## Squeezing out the oddness

The material Sr_{2}RuO_{4} has long been thought to exhibit an exotic, odd-parity kind of superconductivity, not unlike the superfluidity in _{3}He. How would perturbing this material's electronic structure affect its superconductivity? Steppke *et al.* put the material under large uniaxial pressure and found that the critical temperature more than doubled and then fell as a function of strain (see the Perspective by Shen). The maximum critical temperature roughly coincided with the point at which the material's Fermi surface underwent a topological change. One intriguing possibility is that squeezing changed the parity of the superconducting gap from odd to even.

*Science*, this issue p. 10.1126/science.aaf9398; see also p. 133

## Structured Abstract

### INTRODUCTION

A central challenge of modern condensed matter physics is to understand the range of possible collective states formed by assemblies of strongly interacting electrons. Most real materials contain high levels of disorder, which can disrupt possible ordered states and so substantially hinder the path to understanding. There is a premium, therefore, on working with extremely clean materials and identifying clean ways to tune their physical properties. Here, we show that uniaxial pressure can induce profound changes in the superconductivity of one of the model materials in the field, Sr_{2}RuO_{4}, and demonstrate using explicit calculations how our findings provide strong constraints on theory.

### RATIONALE

Superconductivity remains arguably the most intriguing collective electron state. All superconductors form from the condensation of pairs of electrons into a single ground state, but in “unconventional” superconductors, a rich variety of qualitatively different ground states is possible. One of the most celebrated examples, and the one with the lowest known levels of disorder, is Sr_{2}RuO_{4}. Previous experimental results suggest that its superconducting condensate has odd parity, that is, its phase is reversed upon inversion of spatial coordinates. A relatively unexplored route to test this possibility is to perturb the assembly of conduction electrons through lattice distortion, which introduces no additional disorder. Electronic structure calculations suggest that if sufficient uniaxial pressure could be applied to compress the lattice along the pressure axis by about 0.8%, the largest Fermi surface of Sr_{2}RuO_{4} would undergo a topological transition. One of the consequences of tuning to this transition would be to substantially lower the velocity of some of charge carriers, and because slow carriers are generally favorable for superconductivity, the superconductivity might be profoundly affected. Although this topological transition has been achieved with other experimental techniques, too much disorder was introduced for the superconductivity to survive.

### RESULTS

Our central experimental result is summarized in the figure. We prepare the sample as a beam and use piezoelectric stacks to compress it along its length. Compressing the *a* axis of the Sr_{2}RuO_{4} lattice drives the superconducting transition temperature (*T*_{c}) through a pronounced maximum, at a compression of ≈0.6%, that is a factor of 2.3 higher than *T*_{c} of the unstrained material. At the maximum *T*_{c}, the superconducting transition is very sharp, allowing precise determination of the superconducting upper critical magnetic fields for fields along both the *a* and *c* directions. The *c*-axis upper critical field is found to be enhanced by more than a factor of 20. We perform calculations using a weak-coupling theory to compare the *T*_{c}’s and upper critical fields of possible superconducting order parameters. The combination of our experimental and theoretical work suggests that the maximum *T*_{c} is likely associated with the predicted Fermi surface topological transition and that at this maximum *T*_{c}, Sr_{2}RuO_{4} might have an even-parity rather than an odd-parity superconducting order parameter. The anisotropic distortion is key to these results: Hydrostatic pressure is known experimentally to decrease *T*_{c} of Sr_{2}RuO_{4}.

### CONCLUSION

Our data raise the possibility of an odd-parity to even-parity transition of the superconducting state of Sr_{2}RuO_{4} as a function of lattice strain and fuel an ongoing debate about the symmetry of the superconducting state even in the unstrained material. We anticipate considerable theoretical activity to address these issues, and believe that the technique developed for these experiments will also have a broader significance to future study of quantum magnets, topological systems, and electronic liquid crystals as well as superconductors.

## Abstract

Sr_{2}RuO_{4} is an unconventional superconductor that has attracted widespread study because of its high purity and the possibility that its superconducting order parameter has odd parity. We study the dependence of its superconductivity on anisotropic strain. Applying uniaxial pressures of up to ~1 gigapascals along a 〈100〉 direction (*a* axis) of the crystal lattice results in the transition temperature (*T*_{c}) increasing from 1.5 kelvin in the unstrained material to 3.4 kelvin at compression by ≈0.6%, and then falling steeply. Calculations give evidence that the observed maximum *T*_{c} occurs at or near a Lifshitz transition when the Fermi level passes through a Van Hove singularity, and open the possibility that the highly strained, *T*_{c} = 3.4 K Sr_{2}RuO_{4} has an even-parity, rather than an odd-parity, order parameter.

The formation of superconductivity by the condensation of electron pairs into a coherent state is one of the most spectacular many-body phenomena in physics. Initially, all known superconducting condensates were of the same basic class, in which electrons paired into spin-singlet states, forming condensates of even parity whose phase ϕ is independent of wave vector **k** (*1*). Condensates of this form are insensitive to the presence of nonmagnetic scattering and so are easier to observe in materials grown with standard levels of disorder. In the last three decades, a richer and more exciting picture has emerged. In the growing number of known unconventional superconductors, both the phase and amplitude of the condensate order parameter have strong **k** dependence. Unconventional superconductors can have both even and odd parity, and are sensitive to the presence of disorder (*2*, *3*). These materials give a unique opportunity to study the collective physics of interacting electrons and the mechanisms by which the condensation from the normal metallic state occurs. However, considerable material and experimental challenges must be overcome.

The subject of the research described in this paper, Sr_{2}RuO_{4} (transition temperature *T*_{c} ≈ 1.5 K) (*4*), is the most disorder-sensitive of all known superconductors (*5*). However, the stringent requirements this places on material purity also bring advantages. The long mean free paths of ~1 μm that are required to observe its superconductivity in the clean limit have also enabled extensive studies of its normal state via the de Haas–van Alphen effect (*6*). This work, combined with angle-resolved photoemission experiments (*7*) and electronic structure calculations (*8*–*10*), has led to a detailed understanding of the quasi–two-dimensional (2D) Fermi surface topography and the effective masses of the Landau Fermi liquid quasiparticles, which pair to form the superconducting condensate.

However, despite more than two decades of work, the superconducting order parameter is not known with certainty. Soon after the discovery of its superconductivity, the similarity of the Landau parameters of Sr_{2}RuO_{4} to those of the famous p-wave superfluid ^{3}He led to the proposal that it might be an odd-parity superconductor with spin-triplet p-wave pairing (*11*). Knight shift measurements (*12*, *13*) and, recently, proximity-induced superconductivity in epitaxial ferromagnetic SrRuO_{3} layers (*14*) provide strong evidence for triplet pairing. Muon spin rotation (*15*) and Kerr rotation (*16*) experiments point to time-reversal symmetry breaking at *T*_{c}, and tunneling spectroscopy to chiral edge states (*17*). Josephson interferometry indicates the presence of domains in the superconducting state and gives evidence for odd parity (*18*, *19*). In combination, these observations suggest the existence of a chiral, spin-triplet superconducting state with an order parameter of the form *p*_{x} ± *ip*_{y}. Although the edge currents predicted for chiral p-wave order are not seen (*20*–*22*), there are proposals to explain why these might be unobservably small in Sr_{2}RuO_{4} (*23*–*26*). More difficult to explain in the context of spin-triplet pairing is why the upper critical field *H*_{c2} for in-plane fields is first-order at low temperatures (*27*) and smaller than predictions for orbital limiting based on anisotropic Ginzburg-Landau theory (*28*). More of the complete reviews of the superconductivity of Sr_{2}RuO_{4} and arguments for and against various order parameters can be found in (*29*–*32*).

The electronic structure of Sr_{2}RuO_{4} is relatively simple compared with those of many unconventional superconductors. Its Fermi surfaces are known with accuracy and precision (*6*), and it shows good Fermi liquid behavior in the normal state (*33*). Therefore, gaining a full understanding of the superconductivity of Sr_{2}RuO_{4} is an important challenge and a benchmark for the field. An approach not extensively explored so far is to perturb the underlying electronic structure as far as possible from its native state and observe the effects on the superconductivity. Partial substitution of La for Sr (*34*, *35*) and epitaxial thin film growth on lattice-mismatched substrates (*36*) have both been used to push one of the Fermi surface sheets of Sr_{2}RuO_{4} through a Lifshitz transition, that is, a topological change in the Fermi surface, and an associated Van Hove singularity (VHS) in the density of states (DOS). This is a major qualitative change in the electronic structure, and it would be interesting to see how the superconductivity responds. However, the disorder sensitivity of the superconductivity of Sr_{2}RuO_{4} is so strong that it was not possible to do either experiment in a sufficiently clean way for any superconductivity to survive.

In principle, uniaxial pressure has the potential for tuning the electronic structure of Sr_{2}RuO_{4} without introducing disorder and destroying the superconductivity. Pressure applied along a 〈100〉 lattice direction, lifting the native tetragonal symmetry of Sr_{2}RuO_{4}, has been shown to increase the bulk *T*_{c} to at least 1.9 K (*37*). There are hints that *T*_{c} ~ 3 K in pure Sr_{2}RuO_{4} is achievable with lattice distortion (*38*, *39*); however, it has only been seen locally, which complicates determination of its origin and properties. By extending the piezoelectric-based compression techniques introduced in (*37*) to achieve much higher compressions, we demonstrate in this work the existence of a well-defined peak in *T*_{c} at 3.4 K, at about 0.6% compression. The Young’s modulus of Sr_{2}RuO_{4} is 176 GPa (*40*), so this compression corresponds to a uniaxial pressure of ~1 GPa. A factor of 2.3 increase in *T*_{c} is accompanied by more than a factor of 20 enhancement of *H*_{c2} for fields along the *c* axis. We complement our experimental observations with two classes of calculation. Density functional theory (DFT) calculations give evidence that the peak in *T*_{c} likely coincides with a Lifshitz transition. Then, to gain insight into the effect of these large strains on possible superconducting order parameters of Sr_{2}RuO_{4}, we use weak-coupling calculations that include spin-orbit and interband couplings, extending the work of (*41*).

## Calculated band structure of Sr_{2}RuO_{4}

For guidance on the likely effect of strain on the electronic structure, we start with the results of the DFT calculations of the band structure of Sr_{2}RuO_{4}. Unstrained lattice parameters were taken from the *T* = 15 K data of (*42*). In the experiment, the sample is a high–aspect ratio bar that is compressed or tensioned along its length, so in the calculation, the longitudinal strain ε_{xx} is an independent variable, and the transverse strains are set, as in the experiment, according to the Poisson’s ratios of Sr_{2}RuO_{4}: ε_{yy} = −*ν*_{xy}ε_{xx} and ε_{zz} = −*ν*_{xz}ε_{xx} (*40*).

The robustness of the results against different standard approximations was verified by calculations with a moderate density of *k* points; more details are given in Materials and Methods. The final calculations, made in the local density approximation (LDA) with spin-orbit coupling (SOC) and apical oxygen position relaxation, were then extended to 343,000 *k* points: Because of proximity of the VHS to the Fermi level, an unusually large number of *k* points were required for convergence. The first Lifshitz transition was found to occur with a compressive strain of ε_{xx} = ε_{VHS} ≈ −0.0075. The calculated Fermi surfaces at ε_{xx} = 0 and ε_{xx} = ε_{VHS} are shown in Fig. 1, where it can be seen that compression along leads to a Lifshitz transition in the γ Fermi surface along *k*_{y}. Because of the low *k*_{z} dispersion, it occurs for all *k*_{z} over a very narrow range of ε_{xx}, starting at ε_{xx} = (−0.75 ± 0.01) × 10^{−2} and finishing by (−0.77 ± 0.01) × 10^{−2}. Cross sections at *k*_{z} = 0 are also shown. In fully 2D approximations of Sr_{2}RuO_{4}, the Lifshitz transition occurs at a single Van Hove point, labeled in the figure and coinciding with the 2D zone boundary of an isolated RuO_{2} sheet. The calculated change in the total DOS as a function of tensile and compressive strains (Fig. 1C) has sharp maxima that indicate Lifshitz transitions. It should be taken as only a qualitative guide to expectations for real Sr_{2}RuO_{4}, in which many-body effects are likely to strengthen the quasiparticle renormalization of *v*_{F} and the DOS in the vicinity of the peaks. The peak on the tension side corresponds to a Lifshitz transition along *k*_{x}, which is not accessible experimentally because samples break under strong tension.

## Measurements of superconducting properties under uniaxial pressure

The experimental apparatus is based on that presented in (*37*, *43*) but modified to achieve the larger strains required for the current project. Samples were cut with a wire saw into high–aspect-ratio bars and annealed at 450°C for 2 days in air to partially relax dislocations created by the cutting. Their ends were secured in the apparatus with epoxy (Fig. 2) (*44*). Piezoelectric actuators push or pull on one end to strain the exposed central portion of the sample; to achieve high strains, 18-mm-long actuators are used, instead of the 4-mm-long ones used previously. Because samples break under strong tension, we worked here almost exclusively with compression. The superconducting transitions were measured magnetically by measuring the mutual inductance between two coils of diameter ~1 mm placed near the center of the sample. The root mean square applied excitation field was ~0.2 Oe, mostly parallel to the *c* axes of the samples, at frequencies between 1 and 20 kHz. Some samples also had electrical contacts for resistivity measurements.

Five samples were measured in total, and all gave consistent results. Figure 3 shows the real part of the magnetic susceptibility χ′ against temperature at various compressive strains for samples 1 and 3, with zero-strain *T*_{c}’s of about 1.4 K. The strains are determined using a parallel-plate capacitive sensor incorporated into the apparatus. This sensor returns the applied displacement, and the sample strain is determined by dividing this displacement by the length of the strained portion of the sample. This strained length is affected, in turn, by elastic deformation of the epoxy that secures the sample. Comparing results from different samples, expected to have the same intrinsic behavior, yields a ~20% uncertainty in the strain determination, whose dominant origin is probably variability and uncertainty in the geometry and elastic properties of the epoxy.

When samples are initially compressed, the transition moves to higher temperature and broadens somewhat. This broadening differs in form and magnitude from sample to sample, so is probably extrinsic. For example, imperfection in the sample mounts is likely to lead to some sample bending as force is applied, imposing a strain gradient across the thickness of the sample, and, in addition, a low density of dislocations and/or ruthenium inclusions may introduce some internal strain disorder. However, despite the likely presence of some strain inhomogeneity, the transition becomes very sharp as it approaches the maximum *T*_{c}, about 3.4 K. Sample 3 could be compressed well beyond this maximum, and *T*_{c} was found to drop rapidly. In checks made on multiple samples, upon releasing the strain and returning to ε_{xx} ~ 0, the χ′(*T*) curves were found to be unchanged [see fig. S4 (*45*)], indicating that the sample deformation is elastic.

The peak in *T*_{c} can be seen in the graph of *T*_{c} against ε_{xx} for samples 1, 3, and 5 (Fig. 4). The strain scales have been normalized in the plot. At the peak, from averaging independent determinations from samples 1, 2, 3, and 5, ε_{xx} is (−0.60 ± 0.06) × 10^{−2}. The graph is based exclusively on magnetic measurements. The maximum *T*_{c} of sample 5, at ≈3.5 K, slightly exceeds that of the other samples. Resistivity measurements can show anomalously high *T*_{c} because of percolation along locally strained paths; however, on samples where the resistivity was measured (samples 3 and 5), the resistive transitions never exceeded the highest magnetic *T*_{c} by more than 0.08 K, confirming that it is the maximum *T*_{c}.

The apparatus made from nonmagnetic materials, allowing measurement of the superconducting critical fields. Sample 4 was mounted in a vector magnet, with the pressure axis (a 〈100〉 lattice direction) parallel to the magnetic *z* axis, allowing the *c* axis and in-plane upper critical fields to be measured in a single cooldown. The very sharp transitions in χ′(*T*) of Sr_{2}RuO_{4} compressed to the peak in *T*_{c} (referred to henceforth as *T*_{c} = 3.4 K Sr_{2}RuO_{4}) make determination of *T*_{c} and *H*_{c2} very simple: In all temperature and field ramps, a sharp cusp in χ′(*T*) was observed, which could be identified as *T*_{c} or *H*_{c2}. Specifically, the transition was identified as the intersection of linear fits to data just below and above the cusp. The in-plane *H*_{c2} of Sr_{2}RuO_{4} is known to be very sensitive to precise alignment of the field with the plane, so for in-plane measurements, the vector field capability was used to align the field to within 0.2° of the *ab* plane. Within the *ab* plane, the alignment to the 〈100〉 direction is with standard ~3° precision. In long field ramps, the magnet was found to have ~0.1 T–scale hysteresis, so when field ramps were performed, the transition was first located approximately and then precisely with up- and down-ramps over a 0.35-T range, for which the magnet hysteresis was found to be ~10 mT.

Results are shown in Fig. 5. The *c*-axis *H*_{c2}, *H*_{c2∥c}, of *T*_{c} = 3.4 K Sr_{2}RuO_{4} is concave and, at *T* → 0, slightly exceeds the 1.5 T limit of the transverse coils of the vector magnet. For in-plane fields, the upper critical field *H*_{c2∥a} reaches 4.7 T as *T* → 0, and both temperature and field ramps show hysteresis below ≈1.8 K, indicating a first-order transition.

A concave *H*_{c2}(*T*) curve is an indication of high gap nonuniformity, that is, substantially different gap magnitudes on different Fermi sheets, or strong variation within each sheet, or both. It has been seen in, for example, MgB_{2} (*46*) and Be(Fe,Co)_{2}As_{2} (*47*). In *T*_{c} = 3.4 K Sr_{2}RuO_{4}, the slope |*dH*_{c2∥c}/*dT*| is found to steadily increase to the lowest temperatures measured, although *H*_{c2∥c}(*T*) must eventually become convex because *dH*_{c2}/*dT* must approach zero as *T* → 0. *H*_{c2∥c} of unstrained Sr_{2}RuO_{4}, from (*48*) (Fig. 5D), is weakly concave at higher temperatures, but only above ~0.7 K, a much higher fraction of *T*_{c} (*H* = 0) than the concave-convex crossover in *T*_{c} = 3.4 K Sr_{2}RuO_{4}. This difference in the *H*_{c2}(*T*) curves indicates that the gap varies more widely across the Fermi surfaces in *T*_{c} = 3.4 K Sr_{2}RuO_{4} than in unstrained Sr_{2}RuO_{4}.

## Gap symmetry in *T*_{c} = 3.4 K Sr_{2}RuO_{4}

The *T* → 0 critical field values for *T*_{c} = 3.4 K Sr_{2}RuO_{4} are striking. *H*_{c2∥c}(*T* → 0) is enhanced by more than a factor of 20 relative to unstrained Sr_{2}RuO_{4}. *H*_{c2∥a}(*T* → 0) of unstrained Sr_{2}RuO_{4} is 1.5 T (*28*), and it is enhanced by a factor of only ≈3 in *T*_{c} = 3.4 K Sr_{2}RuO_{4}. In the simplest picture of a fully 2D triplet superconductor with the spins in the plane, the ratio γ_{s} between *H*_{c2∥a} and *H*_{c2∥c} would be infinite, because neither orbital nor Pauli limiting would apply for in-plane fields (*49*). However, we observe that γ_{s} is reduced from a value of ≈20 in unstrained Sr_{2}RuO_{4} to ≈3 in *T*_{c} = 3.4 K Sr_{2}RuO_{4}. The electronic structure calculations presented in Fig. 1 indicate that Sr_{2}RuO_{4} remains quasi-2D at high strains, a result supported by the observation in Fig. 5 that just below *T*_{c}, the slope |*dH*_{c2∥a}/*dT*| far exceeds |*dH*_{c2∥c}/*dT*|. Therefore, it seems unlikely that such a reduction in γ_{s} could arise from an orbital limiting effect. In contrast, the first-order nature of the transition under strong in-plane field is consistent with a hypothesis of Pauli limiting (*50*), as is the absolute value of *H*_{c2∥a}. In a mean-field superconductor, both *T*_{c} and the Pauli-limited *H*_{c2} are expected to vary linearly with the *T* → 0 gap magnitude |Δ| (*51*). The rise of *H*_{c2∥a} (*T* > 0) from 1.5 to 4.7 T in *T*_{c} = 3.4 K Sr_{2}RuO_{4} is somewhat, but not drastically, faster than linear against *T*_{c}. In combination, these observations motivate investigation of whether the *T*_{c} = 3.4 K state might be an even-parity condensate of spin-singlet pairs.

A qualitative analysis of the enhancement of *H*_{c2∥c} with strain also points to this possibility. In a mean-field superconductor, the orbitally limited *H*_{c2}(*T* → 0) is proportional to a weighted average of [|Δ|*N*(*E*_{F})]^{2}, where *N*(*E*_{F}) is the Fermi surface DOS. Because *T*_{c} is proportional to a *k*-space average of |Δ|, if |Δ(**k**)| is multiplied by a factor and *N*(*E*_{F}) is not modified, the quantity *H*_{c2}/*T*_{c}^{2} should remain constant. However, when Sr_{2}RuO_{4} is pressurized along a 〈100〉 direction, *N*(*E*_{F}) is substantially modified: It increases strongly near the Van Hove point, so if |∆| is large in this region of the Brillouin zone, *H*_{c2}/*T*_{c}^{2} might increase with strain. The Van Hove point is invariant under inversion, so |∆| of an odd-parity order must be zero at the Van Hove point and parametrically small in its vicinity. Qualitatively, one might therefore expect stronger enhancement of *H*_{c2}/*T*_{c}^{2} for even-parity order, for which large |Δ| is allowed near the Van Hove point, than for odd-parity order, where |Δ| must be small in just the regions where *N*(*E*_{F}) is largest.

We observe, based on the data in Fig. 5, that is enhanced by a factor of 3.6 in *T*_{c} = 3.4 K Sr_{2}RuO_{4}. Alternatively, because the form of *H*_{c2}(*T*) is so different between unstrained and *T*_{c} = 3.4 K Sr_{2}RuO_{4}, it may be preferable to take a measure of *H*_{c2} that relies only on data near *T*_{c}, that is, a hypothetical *H*_{c2}(0) for the *T* → *T*_{c} gap structure that excludes anomalous strengthening of the superconductivity at lower temperatures. Applying the Werthamer-Helfand-Hohenberg formula, *H*_{c2}(0) = −0.7(*dH*_{c2}/*dT*)*T*_{c} (*52*), yields 0.70 and 0.056 T, respectively, for sample 4 strained to maximum *T*_{c} and for the unstrained sample of Fig. 5D. If these values are used in place of the actual *H*_{c2∥c}(*T* → 0), the enhancement is 1.8. In terms of the argument discussed above, the enhancement of defined by either criterion seems to favor an even-parity over an odd-parity order parameter for *T*_{c} = 3.4 K Sr_{2}RuO_{4}.

To investigate these qualitative arguments in more depth and on the basis of a realistic calculation taking into account the multisheet Fermi surface of Sr_{2}RuO_{4}, we have extended to strained Sr_{2}RuO_{4} a 2D weak-coupling calculation, presented in (*41*) as an extension of ideas first presented in (*53*). The advantage of the weak-coupling approach is that it allows an unbiased comparison of different possible superconducting order parameters. Although the weak-coupling approximation is questionable in materials, such as Sr_{2}RuO_{4}, in which the Hubbard parameter *U* on the order of the bandwidth (*54*), the key results of (*41*) were recently reproduced in a finite-*U* calculation of Sr_{2}RuO_{4} (*55*), further motivating the use of the weak-coupling approximation here. In our calculations, whose details are discussed further in (*45*), a tight-binding model of all three Fermi surfaces of Sr_{2}RuO_{4} is specified, including the effects of spin-orbit and interband couplings, and fitted to the experimental dispersion. The remaining free parameter is the ratio of Hund’s coupling to Hubbard interaction, *J*/*U*. In (*41*), it was found that two ranges of *J*/*U* give gap anisotropy consistent with specific heat data (*56*): *J*/*U* ~ 0.08 and *J*/*U* ~ 0.06. Both yield odd-parity pairing; the higher range gives helical order (), with |**d**| slightly larger on the α and β sheets, whereas the lower value favors chiral order [**d** ~ (*p*_{x} ± *ip*_{y})**ẑ**], with |**d**| slightly larger on the γ sheet. **d** is the so-called *d* vector that describes a spin-triplet order parameter, including its spin structure. For states of the type considered here, the energy gap |Δ| equals |**d**|.

Here, we present *J*/*U* = 0.06 results for strained Sr_{2}RuO_{4}; the *J*/*U* = 0.08 results are similar (*45*). At zero strain, the point-group symmetry of the lattice is *D*_{4h}, and (*p*_{x} ± *ip*_{y})**ẑ** and are the most favored odd- and even-parity irreducible representations, respectively. At nonzero strain, the point-group symmetry becomes *D*_{2h}. (*p*_{x} ± *ip*_{y})**ẑ** is resolved into the separate irreducible representations *p*_{x}**ẑ** and *p*_{y}**ẑ**, and becomes . Strain is simulated in the calculation by introducing anisotropy into the hopping integrals. The nearest-neighbor hopping *t*, for example, is resolved into *t*_{x} = *t* × (1 + *a*ε_{xx}) and *t*_{y} = *t* × (1 − *aν*_{xy}ε_{xx}), where *a* is chosen such that the Lifshitz transition occurs at ε_{xx} = −0.0075, in agreement with the LDA + SOC calculation.

*p*_{y}**ẑ** and *p*_{x}**ẑ** are the highest *T*_{c}-order parameters under compression and tension, respectively; compression along favors *p*_{y} because it increases the DOS on the sections the Fermi surface where *p*_{y} order has the largest gap magnitude, similarly for tension and *p*_{x}. For *J*/*U* = 0.06, the possible helical orders ( or ) all have lower *T*_{c} at all strains calculated. Results for *T*_{c} against ε_{xx} for *p*_{x}**ẑ**, *p*_{y}**ẑ**, and orders are shown in Fig. 6. To assign numerical values to *T*_{c}, the bandwidth and *U*/*t* are chosen to set *T*_{c}(ε_{xx} = 0) = 1.5 K and *T*_{c}(ε_{xx} = ε_{VHS}) ≈ 3.4 K; by this procedure, *U*/*t* comes to 6.2. *T*_{c} of the *p*_{x} and *p*_{y} orders cross at ε_{xx} = 0, as they must (*57*), and the slope |*dT*_{c}/*d*ε_{xx}| as ε_{xx} → 0 is ~0.3 K/%. This crossing would appear as a cusp in a *T*_{c}(ε_{xx}) curve derived from measurements that detect only the higher *T*_{c}, and to search for this cusp was the primary aim of (*37*). Although no cusp was seen, the resolution of that experiment does not rule out a cusp of this magnitude, and furthermore, a cusp could be rounded by fluctuations (*58*). At higher strains, *T*_{c} of both even- and odd-parity orders is found to peak at ε_{xx} ≈ ε_{VHS}. (The equivalent peaks on the tension side, as noted above, are not accessible experimentally.) Odd-parity order is found to be favored at nearly all strains; however, *T*_{c} of the even-parity order is found to peak more strongly as the VHS is approached, and in the immediate vicinity of the VHS, even- and odd-parity orders are nearly degenerate in this calculation.

The *k*-space structure of the favored odd- and even-parity orders at ε_{xx} = 0 and ε_{VHS} is shown in Fig. 7. For both parities, the structure of Δ(**k**) is quite complicated; *p*_{x} ± *ip*_{y}, *p*_{y}, etc., are labels of the irreducible representation, not accurate descriptions of the full gap structure. At ε_{xx} = ε_{VHS}, the *p*_{y} order has two nodes on the γ sheet: one at (0, π), where the γ sheet touches the zone boundary and odd-parity orders must have zero amplitude, and the other along (*k*_{x}, 0), where *p*_{y} order has zero amplitude by symmetry. Also, whereas at zero strain the odd-parity |Δ| is generally larger on the γ sheet, at ε_{xx} = ε_{VHS} it is larger on the α and β sheets, owing to the frustration for odd-parity order at the Van Hove point on the γ sheet. *T*_{c} still peaks at ε_{VHS} because the small-**q** fluctuations on γ, which diverge at ε_{VHS}, also contribute to superconductivity on α and β through inter-orbital interaction terms. In contrast, even-parity order does not suffer frustration at the Van Hove point. Its gap remains the largest on γ at ε_{xx} = ε_{VHS}, and its *T*_{c} peaks more strongly.

Following (*59*), we calculate the orbital-limited at various applied strains in the semiclassical approximation. The full expression is given in (*45*); an abbreviated form is : 〈 … 〉 is a Fermi surface average, ψ(**k**) ∝ Δ(**k**), μ is a band index, and is a velocity derived from the Fermi velocity. The results support the qualitative arguments made above and are shown in Fig. 8. For *p*_{y} order, the shift of the gap onto the α and β sheets causes a decrease in , because these sheets have lower DOS than the γ sheet. In contrast, the increased DOS around the Van Hove point causes of order to increase toward the VHS. The actual may be enhanced over the weak-coupling results by strengthened many-body effects toward the VHS; however, the results emphasize a strong quantitative disparity between for even- and odd-parity order parameters.

We note that if unstrained Sr_{2}RuO_{4} has *p*_{x} ± *ip*_{y} order at nonzero strain, the low-*T* order is likely still to be chiral, but with different amplitudes of the *p*_{x} and *p*_{y} components. In Fig. 8, the goal is to determine the expected trend in for odd-parity order by comparing the same irreducible representation, *p*_{x} or *p*_{y}, at different strains. If the order is actually *ap*_{x} ± *ibp*_{y}, with *a* ≠ *b*, *H*_{c2∥c} will generally be higher, but a similar trend in is expected.

Although heat capacity data suggest *J*/*U* ~0.06 or ~0.08, we also considered *J*/*U* over a wider range, from 0 to 0.3. The essential qualitative features presented here for *J*/*U* = 0.06, the peak in *T*_{c} at the Lifshitz transition for both even- and odd-parity order, and the enhancement (suppression) of for even (odd) parity are found to occur across this range. Results for *J*/*U* = 0.08, 0, and 0.25 are shown in (*45*).

## DISCUSSION

One long-standing puzzle in the physics of Sr_{2}RuO_{4} has been the origin of the so-called 3-K phase, which is *T*_{c} ~ 3 K superconductivity observed in eutectic crystals containing inclusions of Ru metal in a matrix of Sr_{2}RuO_{4} (*60*). It has been established that this higher *T*_{c} superconductivity has a low volume fraction (*60*, *61*), showing that it occurs at the inclusions rather than the bulk, and further that it occurs on the Sr_{2}RuO_{4} side of Ru-Sr_{2}RuO_{4} interfaces (*62*). Although full proof would require observation of the strain field around Ru inclusions, it now seems very likely that local internal strain is the origin of the 3-K phase. The upper critical fields of the 3-K phase have been obtained through measurement of resistivity along extended inclusions and were found to be ~1 T for *c*-axis and ~3.5 T for in-plane fields (*63*). The similarity of these fields with the critical fields of bulk *T*_{c} = 3.4 K Sr_{2}RuO_{4} further supports the hypothesis that the 3-K phase is a local strain effect, although it is also possible that the observed 3-K phase critical fields are enhanced by the 2D geometry of interface superconductivity (*63*, *64*).

Three-band models in (*54*, *65*), in addition to the calculations presented here, identify the proximity of the γ sheet to a VHS as an important factor in the superconductivity of Sr_{2}RuO_{4}. Parallel to this work, calculations in (*66*, *67*) have found increasing *T*_{c}, at least initially, on tuning toward the VHS with strain. That the peak in *T*_{c} occurs at a similar strain to ε_{VHS} determined from DFT calculations suggests that it coincides with the Lifshitz transition. However, an alternative possibility is that *T*_{c} of an odd-parity order initially increases, because of the increase in DOS induced by compression, but then decreases as frustration at the Van Hove point becomes more important. This is not the behavior indicated by our calculations, where *T*_{c} of *p*_{y} order peaks at ε_{VHS}, but may still be considered a qualitative possibility. A further possibility, from (*67*), is that compression stabilizes competing spin-density wave order that cuts off the superconductivity before ε_{VHS}.

Evidence that the *T*_{c} peak and Lifshitz transition do coincide comes from preliminary transport data. In the normal state, inelastic scattering is generally expected to scale with the Fermi level DOS, so at nonzero temperature, a peak in the resistivity at the Lifshitz transition is expected. The resistivity *ρ*_{xx} at 4.5 K, above the highest *T*_{c}, peaks in the vicinity of the *T*_{c} peak (fig. S3). At higher strains, it falls rapidly to below its zero-strain value. The calculated Fermi surface DOS (Fig. 1C) similarly drops to below its zero-strain value beyond ε_{VHS}. The resistivity does not show the sharp increase generically expected with transitions into phases involving a gap. Further experiments are needed to determine the precise behavior of the normal-state resistivity across the *T*_{c} peak.

Although important, the issue of whether the peak in *T*_{c} coincides with the Lifshitz point does not strongly affect the main conclusions that we draw here, because the substance of the comparison of the critical fields of *T*_{c} = 3.4 K and unstrained Sr_{2}RuO_{4} stands regardless. The weak-coupling calculations yield strongly divergent trends for for even- and odd-parity order at all intermediate strains, not only at the VHS, and because this is a result of frustration of odd-parity order in the vicinity of the Van Hove point, it is unlikely to be strongly model-dependent. Also, the arguments for Pauli limiting of *H*_{c2∥a}(*T* → 0) are unaffected by whether the peak is at the Lifshitz transition. The critical field comparisons raise the possibility that the *T*_{c} = 3.4 K superconductivity has an even-parity, spin-singlet order parameter. It is difficult to understand in a naive analysis how the critical field anisotropy γ_{s} could be only ≈3 without Pauli limiting of *H*_{c2∥a}. However, most current theories of Sr_{2}RuO_{4} are 2D and make no predictions for γ_{s}; we believe that our observations provide strong motivation for extending realistic three-band calculations into the third dimension.

If the 3.4 K superconducting state is even-parity, there are two obvious possibilities, both exciting, for its relationship with the superconductivity of unstrained Sr_{2}RuO_{4}. One is that the evolution of the order parameter is continuous between the two states, and unstrained Sr_{2}RuO_{4} is also an even-parity superconductor. The appearance of a first-order transition at low temperatures for in-plane fields in both *T*_{c} = 3.4 K (Fig. 5A) and unstrained Sr_{2}RuO_{4} (*27*) also argues for this possibility. However, in this case, a substantial body of experimental evidence (*30*) for triplet, chiral order would require alternative explanation. The evidence for chirality could be accommodated by a spin-singlet state, *d*_{xz} ± *id*_{yz} (*68*). This order parameter has horizontal line nodes, which requires interplane pairing, and would be surprising in such a highly 2D material as Sr_{2}RuO_{4}. However, it would, again, be useful to extend calculations into the third dimension so that it could be compared on an equal footing with the more standard candidate order parameters based on intraplane pairing. The other possibility is that there is a transition at an intermediate strain between odd- and even-parity states. At such a transition, a kink, possibly weak, is expected in *T*_{c}(ε_{xx}), and a jump in *H*_{c2∥c}(*T* → 0). Therefore, an important follow-up experiment is measurement of *H*_{c2∥c} at intermediate strains. This has not been done yet because the broadening of the transitions at intermediate strains complicates accurate determination of *H*_{c2}, and higher-precision sample mounting methods may be required.

Consideration of an odd-parity to even-parity transition at intermediate strains is also motivated by evidence for interference between the superconductivity of Ru inclusions and that of bulk Sr_{2}RuO_{4}, and for hysteresis and switching behavior in Ru-Sr_{2}RuO_{4} systems. The possible interference appears as a sharp drop in the critical current *I*_{c} of Pb-Ru-Sr_{2}RuO_{4} junctions at *T*_{c} of Sr_{2}RuO_{4} (*69*, *70*), which has been interpreted as an onset of phase frustration at the Ru-Sr_{2}RuO_{4} interface. However, it could perhaps also be explained by the appearance of an odd-parity–even-parity interface around the Ru inclusion. Similarly, hysteretic *I*_{c} has been seen in Sr_{2}RuO_{4}-Cu-Pb (*18*), Nb-Ru-Sr_{2}RuO_{4} (*71*), and Pb-Ru-Sr_{2}RuO_{4} (*70*) junctions, and microbridges of Sr_{2}RuO_{4} with Ru inclusions (*72*). The former two also showed time-dependent switching noise. All these results have been interpreted as motion of *p*_{x} + *ip*_{y}/*p*_{x} − *ip*_{y} domain walls; however, even/odd domain walls appear to be a viable alternative possibility.

Our observations also give cause for optimism concerning the prospects of finding superconductivity in biaxially strained thin films: A factor of 20 *H*_{c2∥c} enhancement corresponds to a factor of 4.5 reduction in the coherence length, considerably reducing the disorder constraint for unconventional superconductivity. Biaxial lattice expansion preserves tetragonal symmetry and induces Lifshitz transitions at the *X* and *Y* Van Hove points simultaneously, and so may induce qualitatively different superconductivity than tuning to a single Van Hove point with uniaxial pressure.

Finally, our results provide strong motivation for extending the application of piezoelectric-based strain tuning to other materials. Here, we have demonstrated that compressions up to ~1% are possible, with in situ tunability and good strain homogeneity. The fact that we have achieved a factor of 2.3 increase of *T*_{c} of an unconventional superconductor points the way to substantial tuning of properties of other material classes as well.

## Materials and Methods

Relativistic DFT electronic structure calculations were performed using the full-potential local orbital FPLO code (*73*–*75*), version fplo14.00-49. For the exchange-correlation potential, within the local density (LDA) and the the general gradient approximation (GGA) the parametrizations of Perdew-Wang (*76*) and Perdew-Burke-Ernzerhof (*77*) were chosen, respectively. The spin-orbit coupling (SOC) was treated non perturbatively by solving the four component Kohn-Sham-Dirac equation (*78*). Initial calculations were performed on 8000 *k*-points (20×20×20 mesh), both in the LDA and GGA approximations, with and without SOC, and with and without apical oxygen relaxation. All these calculations gave similar results, with the calculated ε_{VHS} between -0.012 (GGA + relaxation) and -0.009 (LDA+SOC+relaxation). However, proximity of the VHS to the Fermi level meant that convergence of the calculated energy of the VHS to within 3% of *E*_{F} required a higher density of *k*-points, so LDA+SOC+relaxation calculations were then carried out on a mesh of 343,000 *k*-points (70×70×70 mesh, 44766 points in the irreducible wedge of the Brillouin zone). This calculation yields ε_{VHS} ≈ -0.0075.

Although we believe that using experimentally determined structural parameters for unstrained Sr_{2}RuO_{4} (as described in the main text) is the most natural starting point for the calculations, we also checked for the effect of fully relaxing the structure in the local density approximation. That relaxation only slightly reduced the cell volume (by 2.7%), preserved the *c*/*a* ratio to within 0.1%, and led to an increase of only 0.001 in ε_{VHS}, so we are confident that use of a relaxed structure gives no substantial systematic change compared to use of the experimental one.

The pressure apparatus is based on that described in (*43*), however there are a few key modifications that merit mention here. (1) The piezoelectric actuators were 18 mm-long Physik Instrumente PICMA linear actuators. (2) The displacement sensor is a parallel-plate capacitor, in place of the strain gauge described in (*37*) and (*43*). The data in this work suggest that the strains determined in (*37*) are ≈35% too low. One very likely contribution to this error is the mechanical resistance imposed by the strain gauge on the motion of the original apparatus. Temperature shifts in the gauge coefficient of the strain gauge may also contribute. Capacitive sensors are less affected by field and temperature, and impose no mechanical resistance, so we have more confidence in the strains reported in this work. (3) The thermal contraction foils have been eliminated, allowing the core of the apparatus to be made as a single piece. The longer actuators have more than sufficient range to overcome differential thermal contraction between the sample and apparatus.

When mounting samples, a small voltage is often applied to the actuators to move the sample mount points slightly further apart. When this voltage is later released, the sample is placed under modest compression. This step reduces the risk that the sample will break during cooling— for example, if temperature inhomogeneity in the apparatus places the sample under inadvertent tension.

To estimate the strain applied to a sample, two pieces of information are required. The first is the origin of the strain scale, the point where the sample is under zero strain. In (*37*) it was determined that *T*_{c} of Sr_{2}RuO_{4} is minimum within experimental error at zero strain, so for most samples the origin can be identified as the minimum in *T*_{c}. Samples 1 and 4 broke during cooling, and could be compressed by closing the crack, but not tensioned. The process of re-engaging the two ends can be gradual, e.g., if the two faces of the crack do not match perfectly, so zero strain cannot be reliably identified by attempting to locate a precise point where *T*_{c}(ε_{xx}) starts to change. Instead, a quadratic fit was made to the *T*_{c}(ε_{xx}) curve over a temperature range near but above the lowest observed *T*_{c}. Zero strain was identified as the minimum of the fitted curve, plus 2 ⋅ 10^{−4} to account for the anomalous flattening of *T*_{c}(ε_{xx}) around ε_{xx} = 0 observed in (*37*). The other piece of information required is an effective strained length: the capacitive sensor measures a displacement, and ε_{xx} is this displacement divided by the effective strained length. Deformation of the sample mounting epoxy means that the effective strained length is typically ~0.4 mm longer than the exposed length of the sample. It is estimated through finite element analysis, as described in (*37*) and (*43*).

The layers of the epoxy that secure the sample are generally 20–40 μm thick, an estimated broad optimum. Thinner layers transmit force to the sample more efficiently (i.e. give a shorter effective strained length), while thicker layers reduce stress concentration in the epoxy and allow greater tolerance in assembly. The dimensions, calculated effective strained length, and estimated ε_{xx} at the peak in *T*_{c} for each sample are given in (*45*).

## SUPPLEMENTARY MATERIALS

## REFERENCES AND NOTES

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
**Acknowledgments:**We thank H. Pfau for experimental contributions; E.-A. Kim, S. Kivelson, S. Raghu, K. Shen, and F.-C. Zhang for stimulating discussions; and E.-A. Kim and F.-C. Zhang for sharing the results of their calculations with us. Hsu*et al*. (*66*) used renormalization group calculations to study*T*_{c}versus uniaxial but mainly biaxial strain, whereas Liu*et al*. (*67*) used functional renormalization group calculations of the strain dependence of*T*_{c}concentrating on the*d*_{xy}-based Fermi surface sheet. On topics where they overlap, the results of those two calculations, as well as the calculations presented in this paper, are qualitatively similar. We acknowledge the support of the Max Planck Society and the UK Engineering and Physical Sciences Research Council under grants EP/1031014/1, EP/G03673X/1, EP/N01930X/1, and EP/I032487/1. L.Z. acknowledges the support of the China Scholarship Council. T.S. acknowledges the support of the Clarendon Fund Scholarship, the Merton College Domus and Prize Scholarships, and the University of Oxford. Y.M. acknowledges the support by the Japan Society for the Promotion of Science Grant-in-Aids on Topological Quantum Phenomena (KAKENHI JP22103002) and on Topological Materials Science (KAKENHI JP15H05852). C.W.H has 31% ownership of Razorbill Instruments, which has commercialized apparatus based on that used in this work. Raw data for all figures in this paper are available at http://dx.doi.org/10.17630/d2b438fc-2f65-4575-89a3-3296ce5c3a17.