, bismuthates, antimonates, and some nickelates are superconductors due to an electron-phonon mechanism related to a charge and bond ordering (CBO)10,11,12,13,14, whose origin is the existence of an unstable formal oxidation state δ (FOS) of B cations in the doping phase diagram. It produces a rock salt pattern of B(δ+1)+ and B(δ−1)+ cations and hence a charge ordering. It is accompanied by a bond disproportionation Boc mode producing a bond ordering (BO). It thus presents a strong electron-phonon coupling (EPC) and is able to produce an insulating phase (see Supplementary Note 1). Starting from this CBO-insulating phase, doping the materials weakens the EPC and produces a metal without any stable CBO. However, this does not imply that the BO mode is not coupled to the electronic structure, since the EPC remains sufficiently large to produce an attractive interaction that binds electrons into Cooper pairs. This mechanism is also at the core of SC in many other non-oxide materials15,16,17,18, showing that the same phonon can be responsible for the insulating and superconducting state in the doping phase diagram.

Nonetheless, the CBO mode is not the only phonon that can produce an insulating phase in oxides. The other type of phonon is the one associated with the Jahn-Teller effect (JTE) and its related Jahn-Teller distortion (JTD)19,20 (see Supplementary Note 1). According to the Jahn and Teller theorem, a degenerate electronic configuration, such as one electron for two or three degenerate orbitals, is unstable and can produce a lattice distortion that breaks the degeneracy21. The JTE then produces occupancy of alternating orbitals between all nearest neighbor B sites, leading to an orbital-ordering (OO). It is accompanied by a so-called Jahn-Teller distortion (JTD)20 deforming octahedra in order to accommodate the OO. Hence, it yields electron localization and can produce an insulating state (Supplementary Note 1). Famous examples fulfilling all modalities for a JTE are KCuF3 or RVO3 (R = Lu-La, Y)20,22,23.

The presence of a degenerate electronic configuration is a prerequisite but not a sufficient condition for a JTE (or disproportionation effects)14,20,24: if the electronic structure is too hybridized, the electronic instability at the core of the JTE is screened, and it cannot enforce an OO and ensuing JTD. In view of the similarity to the situation of SC mediated by charge and bond ordering distortions, it is natural to ask if the JTE and its resulting JTD and OO could produce a SC transition in a metallic compound and if there are any materials that sit in the vicinity of an orbital-ordered phase. To the best of our knowledge, this remains elusive.

Good candidates for a JTE are the ruthenates A2RuO4 (A = Ca, Sr). These compounds are n = 1 members of the RP series with Ru4+ cations in a d4 electronic configuration25,26. The octahedral crystal field (CF) ∆CFo is greater than the exchange splitting ∆ex and lifts the d orbital degeneracy in two groups of degenerate partners (Fig. 1c). Consequently, Ru4+ cations are in a t2g,↑3 t2g,↓1 electronic configuration that is nominally Jahn-Teller active. Nevertheless, due to the A-to-Ru cation size mismatch, Ca2RuO4 and Sr2RuO4 distort from the I4/mmm cells with equivalent co and ao Ru-O bond length in O6 groups (see Fig. 1a) to a distorted cell with distinct co and ao values. Ca2RuO4 adopts a Pbca symmetry25 characterized by a-a-c+ octahedral rotations in Glazer’s notation27 and a co/ao ratio lower than 1 (co/ao = 0.981, see Table 1). The local symmetry breaking induced by steric effects produces an additional CF ∆CF' lifting the degeneracy of t2g levels, stabilizing the dxy orbital over the dxz/dyz doublet (Fig. 1d). It allows the electron localization in the dxy orbital in the minority spin channel and thus a band gap opens. It explains the insulating state of Ca2RuO4 below 357 K19,28,29 with a band gap of at least 0.4 eV30. In contrast, Sr2RuO4 adopts an elongated I4/mmm cell at 300K26 inducing a co parameter greater than ao (co/ao = 1.069, Table 1) and thus here ∆CF' stabilizes the dxz/dyz doublet over the dxy orbital. One is then left with a single electron for two degenerate orbitals in the minority spin channel (Fig. 1e). Although possessing the precursor to a JTE, Sr2RuO4 is metallic and superconducting with a critical temperature Tc of roughly 1.5 K3,31. Fulfilling the condition for a JTE, Sr2RuO4 is a potential candidate for exhibiting SC at the vicinity of an OO phase with an EPC mediated by the subsequent JTD.

Table 1 Key properties of A2RuO4 compounds
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In this article, using parameter-free Density Functional Theory (DFT) calculations, we reveal that Sr2RuO4 is intrinsically in the vicinity of a Jahn-Teller effect due to a hybridized electronic structure. Then, an electron-phonon coupling (EPC) associated with the related Jahn-Teller distortion explains the Tc observed experimentally. The model further reproduces a superconducting state in Ca2RuO4 nanofilms with a Tc of 63–73 K compatible with the experimental value oscillating between 64 or 80 K32. Thus, superconductivity in oxides with partly filled d states might be given by the same electron-phonon interactions that produce the insulating phases19.

Results

We first perform the structural relaxation (atomic positions plus cell parameters) of bulk Ca2RuO4 and Sr2RuO4 considering several magnetic solutions. Table 1 reports the relevant structural parameters as well as the gap amplitude and Ru magnetic moments. Firstly, the non-magnetic (NM) solution is metallic and at least 84 meV/f.u (i.e., 1000 K/f.u) above any spin-polarized solutions for the two compounds. It highlights that the NM approximation is not suitable to describe the properties of ruthenates. Using spin-polarized solutions, Ca2RuO4 is a C-type antiferromagnet (AFM-C) insulator in the ground state with a gap Eg of 0.574 eV, in agreement with experimental results showing an AFM transition at 110 K and a band gap greater than 0.4 eV30,33 (Fig. 2a). The gap is formed between occupied dxy and unoccupied dxz/dyz orbitals due to a computed co/ao ratio lower than one as anticipated in Fig. 1d. In contrast to Ca2RuO4, Sr2RuO4 is metallic with ferromagnetic (FM) interactions between Ru cations, compatible with experimental reports suggesting the presence of short-range FM order34 and metalicity3. Furthermore, the dxz and dyz orbitals mostly contribute to states at the Fermi level EF, in agreement with the situation for a computed co/ao ratio greater than one displayed on Fig. 1e. In both compounds, the computed magnetic moments, m = 1.413 µB/Ru and m = 1.364 µB/Ru in Ca2RuO4 and Sr2RuO4, respectively, are compatible with a Ru4+ cations in a low spin state (\({{{\rm{t}}}}_{2{{\rm{g}}}\uparrow }^{3}{{{\rm{t}}}}_{2{{\rm{g}}}\downarrow }^{1}\), S = 1) albeit the amplitudes deviate from the expected value of 2 µB/Ru. This discrepancy originates from a hybridized electronic structure between Ru-d and O-p states producing spillage of magnetic moments on surroundings O anions (Fig. 2). The values are however in agreement with the experimental value of 1.3 µB in Ca2RuO433 and previously computed to 1.38 µB in Sr2RuO435.

Fig. 2: Electronic structure of Ca2RuO4 and Sr2RuO4 ground states.
figure 2

Density of states (in states/eV/f.u) for spin up (positive values) and spin down (negative values) channels as a function of the energy (in eV) projected on Ru d (black line) and O p (blue filled area) states for Ca2RuO4 (a) and Sr2RuO4 (b) ground states. Projection on dxy (red line), dxz and dyz (grey line), and dx2-y2 and dz2 (orange line) orbitals around the valence and maximum is provided as inserts. The AFM-C and FM orders are used for Ca2RuO4 and Sr2RuO4, respectively. The black dashed vertical line represents the valence band maximum or the Fermi level.

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The structural relaxations correctly predict the Pbca space group for all spin-polarized solutions with an a−a−c+ octahedral rotation pattern in Ca2RuO4, with amplitudes of distortions in agreement with experimental data available in literature25 (Table 1), at the exception of the PM order. The lower symmetry exhibited by the PM phase is explained by the introduced spin disorder in our simulations that removes symmetry operations (see methods). Nevertheless, the relaxed structure is still characterized by the a−a−c+ octahedral rotations with amplitudes compatible with experiments. In Sr2RuO4, we identify that the 0 K relaxed structure adopts a Cmca space group characterized by a0a0c+ O6 groups rotation for all magnetic orders at the exception of AFM-C (Pbam) and PM (Pm) orders that have lower space groups. The Cmca contrasts with the I4/mmm cell proposed experimentally26. However, the experimental structure is obtained at 300 K, and no structures at low temperatures are reported so far for Sr2RuO4 to the best of our knowledge, hindering the comparison with our first-principles 0 K structure. The lower symmetry exhibited by the DFT-PM relaxation of Sr2RuO4 is again explained by the disorder introduced on the spin-ordering. The Pbam symmetry, a lower space group to Cmca, exhibited by the AFM-C order is discussed below.

The correct trend of co/ao ratio is obtained from the DFT simulations with respect to experiments for Ca2RuO4 and Sr2RuO4 regardless of the imposed magnetic order with the exception of the NM solution in Ca2RuO4. In Ca2RuO4, the co/ao ratio is lower than one and yields an insulating state whatever the spin-polarized solution. In contrast, although the co/ao is greater than 1, whatever the spin orderings in Sr2RuO4, all magnetic orders yield a metal except the AFM-C order, which produces a gap Eg of 0.1 eV. Using a symmetry mode analysis of the relaxed structure with respect to a high symmetry undistorted I4/mmm cell, we observe a sizable Jahn-Teller distortion only for the AFM-C order. It corresponds to a Q2-type mode with two Ru-O bond lengths contraction and extension, alternating on neighboring octahedra20,36,37,38 (see Fig. 1b). Ca2RuO4 also exhibits a non-zero JTD but for all magnetic orders. However, this is simply a consequence of a coupling between the JTD and the a-a-c+ type of octahedral rotations in Glazer’s notations27 and hence this JTD has an improper origin rather than a proper, electronic, origin (see Supplementary Note 2). Nevertheless, the absence of the appropriate octahedral rotations in Sr2RuO4 cannot explain its appearance only for the AFM-C order. Hence, a degeneracy of these two orbitals exists in Sr2RuO4.

In order to understand the appearance or not of a Jahn-Teller mode as a function of the magnetic orders in Sr2RuO4, we report on Fig. 3a the potential energy surface associated with the JTD for a NM, FM, and AFM-C solutions. To that end, we start from a perfectly undistorted cell of Sr2RuO4 with I4/mmm symmetry and freeze-in some amplitudes Q of the JTD. For the NM and FM solutions, we observe a single well potential whose minimum is located at QJTD = 0. Therefore, the Jahn-Teller mode is not willing to appear in the material. However, a shifted single well potential for the AFM-C order whose minimum is at QJTD ≠ 0 is observed with an energy gain of -49 meV/f.u with respect to the high symmetry I4/mmm cell. This signals the presence of an electronic instability associated with a single electron for two degenerate partners. Should the material be characterized by dominant AFM interactions, Sr2RuO4 would thus exhibit a Jahn-Teller effect, producing an insulating orbital ordered phase accompanied by a JTD with the Ru d electron in the minority spin channel located either in the dxz or dyz orbital on alternating first-nearest neighbors (see Fig. 3c and d). The lower Pbam symmetry observed for the AFM-C order in Sr2RuO4 with respect to all other spin-polarized solutions is related to a structural symmetry breaking induced by the JTE and ensuing JTD. It produces a band gap Eg of 0.12 eV (Table 1).

Fig. 3: Jahn-Teller effect in Sr2RuO4.
figure 3

a Energy gain ∆E (in mev/f.u) associated with the condensation of different amplitude QJTD (in Å/f.u) of the Q2-type JTD in Sr2RuO4 for the NM (red filled squares), FM (blue filled circles), AFM-C (green filled diamonds) and FM with a negative Pressure (FM-nP, orange filled circles) starting from an undistorted I4/mmm cell. b Density of states of Ru t2g states in the minority spin channel for the FM (upper part, blue filled area), FM + negative pressure (upper part, green line), and AFM-C order (lower panel, green filled area) within a totally undistorted I4/mmm cell. c Density of states (in states/eV/f.u) for spin up (positive values) and spin down (negative values) channels as a function of the energy (in eV) projected on Ru t2g (black line) and O p (blue filled area) states in Sr2RuO4 ground state with the AFM-C order. d Partial charge density map associated with states around the Fermi level indicated by the orange area in (c) for Sr2RuO4 with the AFM-C order.

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The discrepancy between FM and AFM-C orders can be understood by their different electronic structures. We report on Fig. 3b the projected density of states Ru t2g states in the minority spin channel in the high symmetry I4/mmm cells for FM and AFM-C orders. These two magnetic orders are characterized by very distinct bandwidth Wt2g associated with t2g levels, estimated to 2.04 eV and 1.08 eV for FM and AFM-C orders, respectively. Recalling that (i) electronic instabilities strongly depend on bandwidth—i.e., localized states/correlation strength– and (ii) that AFM orders improve band compacities14,20,24, the AFM-C order preserves the electronic instability toward JTE while it is screened with an FM order. This is confirmed by performing a calculation of the JTD potential with an FM order in which the Ru d states band compacity is improved by increasing the lattice parameters by 8%, thus creating a negative pressure effect (FM-nP, Wt2g = 1.48 eV, Fig. 3b) that yields a strong softening of the JTD potential (Fig. 3a). A DFT + U scheme with a U potential on Ru d states preventing electron delocalization leads to the very same conclusion (see Supplementary Note 3). Hybridizations were recently observed to have a similar role on disproportionation effects14. We conclude here that Sr2RuO4 is intrinsically at the vicinity of a JTE and subsequent orbital-ordered phase.

We report in Fig. 4a the band structure unfolded to the primitive I4/mmm cell of Sr2RuO4 in the FM ground state with a Cmca symmetry. The majority spin channel is gapped between occupied t2g and unoccupied eg states, as expected for a half-filled electronic configuration19. In the minority spin channel, three bands cross the Fermi level with (i) a parabola centered at the Γ point dispersing on roughly 3.52 eV with a dominant dxy character and (ii) two degenerate bands dominated by the dxz and dyz orbitals but dispersing only on 1.72 eV. No energy gaps are observed in the band dispersion, indicating that electrons are not experiencing interactions from the lattice.

Fig. 4: Band structure of Sr2RuO4 superconductors.
figure 4

a Band structure of the FM ground state of Sr2RuO4 (Cmca symmetry) unfolded to primitive I4/mmm cell and projected on O (blue), Sr (green), Ru dxy (red), dxz+dyz (grey) and dx2-y2 and dz2 (orange) states. Majority (left panel) and minority (right panel) are reported. b Band splitting ∆Eg induced by freezing a displacement uJTD of 0.066 Å/atom associated with the JTD Q2 -type mode in Sr2RuO4 ground state. c Evolution of ∆Eg as a function of the displacement uJTD (Å per atom) associated with the JTD Q2 -type mode. The computed REPME is D = 7.76 eV. Å−1. d Total density of states (in states/eV/f.u/spin, grey area) as a function of the energy using the DFT calculation (lower part) and atomic-like Wannier functions (upper part). The WFs allow to extract the contributions from the sole dxz and dyz orbitals (red line). The contribution from each dxz or dyz band is then of N(EF) = 0.291 states/eV/Ru/spin/band. The k-mesh is sampled with 12x12x4 points for the DFT calculation and with 256x256x64 points for the WFs.

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Although the EPC associated with the JTD is not strong enough to drive a metal-insulator transition with localization of electrons, it is straightforward to ask if the EPC can remain sufficiently large to mediate the Cooper pairs formation in Sr2RuO4. To that end, we evaluate the electron phonon-coupling constant λ associated with the JTD Q2 -type mode through the following formula39 \(\lambda=N\left({E}_{{{\rm{F}}}}\right)\frac{{{{\hslash }}}^{2}}{2M{\omega }_{{{\rm{JTD}}}}^{2}}{D}^{2}\) where N(EF) is the density of states at the Fermi level EF associated with a band, ωJTD is the frequency of the JTD, M is the mass of the moving atoms (O in the present situation) and D is the reduced electron-phonon matrix element (REPME) quantifying the change in the electronic structure according to a vibration (see methods).

The frequency of the JTD mode is computed by a frozen displacement approach and yields a value of 81 meV (see methods and Supplementary Note 4). The condensation of the JTD in the ground state structure produces band splitting near EF along the Γ − X, Γ − N, Γ − M, and M − S path for the two bands involving the dxz and dyz orbitals (Fig. 4b). We have computed D for different uJTD values and we observe a linear response between ∆Eg and uJTD. Through a linear fit of the ∆Eg vs. uJTD curve, we extract an average D of 7.76 eV.Å−1. In order to extract the density of states (dos) at EF for these two contributing bands, we have built the Wannier Functions (WFs) associated with the minority spin channel aiming to extract atomic like WFs with dxy, dxz and dyz characters centered on Ru cations (see methods and Supplementary Note 5). It further allows to converge the dos on a very fine kmesh and extracts contribution from bands altered by the JTD—i.e., dxz and dyz. We end up with N(EF) = 0.2908 states/eV/f.u/spin for both dxz and dyz bands. Using these quantities, we obtain an EPC of λ = 0.35 yielding a Tc oscillating between 1.65 K and 0.5 K for usual screened Coulomb potential µ* of 0.1 and 0.15, respectively. This value is reminiscent of the experimental Tc of ~1.5 K. We emphasize that the calculation of λ with a non-spin polarized solution (NM) yields a lower value of λ = 0.08 due to (i) an intrinsic underestimation D to 3.56 eV.Å−1 and (ii) a hardening of the JTD to 98 meV (Supplementary Note 6). However, it is now well known that NM underestimates and/or fails at predicting bands splitting and entangled electron-phonon features by neglecting the basic Hund’s rule12,19. We have checked that the spin-orbit interaction has marginal effects on the electronic properties and we identify that the EPC is unchanged (see Supplementary Note 7). We have finally tested other type of Jahn-Teller distortion that could be observed in other materials such as the Q3 distortion36 with planar Ru-O bond length contraction and out of plane Ru-O bond length elongation, with opposite motion on neighboring Ru sites (see Supplementary Note 8). This JTD Q3 -type mode is identified to have a negligible coupling to the electronic structure (of λ = 0.14) and hence cannot bind electrons into Cooper pairs.

A recent study revealed a possible SC state with a Tc of roughly 64 K coexisting with ferromagnetism in Ca2RuO4 nanofilm single crystals32. The nanofilm corresponds to a pressured materials allowing to tune the co/ao ratio. The consequential effect is to restore the vicinity of a JTE in Ca2RuO4 for co/ao > 1. We performed additional DFT simulations on Ca2RuO4 by using the lattice parameter of Ref. 32 (i.e., a = 5.343 Å, b = 5.350 Å and c = 12.778 Å). We then perform the relaxation of atomic positions at fixed lattice parameters. We end up with a Pbca symmetry similar to the bulk but with a co/ao ratio of 1.05 and the material is found metallic with a FM order. Thus, the JTE is screened in the Ca2RuO4 nanofilms in the spirit of Sr2RuO4 bulk compounds. Following the very same procedure as in Sr2RuO4 with the same JTD Q2 -type mode, the EPC is computed to 1.68, yielding a Tc between 73 and 63 K for µ* of 0.1 and 0.15, respectively (see Fig. 5). It hints at the experimental value with a possible Tc = 64 K32, thereby confirming the model proposed to explain SC in bulk Sr2RuO4.

Fig. 5: Superconducting properties of Ca2RuO4 as a nanofilm.
figure 5

a Band structure of Ca2RuO4 as a nanofilm using a FM order unfolded to primitive I4/mmm cell and projected on O (blue), Ca (green), Ru dxy (red), dxz+dyz (grey) and dx2-y2 and dz2 (orange) states. Majority (left panel) and minority (right panel) are reported. b Band splitting ∆Eg induced by freezing a displacement uJTD (in Å/atom) associated with the JTD Q2 -type mode in Ca2RuO4 nanofilm in the ground state with octahedral rotations (left panels) and without O6 group rotations (right panels). c Evolution of ∆Eg as a function of the displacement uJTD (in Å/atom) associated with the JTD Q2 -type mode. The resulting Reduced Electron Phonon Matrix Element is estimated to D = 9.46 eV.Å−1.d Total density of states (in states/eV/f.u/spin, grey area) as a function of the energy using the DFT calculation (lower part) and atomic-like Wannier functions (upper part). The WFs further allow to extract only contributions from the dxz and dyz orbitals (red line). The contribution from each dxz or dyz band is then of N(F) = 0.31 states/eV/Ru/spin/band. The k-mesh is sampled with 12 x 12 x 4 points for the DFT calculation and with 256 x 256 x 64 points for the WFs. A ferromagnetic order is used throughout these calculations.

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Ca2RuO4 under pressure and Sr2RuO4 bulk possess the same root underpinning superconductivity. Nevertheless, the EPC is much larger in Ca2RuO4 (λ = 1.68) than in Sr2RuO4 (λ = 0.35). At odds with Sr2RuO4, Ca2RuO4 as a nanofilm material retains its a−a−c0 (labeled \({{\phi }^{-}}\)) and a0a0c+ (labeled \({{\phi }^{+}}\)) octahedral rotations with amplitudes \({Q}_{{{\phi }^{-}}}\) = 0.639 Å/f.u and \({Q}_{{{\phi }^{+}}}\) = 0.479 Å/f.u. By means of a free energy expansion associated with the amplitude of the JTD Q2 -type mode and octahedral rotations \({{\phi }^{-}}\) and \({{\phi }^{+}}\), we identify interesting couplings between the biquadratic term of JTD Q2 -type mode with the rotations: \(F \propto ({{{\rm{a}}}}_{20}+{{{\rm{a}}}}_{22}{Q}_{{{\phi }^{-}}}^{2}){Q}_{{{\rm{JTD}}}}^{2}\) \(\propto {{{\rm{a}}}}_{{{\rm{eff}}}}{Q}_{{{\rm{JTD}}}}^{2}\) where a20 and a22 are coefficients. Recalling that the energy of a harmonic oscillator is \(E=\frac{1}{2}M{\omega }_{{{\rm{JTD}}}}^{2}{Q}_{{{\rm{JTD}}}}^{2}\), we get that \({\omega }_{{{\rm{JTD}}}}=\scriptstyle\sqrt{\frac{2{{{\rm{a}}}}_{{{\rm{eff}}}}}{M}}\). Therefore, it follows that the frequency of the JTD mode can be altered by the presence of octahedral rotations. This is confirmed in our simulations in which octahedral rotations produce a renormalization of the frequency of the JTD to 46 meV in the ground state instead of 100 meV in a phase without any octahedral rotation (see methods and Supplementary Note 9). This observation is very similar to the strong softening of the bond disproportionation (BO) frequency with octahedral rotations amplitude appearing in rare-earth nickelates and bismuthates10,40.

The resulting REPME in Ca2RuO4 is evaluated to D = 9.55 eV.Å−1 without O6 groups rotation and to D = 8.77 eV.Å−1 with octahedral rotations, a value roughly matching that of Sr2RuO4. The density of states at the Fermi level N(EF) associated with the dxz and dyz orbitals remains very similar, with N(EF) = 0.31 states/eV/f.u/spin/band vs N(EF) = 0.29 states/eV/f.u/spin/band in Ca2RuO4 and Sr2RuO4, respectively. Thus the strong softening of the JTD in Ca2RuO4 yields an EPC of 1.68. The presence of octahedral rotations with strongly coupled electron-phonon features is thus a potential lever for tuning the superconducting properties of oxide materials.

Outlook

In summary, ruthenates are proposed to be the first identified oxide materials showing superconductivity mediated by an electron-phonon coupling related to the proximity of orbital and bond ordering instabilities. This mechanism is in fact very similar to many other oxides reaching SC at the vicinity of a charge-ordered phase such as nickelates, bismuthates, or antimonates. The underpinning mechanism yielding SC in correlated oxides likely relates with coupled electron-phonon features—such as JTE and disproportionation effects—opening band gaps19. The global strategy to reach SC appears to sufficiently weaken these EPC to drive the material in a metallic regime albeit the EPC has to remain sufficiently large to mediate Cooper pairs formation.

Methods

Exchange-correlation functional

We use the meta-Generalized Gradient Approximation (GGA) Strongly Constrained and Appropriately Normalized SCAN41 functional that better amends self-interaction errors (SIE) inherent to practiced DFT over classical Local Density Approximation (LDA) and GGA functionals. This functional is able to predict the correct trends in lattice distortions and metal-insulator transitions in bulk ABO3 perovskite oxides and trends in doping effects in nickelates, bismuthates and antimonates10,14,42,43. In addition, SCAN does not require any external parameter as in DFT + U.

Magnetic orders

We used several long-range magnetic orders such as a ferromagnetic (FM), A-type AFM (AFM-A) consisting of Ru spins coupled ferromagnetically in (ab)-planes and then coupled AFM along the c axis and C-type AFM (AFM-C) with Ru spins coupled antiferromagnetically in the (ab)-plane. A paramagnetic (PM) calculation is also performed by using the special quasi-random structure (SQS) method to extract the Ru spin arrangement mimicking a PM state within a given supercell size19,44,45. It allows to get a snapshot of all possible local magnetic configurations (i.e., motifs) for Ru cations that would appear in a real PM phase. A 16 f.u corresponding to a (\(2\sqrt{2},\) \(2\sqrt{2},1\)) supercell with respect to the undistorted I4/mmm cell (2 f.u) is used for these simulations. Spins are only treated at the colinear level. The ATAT package is used for identifying the spin arrangement maximizing the disorder characteristic of a random spin configuration46 within a given supercell size44. A non-spin polarized calculation (NM) is also performed in which the number of electrons with a spin up and a spin down are force to be equal by construction on all Ru cations.

Crystallographic cells, structural relaxations, and analysis

The imposed starting cells correspond to high symmetry I4/mmm and the Pbca tetragonal unit cells for Sr2RuO447 and Ca2RuO425,33, respectively. Sr2RuO4 exhibits only a small a0a0c+ octahedral rotation while Ca2RuO4 exhibits large a−a−c+ O6 group rotations induced by the usual A-to-B cation size mismatch. We proceed to the full structural relaxation (cell parameters and shape as well as atomic positions) until the forces acting on each atom are less than 0.005 eV/Å. Amplitude of the distortions of the relaxed ground states are then extracted using symmetry mode analysis taking as reference the undistorted I4/mmm cell. This is performed with the ISODISTORT tool from the ISOTROPY applications48,49.

Seeking for electronic instabilities

The SCAN functional is a local functional of the density matrix unable to make distinction between occupied and unoccupied states—unlike DFT + U or DFT with a hybrid functional. It is therefore unable to identify electronic instabilities in high symmetry cells with degenerate partners as performed in ref. 19 where one has to impose integer occupancy of a specific degenerate partner such as (1,0) instead of (0.5,0.5) for two degenerate orbitals. Instead, we used the strategy proposed in ref. 24: we plot the potential energy surface associated with a lattice distortion and seek to see the shape of the potential energy surface. A shifted single well potential whose minimum is located at non-zero amplitude of the mode then indicates the presence of an electronic instability as those observed for the Jahn-Teller or bond disproportionation distortions in ref. 24.

Potential energy surface and phonon frequencies

The potential energy surfaces associated with the Jahn-Teller Q2-type distortion are computed by freezing different distortion amplitudes QJTD in the material. This is done starting either from a highly symmetric I4/mmm to extract the propensity of the material to display a Jahn-Teller effect or from the ground state to extract the mode frequency. Due to the presence of potential electronic instabilities that move the single well minimum to non-zero amplitude of QJTD, the evolution of the energy ∆E is given by the following expression:

$$\Delta E={E}_{0}+{{\rm{\alpha }}}{\left({Q}_{{{\rm{JTD}}}}-{Q}_{0}\right)}^{2}+{{\rm{\beta }}}{\left({Q}_{{{\rm{JTD}}}}-{Q}_{0}\right)}^{4}$$
(1)

where α and β are coefficients and Q0 signals the force acting on the electrons even in the absence of QJTD (i.e., the electronic instability). It follows that

$$\Delta E= {E}_{0}+{{\rm{\alpha }}}{Q}_{{{\rm{JTD}}}}^{2}+{{\rm{\alpha }}}{Q}_{0}^{2}-{2\alpha Q}_{{{\rm{JTD}}}}{Q}_{0}+{{\rm{\beta }}}{Q}_{{{\rm{JTD}}}}^{4}+{{\rm{\beta }}}{Q}_{0}^{4}-4{{\rm{\beta }}}{Q}_{{{\rm{JTD}}}}^{3}{Q}_{0}\\ -{4{{\rm{\beta }}}Q}_{{{\rm{JTD}}}}{Q}_{0}^{3}+6{{\rm{\beta }}}{Q}_{{{\rm{JTD}}}}^{2}{Q}_{0}^{2}$$
(2)
$$\Delta E= {E}_{0}+\left({{\rm{\alpha }}}{Q}_{0}^{2}+{{\rm{\beta }}}{Q}_{0}^{4}\right)+(-{2{{\rm{\alpha }}}Q}_{0}-4{{\rm{\beta }}}{Q}_{0}^{3}){Q}_{{{\rm{JTD}}}}+({{\rm{\alpha }}}+6{{\rm{\beta }}}{Q}_{0}^{2}){Q}_{{{\rm{JTD}}}}^{2}\\ -4{{\rm{\beta }}}{Q}_{0}{Q}_{{{\rm{JTD}}}}^{3}+{{\rm{\beta }}}{Q}_{{{\rm{JTD}}}}^{4}$$
(3)
$$\Delta E={E}_{0}^{\prime}+{{\rm{a}}}{Q}_{{{\rm{JTD}}}}+{{\rm{b}}}{Q}_{{{\rm{JTD}}}}^{2}+{{\rm{c}}}{Q}_{{{\rm{JTD}}}}^{3}+{{\rm{d}}}{Q}_{{{\rm{JTD}}}}^{4}$$
(4)

From Eq. 4, we recover the linear and trilinear terms in QJTD signaling the contribution of the electronic instability. Using a fit of the potential energy surfaces as a function of QJTD with a polynomial expression up to the 4th order, we can extract all coefficients of Eq.4 and map them to get the mode frequency through the relation \(\omega=\scriptstyle\sqrt{\frac{2\alpha }{M}}\).

As reported in several works on oxide perovskites20,37,38, a0a0c+ (\({\phi}^{+}\)) and a-a-c0 (\({\phi}^{-}\)) octahedral rotations possess a linear coupling with the JTD Q2 -type mode in the free energy expansion starting from the high symmetry undistorted cell: \(F\propto {{F}^{{\prime} }}_{0}+{{\rm{a}}}^{\prime} {Q}_{{{\rm{JTD}}}}^{2}+{{\rm{b}}}^{\prime} {Q}_{{{\rm{JTD}}}}^{4}+{{\rm{c}}}^{\prime} {Q}_{{{\rm{JTD}}}}+{{\rm{d}}}^{\prime} {Q}_{{{\rm{JTD}}}}^{3}\) where a’, b’, c’ and d’ are coefficients depending on the amplitude of octahedral rotations. By using the invariant module of the isotropy suite of software, we extract at the lowest order in amplitudes of the distortions that:

$${{\rm{a}}}^{\prime}={{{\rm{a}}}}_{20}+{{{\rm{a}}}}_{22}{Q}_{{{\phi }^{-}}}^{2}$$
(5)
$${{\rm{b}}}^{\prime}={{{\rm{b}}}}_{4}$$
(6)
$${{\rm{c}}}^{\prime}={{{\rm{c}}}}_{110}{Q}_{{\mathrm{\phi }}^{+}}+{{{\rm{c}}}}_{130}{Q}_{{{\phi }}^{+}}^{3}+{{{\rm{c}}}}_{112}{Q}_{{\mathrm{\phi }}^{+}}{Q}_{{{\phi }}^{-}}^{2} $$
(7)
$${{\rm{d}}}^{\prime}={{{\rm{d}}}}_{31}{Q}_{{\mathrm{\phi }}^{+}}$$
(8)
$${{F}^{{\prime} }}_{0}={F}_{0}+{{{\rm{e}}}}_{020}{Q}_{{{\phi }}^{+}}^{2}+{{{\rm{e}}}}_{002}{Q}_{{{\phi }}^{-}}^{2}+{{{\rm{e}}}}_{040}{Q}_{{{\phi }}^{+}}^{4}+{{{\rm{e}}}}_{004}{Q}_{{{\phi }}^{-}}^{4}+{{{\rm{e}}}}_{022}{Q}_{{{\phi }}^{-}}^{2}{Q}_{{{\phi }}^{+}}^{2}$$
(9)

Without octahedral rotations, one simply recovers the usual equation \(F\propto {{\rm{a}}}{Q}_{{{\rm{JTD}}}}^{2}+{{\rm{b}}}{Q}_{{{\rm{JTD}}}}^{4}\). The role of octahedral rotations is to tune the different coefficients, notably the a coefficient in front \({Q}_{{{\rm{JTD}}}}^{2}\) that will drive the frequency of the mode. Using fits up to the 4th order in QJTD, one gets the frequency through the relation \(\omega=\sqrt{\frac{2{{{\rm{a}}}}^{{\prime} }}{M}}\). We obtain fits with a coefficient of determination R2 of at least 0.9995 for our different potentials.

Superconducting properties

We have calculated the reduced electron-phonon matrix element (REPME) associated with the JTD by freezing its atomic displacement in the structure. Using the band splitting amplitude ΔEg appearing in the band structure in the first Brillouin zone due to the frozen phonon displacement, we compute the REPME by using the following formula \(D=\frac{\Delta {E}_{{{\rm{g}}}}}{2u}\), where u is the displacement of one O atom for the condensed phonon mode. This standard procedure was successful in BaBiO3, SrBiO3, BaSbO3, or MgB2 compounds10,11,14,50. In order to calculate the electron-phonon coupling λ, we use the following formula \(\lambda=N\left({E}_{{{\rm{F}}}}\right)\frac{{\hslash }^{2}}{2M{\omega }_{{{\rm{JTD}}}}^{2}}{D}^{2}\), where N(EF) is the density of states at the Fermi level per spin channel, formula unit and per contributing band, M is the mass of the moving ion in the phonon mode and ωJTD is the frequency of the JT phonon mode. To obtain the critical temperature Tc, we use the McMillian-Allen equation39

$${T}_{{{\rm{c}}}}=\frac{{\omega }_{\log }}{1.2}\exp \left(\frac{-1.04\left(\lambda+1\right)}{\lambda -{\mu }^{*}\left(1+0.62\lambda \right)}\right)$$
(10)

where ωlog is the logarithmic averaged phonon frequency (expressed in K) and μ* is the screened Coulomb potential with conventional values ranging from 0.1 to 0.15. Since the JTD provides a significant contribution to the electron-phonon coupling, we impose that ωlog = ωJTD for estimating Tc. We used the FM order to compute the superconducting properties since local spin formation is critical for quantifying SC in correlated oxides12. Such FM order is suggested experimentally in Sr2RuO4 and observed in the superconducting of Ca2RuO4 nanofilm32,34,51.

Density of states at E F

In order to have accurate density of states at the Fermi level, we have employed the wannier90 package52,53,54,55 on top of our electronic structure calculations. To that end, we have built the Wannier functions associated with the three t2g bands of Ru cations in the minority spin channel. We have projected the 12 Kohn-Sham states located around the Fermi level (4 Ru cations times 3 d states) on dxy, dxz, and dyz guess orbitals centered on Ru cations. After the wannierization, we end up with 12 well-defined dxy, dxz, and dyz atomic-like Wannier functions centered on each Ru cation. Using these WFs, we then proceed to the calculation of the density of states (dos) on a very large kmesh consisting of 256 × 256 × 64 k points. We checked carefully that the dos matched the initial DFT dos computed on a smaller kmesh of 12 × 12 × 4 k points. Furthermore, the WFs allow to extraction the contribution of dxz and dyz orbitals to the density of state at the Fermi level N(EF) since only these bands contribute to the SC properties.

Other technical details

DFT simulations are performed with the Vienna Ab initio Simulation Package (VASP)56,57. Projector Augmented Wave pseudo potentials58 (PAW) are used taking the 5s24d6, 4s24p65s2, 3s23p64s2, and 2s22p4 as valence electrons for Ru, Sr, Ca, and O atoms, respectively. The energy cut-off is set to 650 eV and is accompanied by a 8 × 8 × 4 (4 × 4 × 2) Gamma centered k-mesh for the structural relaxations with long-range magnetic orders (PM order). It is further increased to 12 × 12 × 4 for calculations of density of states with VASP. Band structures are unfolded to the primitive I4/mmm cell using the VaspBandUnfold package.