Abstract
Applying inplane uniaxial pressure to strongly correlated lowdimensional systems has been shown to tune the electronic structure dramatically. For example, the unconventional superconductor Sr_{2}RuO_{4} can be tuned through a single Van Hove point, resulting in strong enhancement of both T_{c} and H_{c2}. Outofplane (c axis) uniaxial pressure is expected to tune the quasitwodimensional structure even more strongly, by pushing it towards two Van Hove points simultaneously. Here, we achieve a record uniaxial stress of 3.2 GPa along the c axis of Sr_{2}RuO_{4}. H_{c2} increases, as expected for increasing density of states, but unexpectedly T_{c} falls. As a first attempt to explain this result, we present threedimensional calculations in the weak interaction limit. We find that within the weakcoupling framework there is no single order parameter that can account for the contrasting effects of inplane versus caxis uniaxial stress, which makes this new result a strong constraint on theories of the superconductivity of Sr_{2}RuO_{4}.
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Introduction
Sr_{2}RuO_{4} is a famous exemplar of unconventional superconductivity, due to the quality of the available samples and the precision of knowledge about its normal state, and because the origin of its superconductivity remains unexplained in spite of strenuous effort^{1,2,3,4}. No proposed order parameter is able straightforwardly to account for all the existing experimental observations. The greatest conundrum is posed by evidence that the order parameter combines even parity^{5,6,7,8} with time reversal symmetry breaking^{9,10,11}. This combination of properties implies, if there is no fine tuning, that the superconducting order parameter is d_{xz} ± id_{yz}^{12}. Under conventional understanding, this is not expected because the horizontal line node at k_{z} = 0 implies interlayer pairing, while the electronic structure of Sr_{2}RuO_{4} is highly twodimensional^{13,14}.
This puzzle has led to substantial theoretical activity. Two recent proposals are s ± id^{15,16} and d ± ig^{17,18} order parameters, which require tuning to obtain T_{TRSB} ≈ T_{c} (where T_{TRSB} is the time reversal symmetry breaking temperature), but avoid horizontal line nodes. A mixedparity state^{19} and superconductivity that breaks time reversal symmetry only in the vicinity of extended defects^{20} have been proposed to account for the absence of a resolvable heat capacity anomaly at T_{TRSB}^{21}. Interorbital pairing through Hund’s coupling is also under discussion^{22,23,24,25}; with some tuning of parameters this mechanism could yield d_{xz} ± id_{yz} order. Quasiparticle interference data, on the other hand, give evidence for a \({d}_{{x}^{2}{y}^{2}}\)like gap, and a recent junction experiment shows timereversal invariance^{26,27}.
Uniaxial stress has become an important probe of the superconductivity of Sr_{2}RuO_{4}. When stress is applied along the [100] direction, the largest Fermi surface sheet (the γ sheet— see Fig. 1) distorts anisotropically, and undergoes a Lifshitz transition from an electronlike to an open geometry at −0.75 GPa (where negative values denote compression)^{28}. The effect on the superconductivity is profound: T_{c} increases from 1.5 K in unstressed Sr_{2}RuO_{4} to 3.5 K, while the caxis upper critical field H_{c2} increases by a factor of twenty^{29}. This very strong enhancement is qualitatively consistent with, for example, a \({d}_{{x}^{2}{y}^{2}}\) order parameter. Under compression along the [100] direction, the Lifshitz transition occurs at approximately (k_{x}, k_{y}) = (0, ± π/a), which we label the Y point. As the transition occurs, the Fermi velocity at the Y point falls to nearly zero, which in general is expected to increase T_{c} and H_{c2} of order parameters such as \({d}_{{x}^{2}{y}^{2}}\) in which the gap is large at the Y point. The data under [100] uniaxial stress argue against, for example, a d_{xy} order parameter.
Naively, then, T_{c} and H_{c2} might be expected to rise even further under compression along the c axis. caxis compression raises the energy of the d_{xz} and d_{yz} bands relative to the d_{xy} band, and the resulting transfer of carriers expands the γ sheet, pushing it towards a Lifshitz transition from an electronlike to a holelike geometry^{30}. This transition occurs at both the X and Y points — see Fig. 1c — so the increase in the Fermilevel density of states (DOS) as it is approached is expected to be larger than for the electrontoopen Lifshitz transition induced by inplane stress. Under aaxis compression T_{c} increases strongly well before the Lifshitz transition is reached, and so generically we expect this to occur for caxis stress, too. The weakcoupling renormalization group study of Ref. 31 and functional renormalization group study of Ref. 32 both predict a rapid increase in T_{c} with approach to the electrontohole Lifshitz transition.
The electrontohole transition has been approached, and crossed, in thin films through epitaxial strain, and in bulk crystals by substitution of La for Sr^{33,34,35}, but in both cases the superconductivity was suppressed by disorder. Here, we apply up to 3.2 GPa along the c axis of Sr_{2}RuO_{4}. This is a record uniaxial stress for bulk Sr_{2}RuO_{4}, and was achieved by sculpting samples with a focused ion beam to concentrate stress. H_{c2} increases, as expected from the increasing density of states. However, unexpectedly, T_{c} decreases. In other words, approaching the Lifshitz transition at either the X or Y point dramatically enhances T_{c}, while approaching both suppresses T_{c}. This is a major surprise. In a first attempt to address this issue we present calculations in the limit of weak coupling, that take into account the threedimensional structure of the Fermi surfaces. Although these show that caxis compression reduces the transition temperatures of certain order parameters, no order parameter could be identified for which the effects of both outofplane and inplane pressure were captured. Our experimental finding therefore consitutes a major new constraint on theories of the superconductivity of Sr_{2}RuO_{4}.
Results
Electronic structure calculations
We start with density functional theory (DFT) calculations of Sr_{2}RuO_{4} under caxis compression, as a guide to the likely effects of caxis strain on the electronic structure. Figure 1 shows our results. Panel a shows the Fermi surfaces under 0.75% compression along the a axis, panel b those of the unstrained lattice, and panel c those under 2.5% compression along the c axis. The calculations are done under conditions of uniaxial stress, meaning that the transverse strains are the longitudinal strain times the relevant Poisson’s ratios for Sr_{2}RuO_{4}. DFT calculations reproduce well the changes under [100] uniaxial stress observed in ARPES measurements^{36}. Technical details of the calculation are provided in the Methods section.
The calculations predict that the electrontohole transition will occur at ε_{zz} = −0.025. Under aaxis compression, these calculations predict that the electrontoopen transition occurs at ε_{xx} = −0.0075, whereas it was observed experimentally to occur at ε_{xx} = −0.0044^{28}, so this caxis prediction might similarly overestimate the level of compression required. The uncertainty arises from the fact that the distance to the Lifshitz transition is sensitive to meVlevel energy shifts, likely driven by manybody renormalisation^{28}. Lowtemperature ultrasound data give a caxis Young’s modulus of 219 GPa^{37}, so ε_{zz} = −0.025 corresponds to σ_{zz} ≈ −5.5 GPa. Separately, we note also that while k_{z} warping increases on all the Fermi sheets, as expected for caxis compression, the β sheet has the strongest k_{z} warping both at ε_{zz} = 0 and at the Lifshitz transition; see the Methods section for an illustration.
Experimental results
Four samples were measured. For good stress homogeneity, samples should be elongated along the stress axis, which is a challenge for the c axis because the cleave plane of Sr_{2}RuO_{4} is the ab plane. A plasma focused ion beam, in which material is milled using a beam of Xe ions, was therefore used to shape the samples. Sample 1 was prepared with a uniform cross section, and a large enough stress, σ_{zz} = −0.84 GPa, was achieved to observe a clear change in T_{c}. To go further, the other samples were all sculpted into dumbell shapes, with the wide ends providing large surfaces for coupling force into the sample. FIB microstructuring has been used to achieve large caxis stress in CaFe_{2}As_{2}^{38}, but here we needed to retain sufficient sample volume for highprecision magnetic susceptibility measurements. For measurement of T_{c} in the neck portion, two concentric coils of a few turns each were wound around the neck. Samples 1 and 4 also had electrical contacts, for measurement of the caxis resistivity ρ_{zz}. Photographs of samples 2 and 4 are shown in the Methods section.
Sample 4 was measured in apparatus that incorporated a sensor of the force applied to the sample^{39}, from which the stress in the sample could be accurately determined. Samples 1–3 were mounted into apparatus that had a sensor only of the displacement applied to the sample, which is an imperfect measure of the sample strain because the measured displacement includes deformation in the epoxy that holds the sample. Therefore, a displacementtostress conversion was applied to samples 1–3 to bring the rate of change of T_{c} over the stress range 0.92 < σ_{zz} < − 0.20 GPa into agreement with that of sample 4. In other words, we impose on our data an assumption that the initial rate of decrease in T_{c} is the same in all the samples, which is reasonable because their zerostress T_{c}’s are very similar: all are between 1.45 and 1.50 K.
We begin by showing resistivity data, in Fig. 2. The plotted resistivities are corrected for the expected stressinduced change in sample geometry (reduced length and increased width), using the lowtemperature elastic moduli reported in Ref. 37, and making the assumption that stress and strain are linear over this entire range. At zero stress the resistivity of sample 4 shows a sharp transition into the superconducting state at 1.55 K. This sharpness, and the fact that it only slightly exceeds the transition temperature seen in susceptibility, indicate high sample quality. With compression, T_{c} decreases. The normalstate resistivity also decreases, following the general expectations that caxis compression should increase k_{z} dispersion.
We find elastoresistivities (1/ρ_{zz})dρ_{zz}/dε_{zz}, obtained from linear fits over the range − 0.5 < σ_{zz} < 0 GPa, of 37 and 32 for samples 1 and 4, respectively. Sample 4 was compressed to −1.7 GPa, and its resistivity does not show any major deviation from linearity over this range. The scatter in the data at strong compression may be a consequence of cracking in the electrical contacts— we show below that the sample deformation was almost certainly elastic.
We now show the effects of caxis compression on magnetic susceptibility. Figure 3a–c shows the transitions of samples 2–4 in susceptibility; the data shown are the mutual inductance M of the sense coils versus temperature. To check that sample deformation remained elastic, we repeatedly cycled the stress to confirm that the form of the M(T) curves remained unchanged; see the Methods section for examples. For samples 3 and 4, the transition remained narrow as stress was applied, indicating high stress homogeneity. For sample 2, there was a tail on the hightemperature side of the transition, that was stronger at higher compressions. We attribute it to inplane strain, possibly originating in the fact that sample 2 was not as well aligned as samples 3 and 4. A similar, though weaker, tail is also visible for sample 3.
We note that the width of the transitions in Fig. 3a–c — ≈ 50 mK — will be a consequence of defects and/or an internal distribution in the inplane strain. Although there will also be inhomogeneity in ε_{zz}, this is not the driver of the transition width: the distribution would have to have a width of ~1 GPa, which is not plausible.
Figure 3 (d) shows T_{c} versus stress for all the samples. T_{c} is taken as the temperature where M crosses a threshold. For samples 1, 3, and 4, we select a threshold at ≈ 50% of the height of the transition, and for sample 2, 20%, in order to minimize the influence from the hightemperature tail. T_{c} is seen to decrease almost linearly out to σ_{zz} ≈ − 1.8 GPa. For sample 4 (to which, as described above, the other samples are referenced), dT_{c}/dσ_{zz} in the limit σ_{zz} → 0 is 76 ± 5 mK/GPa. The error is 6%: we estimate a 5% error on the calibration of the force sensor of the cell, and a 3% error on the crosssectional area of the sample (155 × 106 μm^{2}).
At σ_{zz} ≲ − 1.8 GPa, the stress dependence of T_{c} flattens markedly. In sample 3, T_{c} resumes its decrease for σ_{zz} < − 3 GPa. We show in the Methods section that both the flattening and this further decrease reproduce when the stress is cycled, which, in combination with the narrowness of the transitions, shows that this behavior is intrinsic, not an artefact of any drift or nonelastic deformation in the system.
Figure 4 shows measurements of the caxis upper critical field. M(H) for samples 2 and 3 at constant temperature T ≈ 0.3 K is shown in panels a and b. In Fig. 4c, we plot H_{c2} versus stress, taking H_{c2} as the fields at which M crosses the thresholds indicated in panels a–b. H_{c2} increases as stress is applied, as generally expected when the density of states increases. The increase is faster for sample 2 than sample 3, which may be an artefact of the tail on the transition for sample 2.
The quantity \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) is particularly informative: if pairing strength were modified without changing the gap structure, H_{c2} would be proportional to \({T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\). As shown in Fig. 4d, we observe \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) to increase by 40% by σ_{zz} = − 3.0 GPa. Slower quasiparticles are less strongly affected by magnetic field, so what this result means is that the Fermi velocity decreases on portions of the Fermi surface where the gap is large.
We conclude this section with a note on the peak effect — the small maximum in the susceptibility just below H_{c2}, visible in Fig. 4 (a, b). The peak effect occurs when there is a range of temperature below T_{c} where vortex motion is uncorrelated, allowing individual vortices to find deeper pinning sites^{40}. The peak is suppressed by caxis compression, and it is suppressed downward rather than by being smeared horizontally along the H axis, which means that this suppression is not an artefact of a spread in H_{c2} due to strain inhomogeneity. It could indicate stronger pinning, due to the reduction in the coherence length.
Weakcoupling calculations
As explained in the introduction, there is an apparent contradiction between the increase of T_{c} under aaxis strain reported in previous work (which suggests antinodes at the X and Y points) and the decrease of T_{c} under caxis pressure (which suggests nodes at the X and Y points). To see if this puzzle has a straightforward solution, we perform weakcoupling calculations for repulsive Hubbard models, as developed in Refs. 41,42,43,44,45,46,47,48,49,50. To capture possible changes in the 3D gap structure, we employ threedimensional Fermi surfaces^{51}. These are described by a threeband (4d xy, xz, and yz) tightbinding model. The hopping integrals are derived from the Rucentred Wannier functions obtained in the DFT calculation presented above. Our tightbinding model takes the form
\({{{{{{{{\boldsymbol{\psi }}}}}}}}}_{s}({{{{{{{\bf{k}}}}}}}})={[{c}_{xz,s}({{{{{{{\bf{k}}}}}}}}),{c}_{yz,s}({{{{{{{\bf{k}}}}}}}}),{c}_{xy,\bar{s}}({{{{{{{\bf{k}}}}}}}})]}^{T}\), and \({{{{{{{{\mathcal{H}}}}}}}}}_{s}({{{{{{{\bf{k}}}}}}}})\) incorporates spinorbit coupling, interorbital and intraorbital terms. The complete set of tightbinding parameters retained here is given in the Methods section.
In Fig. 5a, we show the tightbinding Fermi surfaces at ε_{zz} = 0 and − 0.02. In Fig. 5b, we show the orbital weight on the γ sheet at k_{z} = 0. As the γ sheet expands, the orbital mixing around its avoided crossings with the β sheet is reduced, and it becomes more dominated by xy orbital weight.
To H_{0} we add onsite Coulomb terms projected onto the t_{2g} orbitals^{52} (Methods Eq. (8)) and study the solutions to the linearized gap equation in the weakcoupling limit U/t ≪ 1, where U is the intraorbital Coulomb repulsion and t is the leading tightbinding term. We take the interorbital onsite Coulomb repulsion to be \({U}^{\prime}=U2J\), where J is the Hund’s coupling, and the pairhopping Hund’s interaction \({J}^{\prime}\) to be equal to the spinexchange Hund’s interaction J. Under these assumptions, the remaining free parameter is J/U. We take J/U = 0.15, which is close to the value J/U = 0.17 found in Refs. 53,54. The linearized gap equation reads
where μ and ν are band indices, ∣S_{ν}∣ is the area of Fermi surface sheet ν, and \(\bar{{{\Gamma }}}\) is the twoparticle interaction vertex calculated consistently to order \({{{{{{{\mathcal{O}}}}}}}}({U}^{2}/{t}^{2})\). Solutions to Eq. (2) with λ < 0 signal the onset of superconductivity, at the critical temperature \({T}_{c} \sim W\exp (1/\lambda)\), where W is the bandwidth.
In a pseudospin basis each eigenvector φ belongs to one of the ten irreducible representations of the crystal point group D_{4h}^{48,55}. We calculate the leading eigenvalues in four evenparity channels, B_{1g}, B_{2g}, A_{1g}, and A_{2g} — see the legend of Fig. 5c–d. The E_{g} channel — d_{xz} ± id_{yz} — has been found to be strongly disfavored in weakcoupling calculations^{51}, and so is not considered here.
A subset of the present authors have found, in a previous weakcoupling calculation, that the oddparity order parameters track each other closely as J/U is varied, with a splitting that is small compared with that between the evenparity orders, and between the odd and evenparity orders^{51}. Ref. 56, likewise, finds the splitting between the oddparity orders to be small. For this reason, we calculate only one oddparity channel, E_{u} (p_{x} ± ip_{y}), and its behaviour can safely be taken to represent the qualitative behaviour of oddparity order.
The leading eigenvalues in each channel as a function of ε_{zz} are shown in Fig. 5c. Although, as in Ref. 51, oddparity order is found to be favored, calculations in the random phase approximation at similar J/U tend to favor evenparity order^{15,56}. A tendency towards oddparity order appears to be a feature of calculations in the weakcoupling limit. We note also that the ordering of the channels differs from what was found in Ref. 51, due to a different tightbinding parametrisation. The ordering is sensitive to the parametrisation, and so we focus discussion here on trends with applied strain.
The weakcoupling results show a dichotomy in the strain dependence of T_{c}: T_{c} in the channels that have symmetryimposed nodes at the X and Y points (E_{u}, A_{2g}, and B_{2g}) decreases with initial caxis compression. These nodes coincide with the regions of highest local density of states, and this result is an indication that order parameters in these channels are less able to take advantage of the increase in Fermilevel density of states induced by caxis compression. However, under stronger compression T_{c} increases modestly in all channels.
In the weakcoupling calculations of Ref. 29, the contrast in the response to aaxis uniaxial stress between order parameters with and without nodes at the X and Y points was found to be stronger in \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) than T_{c}, and so we also calculate \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\), following the procedure in Ref. 29. Results are shown in Fig. 5d. We find that changes in \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) correlate closely with shifts in the gap weight onto the γ sheet, which has the lowest Fermi velocity. For example, gap weight in the B_{2g} channel, shown in Fig. 5e, shifts from the β to the γ sheet as stress is initially applied, and \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) correspondingly increases. At strong compression, gap weight shifts back to the β sheet, and \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) decreases. This occurs because as the γ sheet expands it comes closer to its copies in adjacent zones, which disfavours a large gap on this sheet because in the B_{2g} channel the gap changes sign across the zone boundary. Among the evenparity channels, for ε_{zz} < − 0.015\({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) increases for those without nodes along the ΓX and ΓY lines, and decreases for those with. The complete set of calculated gap structures is shown in the Methods section.
We conclude this section by noting that although a k_{z} dependence of the gap structure is seen in all channels, we do not find dramatic stressinduced changes in the k_{z} dependence in any channel. Separately, in the A_{2g} channel there is a level crossing between ε_{zz} = 0 and −0.0075. We plot only the leading eigenvalues in Fig. 5; this level crossing causes a large change in the leading gap structure and an anomalously large increase in \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\).
Discussion
The unexpected decrease of T_{c} as two Van Hove points in kspace are approached under caxis compression is the key experimental result that we report. It might provide a vital clue about the nature of the superconducting state in Sr_{2}RuO_{4}, because it is so different to the response to inplane, aaxis pressure. The DFT calculations indicate that our largest achieved stress, −3.2 GPa, is around 60% of the way to the Lifshitz transition, and if the calculations overestimate the compression required to reach the Lifshitz transition, as they did for inplane stress, we might have come even closer. The contrast between the effects of a and caxis stress is unmistakable: 60% of the way to the electrontoopen Lifshitz transition under aaxis stress, T_{c} is 0.7 K higher than in the unstressed sample^{28}.
The response of T_{c} under caxis compression allows resolution of the stress dependence of T_{c} into components through comparison with the the effect of hydrostatic compression, which also suppresses T_{c}. We obtain the coefficients α and β in the expression
where ΔV/V = ε_{xx} + ε_{yy} + ε_{zz} is the fractional volume change of the unit cell, and ε_{zz} − (ε_{xx} + ε_{yy})/2 is a volumepreserving tetragonal distortion. Refs. 12,57,58 report dT_{c}/dσ_{hydro} = 0.22 ± 0.02, 0.24 ± 0.02, and 0.21 ± 0.03 K/GPa; we take dT_{c}/dσ_{hydro} = 0.23 ± 0.01 K/GPa. Employing the lowtemperature elastic moduli from Ref. 37 to convert stress to strain, we find α = 34.8 ± 1.6 K and β = − 2.2 ± 1.2 K. (Under hydrostatic stress, σ_{zz}/ε_{zz} = 396 GPa and ε_{xx} = 0.814ε_{zz}.) The small value of β means that a volumepreserving reduction in the lattice parameter ratio c/a would have little effect on T_{c}: the increase in density of states by approaching the electrontohole Lifshitz transition is balanced, somehow, by weakening of the pairing interaction. The challenge for theory is to understand why that weakening takes place.
In the threedimensional weakcoupling calculations presented here, it is the A_{2g} and B_{2g} channels, both of which have nodes along the ΓX and ΓY lines, that best match observations. Due to differences between the actual and tightbinding electronic structure the ε_{zz} = 0 point in the calculations should not be considered too literally as equivalent to ε_{zz} = 0 in reality, and the key point is that it is only in the A_{2g} and B_{2g} channels that T_{c} is found to decrease and \({H}_{{{{{{{{\rm{c2}}}}}}}}}/{T}_{{{{{{{{\rm{c}}}}}}}}}^{2}\) to increase over some range of strain. However, as we have noted, A_{2g} and B_{2g} order parameters do not appear to be consistent with data under aaxis stress.
In other words, weakcoupling calculations do not explain the contrasting responses to a versus caxis stress, and this provides an opportunity: models of pairing in Sr_{2}RuO_{4} should be tested against this feature, for it might provide substantial resolving power between different models. There may, for example, be stressdriven changes in the interactions that drive superconductivity, though to attempt to calculate this is beyond the scope of this paper.
We highlight two other possible explanations. One is interorbital pairing^{22,23,24,25}. The superconducting energy scale is too weak to induce substantial band mixing, and so these models depend on the proximity of the γ and β sheets, and the resulting mixing of xy and xz/yz orbital weight over substantial sections of Fermi surface^{23}. We have noted that caxis compression reduces this mixing, by pushing the γ and β sheets apart, which could then suppress T_{c}^{59}. In contrast, under inplane uniaxial compression these sheets are pushed closer together along one direction and further apart along the other^{36}.
The other is threedimensional effects. Another feature of the electronic structure that varies oppositely under a versus caxis compression is the interlayer coupling. Under aaxis compression, the RuO_{2} sheets are pushed further apart, and under caxis compression, closer together. For example, an increase in warping of the Fermi surfaces along k_{z} under caxis compression could reduce the quality of nesting and so weaken spin fluctuations in Sr_{2}RuO_{4}, and the weakcoupling calculations here might not have fully captured the effect on the superconductivity.
In summary, we have demonstrated methods to apply uniaxial stress of multiple GPa along the interlayer axis of layered materials in samples large enough to permit highprecision magnetic susceptibility measurements. Under such a compression, we find that T_{c} decreases even though the Fermilevel DOS increases, in striking contrast to the effect of inplane uniaxial stress. Weakcoupling calculations do not provide a clear answer to this puzzle, which makes it important for models of superconductivity in Sr_{2}RuO_{4} to be tested against application of both types of stress. At a more general level, our findings motivate the use of outofplane stress as a powerful tool for investigation of other low dimensional strongly correlated system in which the strength of the interlayer coupling is suspected of playing an important role in their electronic properties.
Methods
Density functional theory calculation
DFT structure calculations were performed using the fullpotential local orbital FLPO^{60,61} version fplo 18.0052 (http://www.fplo.de). For the exchangecorrelation potential, the local density approximation applying the parametrizations of PerdewWang^{62} was chosen. Spinorbit coupling was treated nonperturbatively by solving the fourcomponent KohnShamDirac equation^{63}. To obtain precise band structure and Fermi surface information in the presence of a Van Hove singularity close to the Fermi level, the final calculations were carried out on a wellconverged mesh of 343,000 k points. (70 × 70 × 70; 23,022 points in the in the irreducible wedge of the Brillouin zone). As a starting point, for the unstrained lattice structure the structural parameters at 15 K from Ref. 64 were used. Longitudinal strain ε_{zz} is taken as the independent variable, and ε_{xx} and ε_{yy} are set following the lowtemperature Poisson’s ratio from Ref. 37, which is 0.223 for stress along the c axis. The apical oxygen position was relaxed independently at each strain, by minimising the force to below 1 meV/Å. However, the effect of relaxing this internal parameter is small in comparison with the effect of the stressdriven change in lattice parameters.
The calculated Fermi surfaces of unstressed Sr_{2}RuO_{4} and under interlayer compression, including the warping along k_{z} and the Fermi velocities, are shown in Fig. 6. The β sheet is the most strongly warped both at zero stress and at ε_{zz} = −0.025.
Experimental details
Sr_{2}RuO_{4} samples were grown using a floatingzone method^{65,66}. The four samples here were taken from the same original rod, and from a portion that we verified to have high T_{c} and a lowdensity of Ru inclusions; our aim in taking multiple samples was to test reproducibility in sample preparation and mounting.
Uniaxial stress was applied using piezoelectricdriven apparatus^{39,67}, and precision in sample mounting is important because Sr_{2}RuO_{4} is much more sensitive to inplane than caxis uniaxial stress: T_{c} decreases by 0.13 K under a caxis stress of σ_{zz} = − 3.0 GPa, but increases by 0.13 K under an inplane uniaxial stress of only 0.2 GPa^{29}. Applying caxis pressure could generate inplane stress through bending and/or sample inhomogeneity. In a previous experiment^{68}, caxis compression raised T_{c} and broadened the transition. However, the stress was applied at room temperature, where the elastic limit of Sr_{2}RuO_{4} is low^{39}, so these effects may have been a consequence of inplane strain due to defects introduced by the applied stress.
Samples 2–4 were mounted into twopart sample carriers; that for samples 2 and 3 is diagrammed in Fig. 7a. The purpose was to protect samples from inadvertent application of tensile stress. Samples are mounted across a gap between a fixed and a moving portion of part B of the carrier, and can be compressed, but not tensioned, by bringing part A into contact with part B. In Fig. 7b, we show T_{c} of samples 2 and 3 versus applied displacement, and the point where parts A and B come into contact and T_{c} starts changing is clearly visible. For sample 2 the point of contact is rounded on the scale of a few microns, due to roughness and/or misalignment of the contact faces, and in all figures below we exclude data points that we estimate to be affected by this rounding.
The samples were mounted with Stycast 2850. This epoxy layer constitutes a conformal layer that ensures even application of stress^{67}. Photographs of samples 2 and 4 are shown in Fig. 7c and d. The carrier for sample 4, which has a different design to those used for samples 2 and 3, is shown in Fig. 7e. Where electrical contacts were made, DuPont 6838 silver paste annealed at 450^{∘} for typically 30 min was used. This is longer than usual, in order to penetrate a thin insulating layer deposited during the ion beam milling.
As noted above, samples 1–3 were mounted in apparatus that had a sensor only of the displacement applied to the sample, while for sample 4 there was also a force sensor. Displacement sensors are less reliable as sensors of the state of the sample, because they also pick up deformation of the epoxy that holds the sample. In Fig. 7f the complete set of measurements of T_{c} of sample 3, plotted against applied displacement, are shown. Data points are colored by the order in which they were collected. The data drifted leftward over time: stronger compression was needed to reach the same T_{c}. However, the qualitative form of the curve — initial decrease in T_{c}, then a flattening, and then further decrease — reproduced over multiple stress cycles, and in Fig. 7g it is shown that the form of the transition was the same before and after application of the strongest compression. (We attribute the small apparent shift in T_{c} to an artefact of inadvertent mechanical contact between the stress cell and inner vacuum can of the cryostat).
We therefore conclude that the sample deformed elastically and that it was the epoxy holding the sample that was compressed nonelastically; plastic deformation has previously been observed to broaden the superconducting transition^{69} of Sr_{2}RuO_{4}. In Fig. 3, in the main text, we show only the data taken after the epoxy was maximally compressed. Force versus displacement data for sample 4 are shown in Fig. 7h–i, and here it can be seen that there was very substantial nonelastic compression of the epoxy. As with sample 3, the shape of the superconducting transition in the Sr_{2}RuO_{4} was the same before and after application of large stress. Over regions where the sample and epoxy deformed elastically, the combined spring constant was 1.45 N/μm. The spring constant of the flexures in the carrier, on the other hand, is calculated to be ~0.03 N/μm, meaning that almost all of the applied force was transferred to the sample.
Calculated gap structure in other channels
In Fig. 8 the calculated gap structures in the A_{1g}, A_{2g}, and B_{1g} channels are shown. [The B_{2g} gap structures are shown in Fig. 5e.] caxis compression favors large gaps on the γ sheet in all channels. In the A_{2g} channel, this shift occurs as a firstorder change in gap structure between ε_{zz} = 0 and ε_{zz} = − 0.0075. At the largest compression reached, gap weight in the A_{2g} channel shifts back away from the γ sheet, as it does in the B_{2g} channel. This does not occur in the A_{1g} and B_{1g} channels.
Details of the weakcoupling calculation
The tightbinding Hamiltonian from Eq. (1) takes the form
where we used the Ru orbital shorthand notation A = xz, B = yz, C = xy, and where \(\bar{s}=s\) (s being spin). In Eq. (4) the energies ε_{AB}(k) account for intraorbital (A = B) and interorbital (A ≠ B) hopping, and η_{1}, η_{2} parametrize the spinorbit coupling. We define ε_{AA}(k) = ε_{1D}(k_{x}, k_{y}, k_{z}), ε_{BB}(k) = ε_{1D}(k_{y}, k_{x}, k_{z}), and ε_{CC}(k) = ε_{2D}(k_{x}, k_{y}, k_{z}), and we retain the following terms in the matrix elements:
Here the first Brillouin zone is defined as BZ = [−π, π]^{2} × [ − 2π, 2π]. For the four values of caxis compression ε_{zz} = 0, − 0.0075, − 0.015, − 0.020 we extract the entire set of parameters from DFT calculations consistent with Fig. 1; see Table 1.
For the interactions we use the (onsite) Hubbard–Kanamori Hamiltonian
where i is site, a is orbital, and \({n}_{ias}={c}_{ias}^{{{{\dagger}}} }{c}_{ias}\) is the density operator. We further assume that \({U}^{\prime}=U2J\) and \({J}^{\prime}=J\)^{52}. In the weakcoupling limit this leaves J/U as a single parameter fully characterizing the interactions.
In the linearized gap equation (2) the (dimensionless) twoparticle interaction vertex \(\bar{{{\Gamma }}}\) is defined as^{48}
where \({\rho }_{\mu }=\vert {S}_{\mu }/[{\bar{v}}_{\mu }{(2\pi )}^{3}]\) is the density of states, and \(1/{\bar{v}}_{\mu }={\int}_{{S}_{\mu }}{{{{{{{\rm{d}}}}}}}}{{{{{{{\bf{k}}}}}}}}/\left({S}_{\mu }{v}_{\mu }({{{{{{{\bf{k}}}}}}}})\right)\). Here, Γ is the irreducible twoparticle interaction vertex which to leading order retains the diagrams shown in Fig. 9.
An eigenfunction φ of Eq. (2) corresponding to a negative eigenvalue λ yields the superconducting order parameter
In the chosen pseudospin basis each eigenvector φ belongs to one of the ten irreducible representations of the crystal point group D_{4h}^{48,55}.
Data availability
The data that support the findings of this study are openly available from the Max Planck Digital Library^{70}.
References
Maeno, Y. et al. Superconductivity in a layered perovskite without copper. Nature 372, 532 (1994).
Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr_{2}RuO_{4} and the physics of spintriplet pairing. Rev. Mod. Phys. 75, 657 (2003).
Maeno, Y., Kittaka, S., Nomura, T., Yonezawa, S. & Ishida, K. Evaluation of SpinTriplet Superconductivity in Sr_{2}RuO_{4}. J. Phys. Soc. Jpn. 81, 011009 (2012).
Mackenzie, A. P., Scaffidi, T., Hicks, C. W. & Maeno, Y. Even odder after twentythree years: the superconducting order parameter puzzle of Sr_{2}RuO_{4}. npj Quant. Mat. 2, 40 (2017).
Pustogow, A. et al. Constraints on the superconducting order parameter in Sr_{2}RuO_{4} from oxygen17 nuclear magnetic resonance. Nature 574, 72 (2019).
Ishida, K., Manago, M., Kinjo, K. & Maeno, Y. Reduction of the 17O Knight Shift in the Superconducting State and the Heatup Effect by NMR Pulses on Sr_{2}RuO_{4}. J. Phys. Soc. Jpn. 89, 034712 (2020).
Chronister, A. et al. Evidence for even parity unconventional superconductivity in Sr_{2}RuO_{4}. PNAS 118, e2025313118 (2021).
Petsch, A. N. et al. Reduction of the Spin Susceptibility in the Superconducting State of Sr_{2}RuO_{4} Observed by Polarized Neutron Scattering. Phys. Rev. Lett. 125, 217004 (2020).
Luke, G. M. et al. Timereversal symmetrybreaking superconductivity in Sr_{2}RuO_{4}. Nature 394, 558 (1998).
Grinenko, V. et al. Split superconducting and timereversal symmetrybreaking transitions in Sr_{2}RuO_{4} under stress. Nat. Phys. 17, 748 (2021).
Xia, J., Maeno, Y., Beyersdorf, P. T., Fejer, M. M. & Kapitulnik, A. High Resolution Polar Kerr Effect Measurements of Sr_{2}RuO_{4}: Evidence for Broken TimeReversal Symmetry in the Superconducting State. Phys. Rev. Lett. 97, 167002 (2006).
Grinenko, V. et al. Unsplit superconducting and time reversal symmetry breaking transitions in Sr_{2}RuO_{4} under hydrostatic pressure and disorder. Nat. Comm. 12, 3920 (2021).
Bergemann, C., Mackenzie, A. P., Julian, S. R., Forsythe, D. & Ohmichi, E. Quasitwodimensional Fermi liquid properties of the unconventional superconductor Sr_{2}RuO_{4}. Adv. Phys. 52, 639–725 (2003).
Ohmichi, E. et al. Magnetoresistance of Sr_{2}RuO_{4} under high magnetic fields parallel to the conducting plane. Phys. Rev. B 61, 7101 (2000).
Rømer, A. T., Scherer, D. D., Eremin, I. M., Hirschfeld, P. J. & Andersen, B. M. Knight Shift and Leading Superconducting Instability from Spin Fluctuations in Sr_{2}RuO_{4}. Phys. Rev. Lett. 123, 247001 (2019).
Rømer, A. T., Hirschfeld, P. J. & Andersen, B. M. Superconducting state of Sr_{2}RuO_{4} in the presence of longerrange Coulomb interactions. Phys. Rev. B 104, 064507 (2021).
Kivelson, S. A., Yuan, A. C., Ramshaw, B. & Thomale, R. A proposal for reconciling diverse experiments on the superconducting state in Sr_{2}RuO_{4}. npj Quantum Mater. 5, 43 (2020).
Wagner, G., Røising, H. S., Flicker, F. & Simon, S. H. A microscopic GinzburgLandau theory and singlet ordering in Sr_{2}RuO_{4}. Phys. Rev. B 104, 134506 (2021).
Scaffidi, T. Degeneracy between even and oddparity superconductivity in the quasi1D Hubbard model and implications for Sr_{2}RuO_{4}. Preprint at https://arxiv.org/abs/2007.13769 (2020).
Willa, R., Hecker, M., Fernandes, R. M. & Schmalian, J. Inhomogeneous timereversal symmetry breaking in Sr_{2}RuO_{4}. Phys. Rev. B 104, 024511 (2021).
Li, Y.S. et al. Highsensitivity heatcapacity measurements on Sr_{2}RuO_{4} under uniaxial pressure. PNAS 118, e2020492118 (2021).
Suh, H.G. et al. Stabilizing evenparity chiral superconductivity in Sr_{2}RuO_{4}. Phys. Rev. Res. 2, 032023(R) (2020).
Clepkens, J., Lindquist, A. W. & Kee, H.Y. Shadowed triplet pairings in Hund’s metals with spinorbit coupling. Phys. Rev. Res. 3, 013001 (2021).
Gingras, O., Nourafkan, R., Tremblay, A.M. S. & Côté, M. Superconducting symmetries of Sr_{2}RuO_{4} from firstprinciples electronic structure. Phys. Rev. Lett. 123, 217005 (2019).
Ramires, A. & Sigrist, M. Superconducting order parameter of Sr_{2}RuO_{4}: a microscopic perspective. Phys. Rev. B 100, 104501 (2019).
Sharma, R. et al. Momentumresolved superconducting energy gaps of Sr_{2}RuO_{4} from quasiparticle interference imaging. PNAS 117, 5222 (2020).
Kashiwaya, S. et al. Timereversal invariant superconductivity of Sr_{2}RuO_{4} revealed by Josephson effects. Phys. Rev. B 100, 094530 (2019).
Barber, M. E. et al. Role of correlations in determining the Van Hove strain in Sr_{2}RuO_{4}. Phys. Rev. B 100, 245139 (2019).
Steppke, A. et al. Strong peak in T_{c} of Sr_{2}RuO_{4} under uniaxial pressure. Science 355, eaaf9398 (2017).
Burganov, B. et al. Strain control of fermiology and manybody interactions in twodimensional ruthenates. Phys. Rev. Lett. 116, 197003 (2016).
Hsu, Y.T. et al. Manipulating superconductivity in ruthenates through Fermi surface engineering. Phys. Rev. B 94, 045118 (2016).
Liu, Y.C., Wang, W.S., Zhang, F.C. & Wang, Q.H. Superconductivity in Sr_{2}RuO_{4} thin films under biaxial strain. Phys. Rev. B 97, 224522 (2018).
Kikugawa, N. et al. Rigidband shift of the Fermi level in the strongly correlated metal: Sr_{2−y}La_{y}RuO_{4}. Phys. Rev. B 70, 060508(R) (2004).
Kikugawa, N., Bergemann, C., Mackenzie, A. P. & Maeno, Y. Bandselective modification of the magnetic fluctuations in Sr_{2}RuO_{4}: A study of substitution effects. Phys. Rev. B 70, 134520 (2004).
Shen, K. M. et al. Evolution of the Fermi surface and quasiparticle renormalization through a van Hove singularity in Sr_{2−y}La_{y} RuO_{4}. Phys. Rev. Lett. 99, 187001 (2007).
Sunko, V. et al. Direct Observation of a Uniaxial Stressdriven Lifshitz Transition in Sr_{2}RuO_{4}. npj Quantum Mater. 4, 46 (2019).
Ghosh, S. et al. Thermodynamic Evidence for a TwoComponent Superconducting Order Parameter in Sr_{2}RuO_{4}. Nat. Phys. 17, 199 (2021).
Sypek, J. T. et al. Superelasticity and cryogenic linear shape memory effects of CaFe_{2}As_{2}. Nat. Comm. 8, 1083 (2017).
Barber, M. E., Steppke, A., Mackenzie, A. P. & Hicks, C. W. Piezoelectricbased uniaxial pressure cell with integrated force and displacement sensors. Rev. Sci. Inst. 90, 023904 (2019).
Yamazaki, K. et al. Angular dependence of vortex states in Sr_{2}RuO_{4}. Phys. C. 378381, 537 (2002).
Kohn, W. & Luttinger, J. M. New Mechanism for Superconductivity. Phys. Rev. Lett. 15, 524–526 (1965).
Baranov, M. A. & Kagan, M. Y. Dwave pairing in the twodimensional Hubbard model with low filling. Z. Phys. B 86, 237–239 (1992).
Kagan, M. Y. & Chubukov, A. Increase in superfluid transition in polarized Fermi gas with repulsion. JETP Lett. 50, 517 (1989).
Chubukov, A. V. & Lu, J. P. Pairing instabilities in the twodimensional Hubbard model. Phys. Rev. B 46, 11163–11166 (1992).
Baranov, M. A., Chubukov, A. V. & Kagan, M. Y. Superconductivity and superfluidity in Fermi systems with repulsive interactions. Int. J. Mod. Phys. A 06, 2471–2497 (1992).
Fukazawa, H. & Yamada, K. Third Order Perturbation Analysis of Pairing Symmetry in TwoDimensional Hubbard Model. J. Phys. Soc. Jpn. 71, 1541–1547 (2002).
Hlubina, R. Phase diagram of the weakcoupling twodimensional \(t{t}^{\prime}\) Hubbard model at low and intermediate electron density. Phys. Rev. B 59, 9600–9605 (1999).
Raghu, S., Kivelson, S. A. & Scalapino, D. J. Superconductivity in the repulsive Hubbard model: An asymptotically exact weakcoupling solution. Phys. Rev. B 81, 224505 (2010).
Raghu, S., Kapitulnik, A. & Kivelson, S. A. Hidden QuasiOneDimensional Superconductivity in Sr_{2}RuO_{4}. Phys. Rev. Lett. 105, 136401 (2010).
Scaffidi, T., Romers, J. C. & Simon, S. H. Pairing symmetry and dominant band in Sr_{2}RuO_{4}. Phys. Rev. B 89, 220510 (2014).
Røising, H. S., Scaffidi, T., Flicker, F., Lange, G. F. & Simon, S. H. Superconducting order of Sr_{2}RuO_{4} from a threedimensional microscopic model. Phys. Rev. Res. 1, 033108 (2019).
Dagotto, E., Hotta, T. & Moreo, A. Colossal magnetoresistant materials: the key role of phase separation. Phys. Rep. 344, 1–153 (2001).
Mravlje, J. et al. Coherenceincoherence crossover and the massrenormalization puzzles in Sr_{2}RuO_{4}. Phys. Rev. Lett. 106, 096401 (2011).
Tamai, A. et al. HighResolution Photoemission on Sr_{2}RuO_{4} Reveals CorrelationEnhanced Effective SpinOrbit Coupling and Dominantly Local SelfEnergies. Phys. Rev. X 9, 021408 (2019).
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).
Wang, Z., Wang, X. & Kallin, C. Spinorbit coupling and spintriplet pairing symmetry in Sr_{2}RuO_{4}. Phys. Rev. B 101, 064507 (2020).
Forsythe, D. et al. Evolution of FermiLiquid Interactions in Sr_{2}RuO_{4} under Pressure. Phys. Rev. Lett. 89, 166402 (2002).
Svitelskiy, O. et al. Influence of hydrostatic pressure on the magnetic phase diagram of superconducting Sr_{2}RuO_{4} by ultrasonic attenuation. Phys. Rev. B 77, 052502 (2008).
Beck, S., Hampel, A., Zingl, M., Timm, C. & Ramires, A. Effects of strain in multiorbital superconductors: The case of Sr_{2}RuO_{4}. Phys. Rev. Res. 4, 023060 (2022).
Koepernik, K. & Eschrig, H. Fullpotential nonorthogonal localorbital minimumbasis bandstructure scheme. Phys. Rev. B 59, 1743 (1999).
Opahle, I., Koepernik, K. & Eschrig, H. Fullpotential bandstructure calculation of iron pyrite. Phys. Rev. B 60, 14035 (1999).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Eschrig, H., Richter, M. and Opahle, I. Relativistic solid state calculations. Relativistic Electronic Structure Theory, Part 2. Applications 13 (2014).
Chmaissem, O., Jorgensen, J. D., Shaked, H., Ikeda, S. & Maeno, Y. Thermal expansion and compressibility of Sr_{2}RuO_{4}. Phys. Rev. B 57, 5067 (1998).
Mao, Z. Q., Maeno, Y. & Fukazawa, H. Crystal growth of Sr_{2}RuO_{4}. Mater. Res. Bull. 35, 1813 (2000).
Bobowski, J. S. et al. Improved singlecrystal growth of Sr_{2}RuO_{4}. Condens. Matter 4, 6 (2019).
Hicks, C. W., Barber, M. E., Edkins, S. D., Brodsky, D. O. & Mackenzie, A. P. Piezoelectricbased apparatus for strain tuning. Rev. Sci. Inst. 85, 065003 (2014).
Kittaka, S., Taniguchi, H., Yonezawa, S., Yaguchi, H. & Maeno, Y. HigherT_{c} superconducting phase in Sr_{2}RuO_{4} induced by uniaxial pressure. Phys. Rev. B 81, 180510 (2010).
Taniguchi, H., Nishimura, K., Goh, S. K., Yonezawa, S. & Maeno, Y. HigherT_{c} superconducting phase in Sr_{2}RuO_{4} induced by inplane uniaxial pressure. J. Phys. Soc. Jpn. 84, 014707 (2015).
Data available at the Max Planck Digital Library: https://doi.org/10.17617/3.3O1ZM2.
Acknowledgements
We thank Aline Ramires and Carsten Timm for helpful discussions, Markus König for training on the focused ion beam, and Felix Flicker for help with development and running of the code. H.R. thanks U. Nitzsche for technical support. F.J., A.P.M., and C.W.H. acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)  TRR 288  422213477 (project A10). H.S.R. and S.H.S. acknowledge the financial support of the Engineering and Physical Sciences Research Council (UK). H.S.R. acknowledges support from the Aker Scholarship. T.S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), in particular the Discovery Grant [RGPIN202005842], the Accelerator Supplement [RGPAS202000060], and the Discovery Launch Supplement [DGECR202000222]. This research was enabled in part by support provided by Compute Ontario (http://www.computeontario.ca) and Compute Canada (http://www.computecanada.ca). N.K. is supported by a KAKENHI GrantsinAids for Scientific Research (Grant Nos. 17H06136, 18K04715, and 21H01033), and CoretoCore Program (No. JPJSCCA20170002) from the Japan Society for the Promotion of Science (JSPS) and by a JSTMirai Program (Grant No. JPMJMI18A3).
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F.J., C.W.H., and A.P.M. designed the research project; F.J. and A.S. performed the measurements; N.K. and D.A.S. grew the samples; H.R. performed the density functional theory and H.S.R., T.S. and S.H.S. the weakcoupling calculations; F.J. and C.W.H. analyzed data; and F.J. and C.W.H. wrote the paper with contributions from the other authors.
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Jerzembeck, F., Røising, H.S., Steppke, A. et al. The superconductivity of Sr_{2}RuO_{4} under caxis uniaxial stress. Nat Commun 13, 4596 (2022). https://doi.org/10.1038/s41467022321774
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DOI: https://doi.org/10.1038/s41467022321774
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