Research Article

Strong peak in Tc of Sr2RuO4 under uniaxial pressure

See allHide authors and affiliations

Science  13 Jan 2017:
Vol. 355, Issue 6321, eaaf9398
DOI: 10.1126/science.aaf9398

Squeezing out the oddness

The material Sr2RuO4 has long been thought to exhibit an exotic, odd-parity kind of superconductivity, not unlike the superfluidity in 3He. 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


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, Sr2RuO4, and demonstrate using explicit calculations how our findings provide strong constraints on theory.


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 Sr2RuO4. 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 Sr2RuO4 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.


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 Sr2RuO4 lattice drives the superconducting transition temperature (Tc) through a pronounced maximum, at a compression of ≈0.6%, that is a factor of 2.3 higher than Tc of the unstrained material. At the maximum Tc, 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 Tc’s and upper critical fields of possible superconducting order parameters. The combination of our experimental and theoretical work suggests that the maximum Tc is likely associated with the predicted Fermi surface topological transition and that at this maximum Tc, Sr2RuO4 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 Tc of Sr2RuO4.


Our data raise the possibility of an odd-parity to even-parity transition of the superconducting state of Sr2RuO4 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.

The rise and fall of Tc of Sr2RuO4.

(Top left) A photograph of the uniaxial pressure apparatus. Pressure is applied to the sample by piezoelectric actuators. (Top middle) A sample, prepared as a beam and mounted in the apparatus. The susceptometer is a pair of concentric coils. (Top right) A schematic of a mounted sample. The piezoelectric actuators compress or tension the sample along its length. (Bottom) Tc of three samples of Sr2RuO4 against strain along their lengths. Negative values of εxx denote compression. Tc is taken as the midpoint of the transition, observed by ac susceptibility. Sample #1 was cracked, and so could be compressed but not tensioned.


Sr2RuO4 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 (Tc) 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 Tc 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, Tc = 3.4 K Sr2RuO4 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, Sr2RuO4 (transition temperature Tc ≈ 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 (810), 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 Sr2RuO4 to those of the famous p-wave superfluid 3He 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 SrRuO3 layers (14) provide strong evidence for triplet pairing. Muon spin rotation (15) and Kerr rotation (16) experiments point to time-reversal symmetry breaking at Tc, 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 px ± ipy. Although the edge currents predicted for chiral p-wave order are not seen (2022), there are proposals to explain why these might be unobservably small in Sr2RuO4 (2326). More difficult to explain in the context of spin-triplet pairing is why the upper critical field Hc2 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 Sr2RuO4 and arguments for and against various order parameters can be found in (2932).

The electronic structure of Sr2RuO4 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 Sr2RuO4 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 Sr2RuO4 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 Sr2RuO4 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 Sr2RuO4 without introducing disorder and destroying the superconductivity. Pressure applied along a 〈100〉 lattice direction, lifting the native tetragonal symmetry of Sr2RuO4, has been shown to increase the bulk Tc to at least 1.9 K (37). There are hints that Tc ~ 3 K in pure Sr2RuO4 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 Tc at 3.4 K, at about 0.6% compression. The Young’s modulus of Sr2RuO4 is 176 GPa (40), so this compression corresponds to a uniaxial pressure of ~1 GPa. A factor of 2.3 increase in Tc is accompanied by more than a factor of 20 enhancement of Hc2 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 Tc likely coincides with a Lifshitz transition. Then, to gain insight into the effect of these large strains on possible superconducting order parameters of Sr2RuO4, we use weak-coupling calculations that include spin-orbit and interband couplings, extending the work of (41).

Calculated band structure of Sr2RuO4

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 Sr2RuO4. 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 εxx is an independent variable, and the transverse strains are set, as in the experiment, according to the Poisson’s ratios of Sr2RuO4: εyy = −νxyεxx and εzz = −νxzεxx (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 εxx = εVHS ≈ −0.0075. The calculated Fermi surfaces at εxx = 0 and εxx = εVHS are shown in Fig. 1, where it can be seen that compression along Embedded Image leads to a Lifshitz transition in the γ Fermi surface along ky. Because of the low kz dispersion, it occurs for all kz over a very narrow range of εxx, starting at εxx = (−0.75 ± 0.01) × 10−2 and finishing by (−0.77 ± 0.01) × 10−2. Cross sections at kz = 0 are also shown. In fully 2D approximations of Sr2RuO4, the Lifshitz transition occurs at a single Van Hove point, labeled in the figure and coinciding with the 2D zone boundary of an isolated RuO2 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 Sr2RuO4, in which many-body effects are likely to strengthen the quasiparticle renormalization of vF and the DOS in the vicinity of the peaks. The peak on the tension side corresponds to a Lifshitz transition along kx, which is not accessible experimentally because samples break under strong tension.

Fig. 1 DFT calculation results.

(A) Calculated Fermi surfaces of unstrained Sr2RuO4, colored by the Fermi velocity vF, at zero strain. The three surfaces are labeled α, β, and γ. A cross section through kz = 0 is also shown. The dashed lines indicate the zone of an isolated RuO2 sheet; in 2D models of Sr2RuO4, the Van Hove point is located on this zone boundary. (B) Calculated Fermi surfaces at εxx = − 0.0075. (C) Calculated total DOS against εxx.

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.

Fig. 2 Apparatus and sample configuration.

(Top) Apparatus configuration. Extending the outer two piezoelectric actuators tensions the sample, and extending the central actuator compresses the sample. (Middle) Sample configuration. The ends are secured with epoxy. Some samples have contacts (shown schematically) for resistivity measurements. (Bottom) Photograph of sample 3. On top of the sample, mounted on a flexible cantilever, are two concentric coils, used for measuring magnetic susceptibility.

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 Tc’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.

Fig. 3 Susceptibility against temperature.

(Top) Real part of the susceptibility χ against temperature for sample 1 at various εxx. No normalizations or offsets are applied to the curves. (Middle and bottom) Same for sample 3.

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 Tc, about 3.4 K. Sample 3 could be compressed well beyond this maximum, and Tc was found to drop rapidly. In checks made on multiple samples, upon releasing the strain and returning to εxx ~ 0, the χ′(T) curves were found to be unchanged [see fig. S4 (45)], indicating that the sample deformation is elastic.

The peak in Tc can be seen in the graph of Tc against εxx 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, εxx is (−0.60 ± 0.06) × 10−2. The graph is based exclusively on magnetic measurements. The maximum Tc of sample 5, at ≈3.5 K, slightly exceeds that of the other samples. Resistivity measurements can show anomalously high Tc 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 Tc by more than 0.08 K, confirming that it is the maximum Tc.

Fig. 4 Tc against strain for samples 1, 3, and 5.

The points are the midpoints (50% levels) of the transitions shown in Fig. 3, and the lines are the 20 and 80% levels, giving a measure of the transition width. The strain scales have been normalized. We estimate an uncertainty of 0.04 × 10−2 on the determination of zero strain of each sample, and the strain at the peak in Tc is determined by averaging independent determinations from four samples to be (−0.60 ± 0.06) × 10− 2. The flat region around εxx = 0 for sample 1 is an artifact: The sample broke during cooldown, meaning that tensile strain could not be applied, and a compressive displacement was required for it to reengage.

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 Sr2RuO4 compressed to the peak in Tc (referred to henceforth as Tc = 3.4 K Sr2RuO4) make determination of Tc and Hc2 very simple: In all temperature and field ramps, a sharp cusp in χ′(T) was observed, which could be identified as Tc or Hc2. Specifically, the transition was identified as the intersection of linear fits to data just below and above the cusp. The in-plane Hc2 of Sr2RuO4 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 Hc2, Hc2∥c, of Tc = 3.4 K Sr2RuO4 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 Hc2∥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.

Fig. 5 Hc2 against temperature.

(A) Hc2 ∥ a and Hc2 ∥ c against temperature for sample 4, compressed to the peak in Tc. Hc2 ∥ a was measured with both field and temperature ramps and found to be hysteretic below ~1.8 K (upper inset). Lower inset: Angle dependence of Hc2 at 990 mK, confirming the field alignment. θ is the angle in the a-c plane, and θ = 0 is the field angle at which the Hc2 ∥ a data were collected. (B) Raw data for χ ' (T) of sample 4 at various εxx. The y axis is the mutual inductance of the measurement coils. (C) Measured χ ' (T) at the peak in Tc, and at fields in 0.1-T increments between 0 and 1.5 T. (D) Data for Hc2 ∥ c of an unstrained Sr2RuO4 sample with slightly suboptimal Tc, from (48).

A concave Hc2(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, MgB2 (46) and Be(Fe,Co)2As2 (47). In Tc = 3.4 K Sr2RuO4, the slope |dHc2∥c/dT| is found to steadily increase to the lowest temperatures measured, although Hc2∥c(T) must eventually become convex because dHc2/dT must approach zero as T → 0. Hc2∥c of unstrained Sr2RuO4, from (48) (Fig. 5D), is weakly concave at higher temperatures, but only above ~0.7 K, a much higher fraction of Tc (H = 0) than the concave-convex crossover in Tc = 3.4 K Sr2RuO4. This difference in the Hc2(T) curves indicates that the gap varies more widely across the Fermi surfaces in Tc = 3.4 K Sr2RuO4 than in unstrained Sr2RuO4.

Gap symmetry in Tc = 3.4 K Sr2RuO4

The T → 0 critical field values for Tc = 3.4 K Sr2RuO4 are striking. Hc2∥c(T → 0) is enhanced by more than a factor of 20 relative to unstrained Sr2RuO4. Hc2∥a(T → 0) of unstrained Sr2RuO4 is 1.5 T (28), and it is enhanced by a factor of only ≈3 in Tc = 3.4 K Sr2RuO4. In the simplest picture of a fully 2D triplet superconductor with the spins in the plane, the ratio γs between Hc2∥a and Hc2∥c would be infinite, because neither orbital nor Pauli limiting would apply for in-plane fields (49). However, we observe that γs is reduced from a value of ≈20 in unstrained Sr2RuO4 to ≈3 in Tc = 3.4 K Sr2RuO4. The electronic structure calculations presented in Fig. 1 indicate that Sr2RuO4 remains quasi-2D at high strains, a result supported by the observation in Fig. 5 that just below Tc, the slope |dHc2∥a/dT| far exceeds |dHc2∥c/dT|. Therefore, it seems unlikely that such a reduction in γs 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 Hc2∥a. In a mean-field superconductor, both Tc and the Pauli-limited Hc2 are expected to vary linearly with the T → 0 gap magnitude |Δ| (51). The rise of Hc2∥a (T > 0) from 1.5 to 4.7 T in Tc = 3.4 K Sr2RuO4 is somewhat, but not drastically, faster than linear against Tc. In combination, these observations motivate investigation of whether the Tc = 3.4 K state might be an even-parity condensate of spin-singlet pairs.

A qualitative analysis of the enhancement of Hc2∥c with strain also points to this possibility. In a mean-field superconductor, the orbitally limited Hc2(T → 0) is proportional to a weighted average of [|Δ|N(EF)]2, where N(EF) is the Fermi surface DOS. Because Tc is proportional to a k-space average of |Δ|, if |Δ(k)| is multiplied by a factor and N(EF) is not modified, the quantity Hc2/Tc2 should remain constant. However, when Sr2RuO4 is pressurized along a 〈100〉 direction, N(EF) is substantially modified: It increases strongly near the Van Hove point, so if |∆| is large in this region of the Brillouin zone, Hc2/Tc2 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 Hc2/Tc2 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(EF) is largest.

We observe, based on the data in Fig. 5, that Embedded Image is enhanced by a factor of 3.6 in Tc = 3.4 K Sr2RuO4. Alternatively, because the form of Hc2(T) is so different between unstrained and Tc = 3.4 K Sr2RuO4, it may be preferable to take a measure of Hc2 that relies only on data near Tc, that is, a hypothetical Hc2(0) for the TTc gap structure that excludes anomalous strengthening of the superconductivity at lower temperatures. Applying the Werthamer-Helfand-Hohenberg formula, Hc2(0) = −0.7(dHc2/dT)Tc (52), yields 0.70 and 0.056 T, respectively, for sample 4 strained to maximum Tc and for the unstrained sample of Fig. 5D. If these values are used in place of the actual Hc2∥c(T → 0), the enhancement is 1.8. In terms of the argument discussed above, the enhancement of Embedded Image defined by either criterion seems to favor an even-parity over an odd-parity order parameter for Tc = 3.4 K Sr2RuO4.

To investigate these qualitative arguments in more depth and on the basis of a realistic calculation taking into account the multisheet Fermi surface of Sr2RuO4, we have extended to strained Sr2RuO4 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 Sr2RuO4, 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 Sr2RuO4 (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 Sr2RuO4 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 (Embedded Image), with |d| slightly larger on the α and β sheets, whereas the lower value favors chiral order [d ~ (px ± ipy)Embedded Image], 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 Sr2RuO4; the J/U = 0.08 results are similar (45). At zero strain, the point-group symmetry of the lattice is D4h, and (px ± ipy) and Embedded Image are the most favored odd- and even-parity irreducible representations, respectively. At nonzero strain, the point-group symmetry becomes D2h. (px ± ipy) is resolved into the separate irreducible representations px and py, and Embedded Image becomes Embedded Image. Strain is simulated in the calculation by introducing anisotropy into the hopping integrals. The nearest-neighbor hopping t, for example, is resolved into tx = t × (1 + aεxx) and ty = t × (1 − xyεxx), where a is chosen such that the Lifshitz transition occurs at εxx = −0.0075, in agreement with the LDA + SOC calculation.

py and px are the highest Tc-order parameters under compression and tension, respectively; compression along Embedded Image favors py because it increases the DOS on the sections the Fermi surface where py order has the largest gap magnitude, similarly for tension and px. For J/U = 0.06, the possible helical orders (Embedded Image or Embedded Image) all have lower Tc at all strains calculated. Results for Tc against εxx for px, py, and Embedded Image orders are shown in Fig. 6. To assign numerical values to Tc, the bandwidth and U/t are chosen to set Tcxx = 0) = 1.5 K and Tcxx = εVHS) ≈ 3.4 K; by this procedure, U/t comes to 6.2. Tc of the px and py orders cross at εxx = 0, as they must (57), and the slope |dTc/dεxx| as εxx → 0 is ~0.3 K/%. This crossing would appear as a cusp in a Tcxx) curve derived from measurements that detect only the higher Tc, 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, Tc of both even- and odd-parity orders is found to peak at εxx ≈ εVHS. (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, Tc 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.

Fig. 6 Weak-coupling calculations: Tc versus strain.

The bandwidth and U/t were set to reproduce the experimental values of Tc(0) = 1.5 K and TcVHS) ~ 3.4 K. px ± ipy and Embedded Image are irreducible representations of the εxx = 0 (that is, tetragonal) lattice. For εxx ≠ 0, px ± ipy is resolved into separate representations px and py, and Embedded Image becomes Embedded Image.

The k-space structure of the favored odd- and even-parity orders at εxx = 0 and εVHS is shown in Fig. 7. For both parities, the structure of Δ(k) is quite complicated; px ± ipy, py, etc., are labels of the irreducible representation, not accurate descriptions of the full gap structure. At εxx = εVHS, the py 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 (kx, 0), where py order has zero amplitude by symmetry. Also, whereas at zero strain the odd-parity |Δ| is generally larger on the γ sheet, at εxx = εVHS 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. Tc still peaks at εVHS because the small-q fluctuations on γ, which diverge at εVHS, 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 εxx = εVHS, and its Tc peaks more strongly.

Fig. 7 Weak-coupling calculations: Order parameters.

(Top) The odd-parity order parameter at εxx = 0 and εxx ≈ εVHS; the VHS is reached at (0, π). The width of the traces is proportional to the energy gap, and the color indicates the phase. For εxx ≠ 0, px ± ipy is no longer an irreducible representation of the lattice, so the py representation, the favored order parameter for εxx < 0, is shown instead. (Bottom) Even-parity order at εxx = 0 and εxx ≈ εVHS.

Following (59), we calculate the orbital-limited Embedded Image at various applied strains in the semiclassical approximation. The full expression is given in (45); an abbreviated form is Embedded Image: 〈 … 〉 is a Fermi surface average, ψ(k) ∝ Δ(k), μ is a band index, and Embedded Image is a velocity derived from the Fermi velocity. The results support the qualitative arguments made above and are shown in Fig. 8. For py order, the shift of the gap onto the α and β sheets causes a decrease in Embedded Image, because these sheets have lower DOS than the γ sheet. In contrast, the increased DOS around the Van Hove point causes Embedded Image of Embedded Image order to increase toward the VHS. The actual Embedded Image 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 Embedded Image for even- and odd-parity order parameters.

Fig. 8 Weak-coupling calculations: Embedded Image versus strain.

The results are normalized to the Embedded Image calculated at zero strain.

We note that if unstrained Sr2RuO4 has px ± ipy order at nonzero strain, the low-T order is likely still to be chiral, but with different amplitudes of the px and py components. In Fig. 8, the goal is to determine the expected trend in Embedded Image for odd-parity order by comparing the same irreducible representation, px or py, at different strains. If the order is actually apx ± ibpy, with ab, Hc2∥c will generally be higher, but a similar trend in Embedded Image 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 Tc at the Lifshitz transition for both even- and odd-parity order, and the enhancement (suppression) of Embedded Image 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).


One long-standing puzzle in the physics of Sr2RuO4 has been the origin of the so-called 3-K phase, which is Tc ~ 3 K superconductivity observed in eutectic crystals containing inclusions of Ru metal in a matrix of Sr2RuO4 (60). It has been established that this higher Tc 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 Sr2RuO4 side of Ru-Sr2RuO4 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 Tc = 3.4 K Sr2RuO4 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 Sr2RuO4. Parallel to this work, calculations in (66, 67) have found increasing Tc, at least initially, on tuning toward the VHS with strain. That the peak in Tc occurs at a similar strain to εVHS determined from DFT calculations suggests that it coincides with the Lifshitz transition. However, an alternative possibility is that Tc 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 Tc of py order peaks at εVHS, 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 εVHS.

Evidence that the Tc 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 ρxx at 4.5 K, above the highest Tc, peaks in the vicinity of the Tc 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 εVHS. 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 Tc peak.

Although important, the issue of whether the peak in Tc 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 Tc = 3.4 K and unstrained Sr2RuO4 stands regardless. The weak-coupling calculations yield strongly divergent trends for Embedded Image 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 Hc2∥a(T → 0) are unaffected by whether the peak is at the Lifshitz transition. The critical field comparisons raise the possibility that the Tc = 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 γs could be only ≈3 without Pauli limiting of Hc2∥a. However, most current theories of Sr2RuO4 are 2D and make no predictions for γs; 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 Sr2RuO4. One is that the evolution of the order parameter is continuous between the two states, and unstrained Sr2RuO4 is also an even-parity superconductor. The appearance of a first-order transition at low temperatures for in-plane fields in both Tc = 3.4 K (Fig. 5A) and unstrained Sr2RuO4 (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, dxz ± idyz (68). This order parameter has horizontal line nodes, which requires interplane pairing, and would be surprising in such a highly 2D material as Sr2RuO4. 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 Tcxx), and a jump in Hc2∥c(T → 0). Therefore, an important follow-up experiment is measurement of Hc2∥c at intermediate strains. This has not been done yet because the broadening of the transitions at intermediate strains complicates accurate determination of Hc2, 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 Sr2RuO4, and for hysteresis and switching behavior in Ru-Sr2RuO4 systems. The possible interference appears as a sharp drop in the critical current Ic of Pb-Ru-Sr2RuO4 junctions at Tc of Sr2RuO4 (69, 70), which has been interpreted as an onset of phase frustration at the Ru-Sr2RuO4 interface. However, it could perhaps also be explained by the appearance of an odd-parity–even-parity interface around the Ru inclusion. Similarly, hysteretic Ic has been seen in Sr2RuO4-Cu-Pb (18), Nb-Ru-Sr2RuO4 (71), and Pb-Ru-Sr2RuO4 (70) junctions, and microbridges of Sr2RuO4 with Ru inclusions (72). The former two also showed time-dependent switching noise. All these results have been interpreted as motion of px + ipy/pxipy 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 Hc2∥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 Tc 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 (7375), 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 εVHS 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 EF 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 εVHS ≈ -0.0075.

Although we believe that using experimentally determined structural parameters for unstrained Sr2RuO4 (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 εVHS, 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 Tc of Sr2RuO4 is minimum within experimental error at zero strain, so for most samples the origin can be identified as the minimum in Tc. 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 Tcxx) starts to change. Instead, a quadratic fit was made to the Tcxx) curve over a temperature range near but above the lowest observed Tc. Zero strain was identified as the minimum of the fitted curve, plus 2 ⋅ 10−4 to account for the anomalous flattening of Tcxx) around εxx = 0 observed in (37). The other piece of information required is an effective strained length: the capacitive sensor measures a displacement, and εxx 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 εxx at the peak in Tc for each sample are given in (45).


Supplementary Text

Figures S1 to S7

Table S1

References (8187)


  1. 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 Tc versus uniaxial but mainly biaxial strain, whereas Liu et al. (67) used functional renormalization group calculations of the strain dependence of Tc concentrating on the dxy-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
View Abstract

Stay Connected to Science

Navigate This Article