Introduction

The superconducting order parameter of Sr2RuO41,2,3,4 is thought to be of unconventional nature, but has proven unexpectedly difficult to identify. Soon after its discovery in 1994, it was proposed as a promising candidate for a chiral p-wave, spin triplet superconductor by analogy to superfluid 3He-A, based in particular on early evidence from NMR5. Muon-spin rotation6 and Kerr effect7 measurements suggested intrinsic time-reversal symmetry (TRS) breaking8 below Tc, consistent with this proposal. Thermodynamic measurements provided clear evidence for low-energy quasiparticle states, however, suggesting the existence of gap nodes or deep minima9,10,11,12.

In 2019, the authors of ref. 13 challenged the chiral p-wave paradigm with in-plane 17O nuclear magnetic resonance measurements that found a significant decrease in the Knight shift below Tc13,14. Spin-triplet pairing was then definitively ruled out by comparison to the change of the entropy from earlier specific heat experiments15. These measurements were accompanied by observations of shifts in the elastic constants16,17 together with experiments under strain18 suggesting a two component nature of the order parameter. All these results led to renewed theoretical attempts to calculate the superconducting ground state of Sr2RuO4 within a spin singlet picture19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37, leading to a variety of proposals, including the even-parity 1D irreducible representation \({B}_{1g}({d}_{{x}^{2}-{y}^{2}})\), multi-component orders such as \({d}_{{x}^{2}-{y}^{2}}+i{g}_{xy({x}^{2}-{y}^{2})}\) and \(s^{\prime} +i{d}_{xy}\), as well as the 2D irreducible representation E1g(dxz + idyz). A two-component state is thought to be required to explain ultrasound measurements16,17 and recent μSR experiments under strain18 (see however refs. 36,38), but other recent measurements provide evidence for a single order parameter component39,40.

While the experiments cited above and many others provide indirect evidence in support of one superconducting pairing channel or another, the community’s ability to definitively identify the order parameter is severely hindered by the difficulty of making direct measurements of the superconducting gap over the Fermi surface. The extremely small gap (∣Δ∣ ≤ 350 μeV)41,42 of the superconducting order parameter Δ(k) in Sr2RuO4 means that angle-resolved photoemission (ARPES) experiments do not currently have the fine energy and momentum resolution to detect spectral features reflecting the gap. Scanning tunneling spectroscopy experiments that detect Bogoliubov quasiparticle interference (BQPI) provide, on the other hand, good momentum resolution and finer energy resolution in the best circumstances. Recently, the BQPI technique was used for Sr2RuO442. This analysis, based on comparison with a simple lattice calculation of the joint density of states, suggested a \({d}_{{x}^{2}-{y}^{2}}\) superconducting gap symmetry for Sr2RuO4. A somewhat more sophisticated calculation involving first-principles surface Wannier functions was also compared to the same data43, but more than one gap function appeared to fit nearly equally well. Nevertheless, BQPI appears to be the best possibility of “directly” measuring the superconducting gap in this canonical unconventional superconductor.

There is however one enduring, poorly understood puzzle regarding scanning tunneling spectroscopy (STS) and scanning tunneling microscopy (STM) studies of superconducting Sr2RuO4. While some STS/STM measurements have reported signatures of superconductivity on the surface of Sr2RuO4 for many years41,42,44, others on apparently equivalent surfaces under similar conditions are unable to detect any gap at all45,46,47,48. Interestingly, superconductivity was reported in samples with regions of RuO2 termination41, while most of the investigations where superconductivity was not detected were performed on the the usual SrO termination. Recognizing that this question was an important one to resolve in order to properly interpret STS/STM data on the Sr2RuO4 surface, the authors of refs. 49,50,51 performed a systematic study of local STS/STM conductance spectra at several distinct types of local topographic structures on the surface with different termination layers. They found that superconducting spectra with coherence peaks appeared only in the region of small nanoscale “pits” on the surface, corresponding to the removal of one RuO surface layer. Since the Sr2RuO4 surface is thought to be reconstructed in a pattern of RuO6 octahedral rotation similar to that of bulk Sr3Ru2O7 and calcium doped bulk Sr2RuO447,52,53,54,55, it is natural to ask if the atomic layer on the surface might be electronically different to that placed immediately underneath it, such that conditions favorable to superconductivity are “masked”. As stated in refs. 47,53, superconductivity is not observed although the surfaces are pristine. In other words, is it possible that superconductivity in Sr2RuO4 can never be observed on a hypothetical perfect but reconstructed Sr2RuO4 surface, but is only revealed when such pits form? Or, alternatively, are there other possible Sr2RuO4 surfaces, with little reconstruction compared to the bulk case, that might actually reveal the superconducting gap as in the bulk? Here we investigate the plausibility of such a scenario by performing first-principles-based electronic structure simulations.

Results

In the following we examine how the surface termination and rotation angles of the RuO6 octahedra in Sr2RuO4 affect its electronic structure employing density functional theory (DFT), which has provided reliable results for the Fermi-liquid normal state of bulk Sr2RuO456. We determine the optimal rotation angle depending on the type of termination, and perform large-scale canyon structure simulations to investigate the above scenario predicting that pits in the surface allow to observe signatures of bulk physics.

To understand better the changes induced by the octahedral rotation in Sr2RuO4 we first focus on a Sr-terminated monolayer for which we consider different RuO6 rotation angles. The effect of the rotation on the band structure, Fermi surface, total density of states (DOS) and projected DOS is visualized in Fig. 1 and is in agreement with previous results57 as well as recently reported ARPES measurements58, with the exception that the pocket opening around the \({\bar{\Gamma}}\) point is at smaller values of the octahedral rotation - which is a known shortcoming of DFT57. As shown in Fig. 1, the octahedral rotation pushes the van-Hove singularity (vHs) – located in the dxy orbital – from a position above the Fermi level at 0∘ rotation, as in the bulk case21,28,56,59,60,61, to a position below the Fermi level at the optimal octahedral rotation value. As a consequence of this, the dxy-orbital dominated Fermi surface section near the \(\bar{M}\)-point vanishes (see the band structure and Fermi surface plotted in Fig. 1a, b. Furthermore, a pocket at the \(\bar{\Gamma }\) point in the dxy orbital opens up at approximately 6∘ octahedral rotation. This feature has been identified as a source of suppressed superconductivity in a recent theoretical investigation by some of the authors62 by performing functional renormalization group (FRG) calculations for wannierized tight-binding models for the surface of Sr2RuO4.

Fig. 1: Electronic structure of Sr Monolayer.
figure 1

a Band structure along the high symmetry path of the folded Brillouin zone (BZ) in Sr2RuO4 and the density of states (DOS) for the rotations 0∘ (lightgrey), 5∘ (grey), 7∘ (darkgrey) and 10.28∘ (black). b Fermi surfaces drawn in the conventional bulk BZ (black solid frame). As inset we depict the new primitive BZ (green dashed square). The \(\bar{X}\) of the band structure plot is located mid of the green dashed lines. For the 0∘ case we included the unfolded Fermi surface in solid red. We observe a Lifshitz transition occurring at octahedral rotation angles between 5∘ (grey) and 7∘ (darkgrey). c d orbitally resolved DOS plotted with different shades of blue. The dxy orbitals are the dominant orbitals at the Fermi level EF.

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In the same spirit and motivated by recent strain dependent experiments18,40,63,64,65 indicating a strong effect of the vHs position on Tc, we investigate different possible realizations of surfaces with the aim to recover bulk-like behavior in the surface layer, which would support the scenario that the detection of superconductivity on the surface is directly related to the surface having similar electronic properties as the bulk. By cleaving such surfaces, surface sensitive probes, like STS, as mentioned in the introduction, can then be utilized to image the superconducting order parameter.

We consider four different charge neutral surface configurations in Sr2RuO4 including the experimentally realized Sr termination47, a Ru termination, and a 2Sr termination (see Fig. 2a). While the latter two terminations, to our knowledge, have not yet been reported as a clean surface, our findings indicate that trying to cleave or grow these is highly desirable in order to restore bulk physics on the surface. As a fourth configuration, we examine a canyon-like structure, i. e. along a the periodicity is shown in Fig. 2b, while along b it continues the structure as it is depicted. This structure is similar to the suggested pit in refs. 49,50 and investigate how the electronic structure of Sr2RuO4 is affected thereby.

Fig. 2: Slab terminations considered in this work.
figure 2

a, b Depict the surface and subsurface layers used for our slab calculations where the shaded grey region on top marks the position of the void for the calculations. The labels on the left (Sr-O or Ru-O) denote the layer composition. Panel (a) shows the Sr-terminated, Ru-terminated and 2Sr-terminated slabs. The rotation of RuO6 octahedra is depicted on top of the Sr-termination figure. b Canyon-like structure with 2Sr termination in the pit (2Sr Canyon). This canyon-like structure has been reported in STM experiments49. The dark blue and red horizontal bars are color indicators for Fig. 4. c Global pattern with the conventional (dashed) and \(\sqrt{2}\times \sqrt{2}\) unit cell (solid). The inset shows the definition of the RuO6 octahedra rotation angle. The depicted coordinate system in all subplots is the one from the conventional unit cell.

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The optimized RuO6 rotation angles for slabs with Sr surface termination (Fig. 2a, left panel) are summarized in Fig. 3 as a function of number of layers n (see also Table S1). We find that the octahedral rotation in both surface and subsurface show rapid convergence to stable values with the number of layers considered in the slab calculation, indicating the presence of small interlayer couplings, as expected from resistivity measurements66. Incorporating spin-orbit coupling (SOC) in the structural slab relaxations is not found to play a crucial role for the rotation angles, in contrast to the role it plays for the electronic properties near the Fermi surface56,59,61 where it was found to be essential. In other words, even though to reproduce the experimental Fermi-surface shape spin-orbit coupling is essential, it has negligible effect on the structural relaxation which is the central focus of this study. The optimal rotation angle of 7.17∘ at the surface layer obtained in non-relativistic calculations is in agreement with experiments53,67 and earlier theoretical investigations57,68. Therefore, for the structure optimization we neglect the effect of SOC in multilayered slabs. Furthermore, we checked that structure optimization using different setups, like surface and subsurface optimization or structural optimization of the full slab with different depths, leads to changes in the rotation angle of the RuO6 octahedra of at most 0.1∘.

Fig. 3: Octahedral rotation θ.
figure 3

(see the inset of Fig. 2c for the angle definition) (a) as a function of number of layers within the slab for both, surface (continuous line) and subsurface (dashed line) for all three terminations in Fig. 2a. b Change of the octahedral rotation angle as a function of the relaxation configuration, i. e. “surface” stands for only the surface, ''subsurface” for the surface and subsurface and “full” uses all of the five layers in our calculations. Note that only as of a three-layer slab a subsurface layer exists. We find a much stronger surface reconstruction in the case of Sr termination (green solid curve) than for the Ru termination (blue, in inset) and the 2Sr termination (red, inset). All subsurface layers show negligible rotations. The exact values of the relaxation can be found in Tables S1, S2 and S3. On the other hand, the octahedral rotation does not change drastically when changing the computational setup of the relaxation, see subfigure (b).

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We next consider slabs with Ru termination (Fig. 2a, middle panel). In this case the surface RuO6 octahedra don’t have top apex oxygens and the surface reconstruction after relaxation is less severe. The optimized rotation angles are therefore much smaller than in the previous case as summarized in Fig. 3 and in Table S2. The slabs exhibit once again a fast convergence in rotation angles with number of layers considered within the slab (see inset of Fig. 3).

We also performed relaxations by considering a 2Sr termination (Fig. 2a, right panel), where the surface is assumed to be terminated at a SrO layer, two layers away from the RuO plane. In this case we observe, similarly to the Ru termination, a strong suppression of the RuO6 octahedral rotation in the most outer RuO surface (see Fig. 3 and Table S2). The rotation angles again converge quickly with the number of layers within the slab. As expected from the electronic structure analysis of the effect of RuO6 octahedral rotations in Sr2RuO4 (Fig. 1), we find bulk-like behavior in the absence of the octahedral rotation for both the 2Sr and Ru terminations.

We proceed now with a more complex slab termination. As reported in ref. 49 imperfections in the cleaving process of the crystal, which create small canyon-like structures, enable the observation of a superconducting state in STM. Here, we perform DFT calculations for one such case, a 2Sr canyon termination (Fig. 2b). For that we choose a trade-off between numerical feasibility and complexity of the structure. The canyon-like structure is built from previously relaxed Sr-terminated slabs. After removing the atoms in order to resemble the structures suggested in ref. 49, we perform a relaxation of the whole structure. We find that near the edges of the canyon, a substantial displacement of the atoms occurs - while at the bottom of the pit, the RuO6 octahedral rotation is nearly zero. Figure 4 shows the orbitally resolved DOS for the 2Sr canyon (Fig. 2b) where the contribution of the surface Ru is shown in shades of dashed blue (Fig. 4a) and the contribution coming from subsurface Ru is shown in shades of solid red (Fig. 4b). Figure 4c depicts the comparison of the dxy orbital projected DOS for the surface and subsurface. By this direct comparison it is visible that the subsurface layer has a clear bulk-like feature, with a clear peak close to the Fermi level (compare, for instance, with ref. 47, and with Fig. 1c left panel where the electronic structure of bulk Sr2RuO4 is emulated). As can be seen in Fig. 4 the influence of the RuO6 octahedral rotation is quite drastic - the electronic structure of the subsurface Ru layer retains the position of the bulk system’s van-Hove singularity while the vHs of the surface Ru is shifted below the Fermi level, again resembling what we observed in the Sr-terminated slab (Fig. 1). This indicates that the subsurface electronic structure is the same as for bulk Sr2RuO4, since it is not affected by the octahedra rotation. Therefore, bulk-like behavior is expected, i.e. the layer should show superconductivity when the tip is placed in the area of the canyon, as indicated by STM measurements49,50.

Fig. 4: DOS for the Ru atoms of the 2Sr canyon plotted in three setups.
figure 4

The colors here correspond to the colors of the layers in Fig. 2b for the canyon. a Shows the DOS of the top layer (blue, dashed) of the canyon for all d orbitals. b It is the same as (a) but for the subsurface layer (red, solid). c Displays the comparison of the most dominant Ru orbital, dxy, at the Fermi level for the surface and subsurface layer. The DOS are normalized to one Ru atom.

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Discussion

In this work, we investigated via DFT calculations the surface reconstruction of different charge-neutral surfaces of Sr2RuO4, including experimentally observed canyon structures. Our motivation was to understand the microscopic origin of contradictory reports of detection of superconductivity in STS/STM measurements of Sr2RuO4. The results of our calculations are condensed into two main quantities which are (i) the structural properties in terms of the behavior of the RuO6 octahedra at the surface and layers below and (ii) the (partial) density of states of the Ru-d states in the corresponding layers. To examine whether to expect a structure to have bulk-like behavior, we compare the bulk and relaxed surface electronic structures. From experiments, we know that the bulk system is superconducting1 while clean SrO-terminated surface samples show no superconductivity45,46,47,48,49. As a probe for whether superconductivity is expected or not, we use the position of the vHs of the band with mainly dxy orbital character relative to the Fermi-level. Experiments under uniaxial strain demonstrated that its position has drastic effects on the critical transition temperature18,40,63,64,65, both allowing an increase of Tc when approaching the Fermi-level and fueling a rapid suppression once the vHs is pushed below EF. In contrast to the strain experiments, the movement of the vHs is not the only change induced by the surface reconstruction, i.e. at large octahedral rotation (θ ≥ 6∘) the van-Hove singularity lies below the Fermi level (EF) and the DOS at EF is predominantly provided by the new dxy pocket opening near the \({\bar{\Gamma}}\) point (see Fig. 1b) right panel), unlike the bulk-like electronic structure (no RuO6 rotation, Fig. 1b) left panel) where this pocket is absent. We stress that while the property of a single orbital is central in determining Tc, the order parameter is a fully multi-orbital object and the gap on the out-of-plane orbitals is sizable. Thereby we expect that STS/STM can detect the order even though the dxy-orbital couples only weakly to the out of plane tunneling. The evolution of the electronic structure under strain was recently reported from ARPES measurements58, where shifts of the surface bands and a sequence of Lifshitz transitions were detected. We note that our finding of the Lifshitz transition is in line with this observation. The interpretation of the shifts of the surface electronic structure in terms of bond length changes rather than octahedral rotation58 is a different picture. We note however that application of strain is qualitatively different than removing a surface layer in a canyon as we simulate in our work, but still could be a potential avenue to expose superconductivity to the surface. We note that the \({\bar{\Gamma}}\) pocket is slightly higher in energy in ARPES measurements58. This additional feature was however recently argued to be irrelevant for the pairing62 and thereby the suppression of superconductivity can be understood in rough analogy with the experiments under uniaxial strain. Considering these arguments, it is to be expected that the modifications in the surface electronic structure also change the tendency towards superconductivity on the surface. As a consequence, searching for a bulk-like electronic structure would potentially enable to observe bulk-like physics. As a last note, while Tc is mainly influenced by specifics of the dxy orbital, the gap is known to be of roughly equal size on all three orbitals from specific heat experiments12,39 which seems at first glance contradictory since the orbitals are only weakly coupled in the kinetic sector. However, as was demonstrated in numerical studies, the interactions in between the orbitals lead to an enhanced amplitude in all orbitals due to the feedback of one to another19,69.

Using this qualitative understanding of how superconductivity is influenced by the structural state of the Sr2RuO4 and by examining the signatures of the density of states, we can indirectly connect the tendency towards superconducting pairing to the structural state of the RuO6 octahedra. We find that the density of states in the layers without rotation exhibits similar positions of the van Hove peaks as that of an unreconstructed (bulk) system; in contrast to layers with significant octahedra rotation, thus connecting the electronic structures in the corresponding layers to the bulk electronic structure. Our calculations corroborate that for Sr-terminated surfaces, the RuO6 octahedra show a rotation of about 7∘, in agreement with previous investigations53,57,67,68. However, for both the rare 2Sr- and Ru-terminated surfaces, no significant octahedral rotation angle was found.

This suggests that if these latter terminations could be successfully grown or cleaved, superconducting surfaces should be observed. This in turn would allow for direct measurements of the superconducting order parameter utilizing Bogoliubov quasiparticle interference. Importantly, since the surface electronic structure in these cases does not essentially deviate from the bulk electronic structure, it is expected that the superconducting order parameter is identical to the bulk as well. In experiments on these proposed surfaces, it will be crucial to ensure that the observed order parameter indeed matches the bulk one by comparing properties of the surface order to the bulk order. Natural consistency checks include those between STS spectra/QPI analysis and the nodal positions expected from bulk transport12,16.

This suggestion is supported by our 2Sr canyon slab simulations resembling reported surface imperfections in ref. 49 where we found that the octahedral rotation in the subsurface is essentially absent. Accordingly, the electronic structure does not differ substantially from bulk - explaining why within these canyons a superconducting gap is observed in STM measurements49,50. Therefore, we argue that STM (or other surface) experiments do not see the superconducting gap if they are located with their tip on the Sr-terminated surface with a reconstructed surface that contains octahedral rotation.

While so far there has been no report of a clean 2Sr or Ru surface termination, our results suggest that fabricating such a termination could help significantly in determining the gap structure in Sr2RuO4. Furthermore, this could settle the longstanding debate of whether there is a second order parameter or not16,17,18,36,39,40,70. While a relatively clean surface might be required to obtain Bogoliubov quasiparticle interference patterns, the observation of a STS/STM gap in canyon defects already allows for a more thorough investigation of the gap structure.

Methods

First principles calculations

We performed ab initio electronic structure calculations within density functional theory (DFT)71,72 by using the Vienna Ab initio Simulation Package (VASP)73,74,75 within the pseudo-potential augmented plane-wave76,77 (PAW) basis set. The calculations were performed with the Perdew-Burke-Ernzerhof (PBE)78 exchange correlation functional as a generalized gradient approximation (GGA), and a plane-wave cutoff of 800 eV was chosen. First the bulk structure was relaxed on a 16 × 16 × 4 k-point mesh using the conventional unit cell of Sr2RuO4. The relaxed unit cell was then transformed to a tetragonal \(\sqrt{2}\times \sqrt{2}\times 1\) unit cell in order to have two inequivalent Ru sites and four inequivalent O sites in the ab plane. The convergence criterion was chosen to be 2 ⋅ 10−3 eV/Å for these slabs and 1 ⋅ 10−3 eV/Å for the bulk.

From the optimized bulk unit cell we generated slabs with up to 5 Ru layers with Sr, 2Sr or Ru termination on both ends. A void of 15 Å was set on top of the surfaces. In these slab unit cells we rotated the RuO6 octahedra starting form their initial position, i. e. 180∘ between two neighboring Ru atoms and the O in between, by adding a displacement dO for the O atoms in the Ru plane dependent on the rotation angle θ

$${{\bf{d}}}_{{\rm{O}}}(\theta )=\left(\begin{array}{ccc}0&\tan (\theta )&0\\ -\tan (\theta )&0&0\\ 0&0&0\end{array}\right)\cdot {{\bf{r}}}_{{\rm{O}}},$$
(1)

where rO is the initial positions of the O atoms in the Ru plane. For all slabs we optimized every pair of Ru layers symmetrically. Crosscheck for non symmetric setups were done, however, these had a higher total energy.

The scheme of optimizing was done carefully for all terminations as follows: (i) We first started with one Ru layer. (ii) This was optimized first with a low resolution energy landscape. (iii) Then we relocated and zoomed in to restart with a higher resolution. (iv) We repeated this procedure until we had a sufficient amount of resolution. (v) As a last step of refinement we let VASP internally relax, to printout the forces and confirm that we are in a minimum. (vi) Restart with an additional Ru layer, and use the previous optimal rotation angle as new starting rotation angle. The k mesh for optimizing these structures was 6 × 6 × 1.

For the 2Sr canyon structure, we started from the relaxed 3 layer Sr-terminated slab and expanded the unit cell by \(\sqrt{2}\times \sqrt{2}\times 1\). This supercell was then extended in one direction 3 × 1 × 1. Finally we dug a hole on both surfaces and started a relaxation. The forces were reduced to 5 ⋅ 10−3eV/Å with a k-mesh of 2 × 6 × 1.