Strained Superconductor
Distorting a material and observing its response can allow insight into its electronic properties. Thin films can be strained by placing them on a substrate with a different lattice constant; bulk samples present more of a challenge. Hicks et al. (p. 283) designed an apparatus to apply both tensile and compressive strain and used it to study the properties of the superconductor Sr_{2}RuO_{4}, which has long been hypothesized to host the unusual p-wave superconductivity. The response of the superconducting transition temperature T_{c} to the applied strain depended on the direction in which the strain was applied, and did not exhibit a cusp predicted to occur around zero strain. As the technique leaves a surface of the probe open to external probes, it could be adopted for a wide range of methods.
Abstract
A sensitive probe of unconventional order is its response to a symmetry-breaking field. To probe the proposed p_{x} ± ip_{y} topological superconducting state of Sr_{2}RuO_{4}, we have constructed an apparatus capable of applying both compressive and tensile strains of up to 0.23%. Strains applied along crystallographic directions yield a strong, strain-symmetric increase in the superconducting transition temperature T_{c}. strains give a much weaker, mostly antisymmetric response. As well as advancing the understanding of the superconductivity of Sr_{2}RuO_{4}, our technique has potential applicability to a wide range of problems in solid-state physics.
The layered perovskite Sr_{2}RuO_{4} is one of the most extensively studied unconventional superconductors (1–4). It has attracted particular attention because of the strong possibility that the pairing is spin-triplet, with an odd-parity, chiral orbital order parameter, p_{x} ± ip_{y}. This would be a superconducting analog of superfluid ^{3}He, but with the key additional feature of being quasi–two-dimensional. This combination of properties, distinct among known superconductors, would, in principle, support topologically protected edge states and half-flux quantum vortices with Majorana zero modes (5, 6). Thus, the study of the superconducting state of Sr_{2}RuO_{4} is of fundamental interest.
The evidence for triplet pairing is strong (7, 8), but definitive demonstration of the orbital order has proved more difficult (9). Some predictions of two-component chiral order have been observed in experiments, including time-reversal symmetry-breaking (10, 11) and complex Josephson interferometry (12). However, chiral order is also expected to result in substantial edge currents (13), splitting of the transition in an in-plane magnetic field, and anisotropy in the upper critical field H_{c2} at temperatures near the transition temperature T_{c} (14). Despite experimental effort, none of these properties have been observed (15–18).
Because the existence of p_{x} ± ip_{y} order in Sr_{2}RuO_{4} would have broad importance, further evaluation is necessary. Under the tetragonal symmetry of the Sr_{2}RuO_{4} lattice, the two components p_{x} and p_{y} have the same transition temperatures. This degeneracy can be lifted by applying a symmetry-breaking field, with in-plane uniaxial strain an ideal choice: Based purely on symmetry considerations, a phase diagram of the form shown in Fig. 1A is expected for a strained p_{x} ± ip_{y} superconductor (19, 20). If the transition temperature T_{c} is measured with a probe sensitive mainly to the upper transition, such as resistivity or susceptibility, then the observed T_{c} versus strain ε is expected to follow the solid line, showing a cusp at ε = 0, and a substantial symmetric response about zero (21, 22).
Sr_{2}RuO_{4} has consistently been found to be sensitive to uniaxial pressure: The 3-K phase, locally higher-T_{c} superconductivity that appears in Sr_{2}RuO_{4}-Ru eutectic samples (23), can be induced in pure Sr_{2}RuO_{4} by uniaxial pressures (24, 25). But the direction dependence of the uniaxial strain effect has not been resolved, and the higher-T_{c} superconductivity has not been induced homogeneously. Furthermore, the expected strain-symmetric response has not been probed because it requires applying both tensile and compressive stresses, a capability that has been lacking.
To address these issues, we constructed the apparatus sketched in Fig. 1B. More details can be found in (26), but the main features are as follows: (i) The sample is epoxied into place, and piezoelectric stacks are used to apply compression and extension; (ii) the stacks are arranged so that their thermal contraction does not, in principle, strain the sample; and (iii) “strain amplification,” meaning that the stacks are longer than the sample, allowing larger strains to be applied to the sample than are achievable on the stacks. In addition, the sample is cut into a long, thin bar to obtain good strain homogeneity toward its center. The apparatus is compact and would be compatible with many cryogenic platforms; for our experiments, it was mounted in an adiabatic demagnetization refrigerator.
As with the piezoelectric-based technique of (27), it is mainly the sample strain, rather than stress, that is the controlled parameter. The pressure on the sample is the strain times the relevant elastic constant. The strain is monitored with a strain gauge mounted beneath the sample but not attached to it. A portion of the applied displacement (27%, on average, for the samples studied here) is taken up by deformation of the sample-mounting epoxy, so for each sample the actual sample strain was estimated by finite element analysis (26, 28, 29).
We measured T_{c} through ac magnetic susceptibility χ, using two concentric coils mounted on top of the sample. Measurement frequencies ranged from 59 to 369 Hz, and the excitation field from ~0.05 to ~0.5 G.
We used single-crystal samples grown by a floating-zone method (1). In Fig. 2, A and B, we show χ(T) for a sample with T_{c} = 1.35 K at zero strain, under compression (A) and tension (B) along a crystal direction. Full superconducting transitions are seen at all strains, with minimal broadening. The extra structure observed at high strains is most likely an effect of strain inhomogeneity.
We measured four samples cut in a direction and three in a direction. The main features we report were reproduced across the entire sample set. The dependence of T_{c} on strains applied along and directions is shown in Fig. 2, C and D. There is some uncertainty in locating zero strain, but the data can be summarized thus: For strain (ε_{100}), there is a large symmetric response about ε_{100} = 0, whereas the response to strain (ε_{110}) is far weaker and mainly linear in ε. The proposed p_{x} ± ip_{y} superconductivity of Sr_{2}RuO_{4} has often been modeled supposing an isotropic, cylindrical Fermi surface, but these data show that the tetragonal symmetry is hugely influential.
Neglecting c-axis strain, the applied strains can be resolved into isotropic dilatation (ε_{xx} = ε_{yy}) and volume-preserving anisotropic distortion (ε_{xx} = –ε_{yy} for ε_{100}, and ε_{xx} = ε_{yy} = 0, ε_{xy} ≠ 0 for ε_{110}) (30). The in-plane Poisson’s ratio for Sr_{2}RuO_{4} is ≈0.40 (29, 31), so for both ε_{100} and ε_{110}, ~30% of the longitudinal strain is dilatation, and ~70% is volume-preserving distortion. Dilatation will, in general, give a ε-linear response for small strains about zero, but under tetragonal crystal symmetry the volume-preserving distortion must give a symmetric response. It seems clear that the dominant effect of strains is through dilatation and of strains through the volume-preserving distortion. Small variations about have a large effect: A 2.5° change in the angle of cut increased the symmetric component of the response considerably, probably by mixing in a small ε_{100} component to the strain (fig. S3).
A key question is the lack of the expected cusp near zero strain. At |ε_{100}| < 0.03%, T_{c}(ε) and dT_{c}/dε are essentially flat, then T_{c} rises strongly at larger strains. Both of these features remain essentially unchanged when the excitation field is varied (fig. S4). As emphasized by the dT_{c}/dε data in Fig. 2E, the increase of T_{c} beyond 0.03% strain is quadratic in strain, not linear. The precise low-strain form of T_{c}(ε_{100}) may be affected by inhomogeneity and requires further investigation, but the strong increase in T_{c} at larger strains is unambiguous, and this is where we wish to focus our attention.
A quadratic form can result from strain-induced mixture of near-degenerate order parameters. The tetragonal crystal symmetry and strong response to strain suggest s- and -wave orders, but these would be in conflict with both triplet pairing and time-reversal symmetry breaking at T_{c} and, thus, are unlikely. Another possibility is that sketched in Fig. 3A: that Sr_{2}RuO_{4} at ε = 0 does, in fact, have degenerate order parameters, but the cusp appears on top of a strong underlying strain response and is too small to be resolved in the present data.
To gain further insight, we used the Wien2k package (32) to calculate the electronic structure in the presence of strain; the Fermi surfaces for ε_{100} and ε_{110} = ±0.5% are shown in Fig. 3, B and C. For each strain, the lattice parameters were set according to the applied strain and the in- and out-of-plane Poisson’s ratios. The Fermi surfaces are altered more dramatically by than strains, a result supported by ultrasound data: Above T_{c}, sound waves that generate xx and yy strains are damped much more than waves generating xy strains (33).
The shifting Fermi surfaces will alter T_{c}. With p_{x} ± ip_{y} order, a strong cusp is expected when the longitudinal and transverse responses to strain are very different. In Sr_{2}RuO_{4}, this occurs most clearly on the {100} sections of the γ sheet, near the van Hove points: Under strains, the Fermi surface and, correspondingly, the density of states change rapidly and oppositely on the (100) and (010) sections. If the superconducting gap were dominated by these sections of Fermi surface, a large cusp would be expected. Based on our structure calculations, we estimate that dT_{c}/dε_{100} would jump by several kelvin per percent across the cusp (29), which would have been easily observable.
However, a large gap is not expected near the van Hove points: By symmetry, with p-wave order the gap must vanish at the van Hove points themselves. Some authors have proposed that the α and β sheets might have the largest gaps (34–36), in which case a weaker cusp is expected, which could have been obscured by a strong underlying response. Finally, p_{x} ± ip_{y} superconductivity may live dominantly on the {110} sections of the γ sheet (37, 38), as indicated by field-angle–dependent specific heat data (39). In this case, a stronger cusp would be expected for than strains. But [110] strain changes the dispersions on the (110) and sections of the γ sheet almost identically, so the cusp might be tiny. We estimate that dT_{c}/dε_{110} would change at the cusp by well under 0.1 K per percent (29), below the resolution of the present experiment.
Therefore, by our estimates our data still permit certain models of p_{x} ± ip_{y} order. The challenge for degenerate p-wave models is to explain simultaneously the apparent weakness of the cusp and the large nonlinear response to strains. A natural hypothesis is that the superconductivity is strongly intertwined with the van Hove points (40), even if this is not where the gap is largest. This is a qualitative observation, however, and further theoretical effort would be very interesting.
Figure 3, B and C, illustrate a final point with implications far beyond the superconductivity of Sr_{2}RuO_{4}: The characteristic energy scale of a 0.5% change in strain is large—roughly that percentage of the bandwidth. For Sr_{2}RuO_{4}, a Zeeman splitting with an equivalent energy scale would require a magnetic field of hundreds of teslas. This method can produce these large changes with directional resolution and precise tunability, without introducing disorder and while leaving the upper surface of the sample open for spectroscopic probes such as angle-resolved photoemission (41, 42), electron energy-loss spectroscopy, and scanning probe microscopy. We therefore believe that the technique we have developed opens the way to a host of new experiments across a wide range of materials.
Supplementary Materials
www.sciencemag.org/content/344/6181/283/suppl/DC1
Materials and Methods
Supplementary Text
Figs. S1 to S7
References and Notes
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- ↵ In addition to p_{x} ± ip_{y}, ground states of unstrained Sr_{2}RuO_{4} that would yield cusps in T_{c} at ε = 0 include d_{xz} ± id_{yz} (22) and the nonchiral orders p_{x} and p_{x} + p_{y}.
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- ↵ Strain, rather than stress, is the controlled parameter if the apparatus spring constant dominates the sample spring constant. The softest part of the apparatus is the sample-mounting epoxy. For all samples, the apparatus exceeded the sample spring constant, but not vastly, so finite element analysis of the sample and epoxy deformation was necessary.
- ↵ More details on our methods and calculations can be found in the supplementary materials on Science Online.
- ↵ We use ε_{100} and ε_{110} to denote the applied strains, implying simultaneous transverse strains following the Poisson’s ratios, whereas ε_{ij} are defined as usual.
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- ↵ Angle-resolved photoemission spectroscopy under uniaxial stress has been performed [see (42)].
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- The Young’s modulus of Stycast 2850FT at 150 K was found to be 11.5 and 16 GPa when cured with Catalysts 9 and 24LV, respectively (45). We used Catalyst 23LV.
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- Acknowledgments: We acknowledge useful discussions with E. Berg, C. Hooley, A. Huxley, S. Kivelson, J. Sauls, and S. Simon. We thank the UK Engineering and Physical Sciences Research Council, the Max Planck Society, and the Royal Society for financial support. Work at Kyoto University was supported by a KAKENHI grant (no. 22103002) from the Ministry of Education, Culture, Sports, Science and Technology.