Strong Increase of Tc of Sr2RuO4 Under Both Tensile and Compressive Strain

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Science  18 Apr 2014:
Vol. 344, Issue 6181, pp. 283-285
DOI: 10.1126/science.1248292

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 Sr2RuO4, which has long been hypothesized to host the unusual p-wave superconductivity. The response of the superconducting transition temperature Tc 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.


A sensitive probe of unconventional order is its response to a symmetry-breaking field. To probe the proposed px ± ipy topological superconducting state of Sr2RuO4, we have constructed an apparatus capable of applying both compressive and tensile strains of up to 0.23%. Strains applied along Embedded Image crystallographic directions yield a strong, strain-symmetric increase in the superconducting transition temperature Tc. Embedded Image strains give a much weaker, mostly antisymmetric response. As well as advancing the understanding of the superconductivity of Sr2RuO4, our technique has potential applicability to a wide range of problems in solid-state physics.

The layered perovskite Sr2RuO4 is one of the most extensively studied unconventional superconductors (14). It has attracted particular attention because of the strong possibility that the pairing is spin-triplet, with an odd-parity, chiral orbital order parameter, px ± ipy. This would be a superconducting analog of superfluid 3He, 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 Sr2RuO4 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 Hc2 at temperatures near the transition temperature Tc (14). Despite experimental effort, none of these properties have been observed (1518).

Because the existence of px ± ipy order in Sr2RuO4 would have broad importance, further evaluation is necessary. Under the tetragonal symmetry of the Sr2RuO4 lattice, the two components px and py 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 px ± ipy superconductor (19, 20). If the transition temperature Tc is measured with a probe sensitive mainly to the upper transition, such as resistivity or susceptibility, then the observed Tc 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).

Fig. 1 Hypothesis and apparatus.

(A) General phase diagram expected for px ± ipy superconductivity in a tetragonal crystal subject to a small, volume-preserving, symmetry-breaking strain εxx – εyy. T, temperature. (B) Sketch of the uniaxial strain apparatus constructed to test this hypothesis.

Sr2RuO4 has consistently been found to be sensitive to uniaxial pressure: The 3-K phase, locally higher-Tc superconductivity that appears in Sr2RuO4-Ru eutectic samples (23), can be induced in pure Sr2RuO4 by uniaxial pressures (24, 25). But the direction dependence of the uniaxial strain effect has not been resolved, and the higher-Tc 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 Tc 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 Tc = 1.35 K at zero strain, under compression (A) and tension (B) along a Embedded Image 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.

Fig. 2 Superconductivity of Sr2RuO4 under strain.

(A and B) In-phase (χ′) and out-of-phase (χ′′) parts of the ac susceptibility, measured at 369 Hz on the Embedded Image-oriented sample shown in the inset of (B). Approximate strains for some curves are indicated. arb. units, arbitrary units. (C) Tc versus Embedded Image-oriented strain ε100 of two samples of Sr2RuO4, one with a zero-strain Tc of 1.35 K [for which the raw data are shown in (A) and (B)] and the other with 1.45 K. ε > 0 indicates tension. Tc is taken as the 50% point of χ′, and the black lines are the 20 and 80% points, giving a measure of the transition width. The error bar on the horizontal axis indicates the error in locating ε = 0 (29). (D) Tc versus Embedded Image strain ε110 for two further samples cut from the same crystals as in (C). The temperature scale is the same as in (C), highlighting the large difference in response between the two directions. (E) dTc/dε for the data in (C).

We measured four samples cut in a Embedded Image direction and three in a Embedded Image direction. The main features we report were reproduced across the entire sample set. The dependence of Tc on strains applied along Embedded Image and Embedded Image 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 Embedded Image strain (ε100), there is a large symmetric response about ε100 = 0, whereas the response to Embedded Image strain (ε110) is far weaker and mainly linear in ε. The proposed px ± ipy superconductivity of Sr2RuO4 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 (εxx = εyy) and volume-preserving anisotropic distortion (εxx = –εyy for ε100, and εxx = εyy = 0, εxy ≠ 0 for ε110) (30). The in-plane Poisson’s ratio for Sr2RuO4 is ≈0.40 (29, 31), so for both ε100 and ε110, ~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 Embedded Image strains is through dilatation and of Embedded Image strains through the volume-preserving distortion. Small variations about Embedded Image 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 ε100 component to the strain (fig. S3).

A key question is the lack of the expected cusp near zero strain. At |ε100| < 0.03%, Tc(ε) and dTc/dε are essentially flat, then Tc rises strongly at larger strains. Both of these features remain essentially unchanged when the excitation field is varied (fig. S4). As emphasized by the dTc/dε data in Fig. 2E, the increase of Tc beyond 0.03% strain is quadratic in strain, not linear. The precise low-strain form of Tc100) may be affected by inhomogeneity and requires further investigation, but the strong increase in Tc 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 Embedded Image strain suggest s- and Embedded Image-wave orders, but these would be in conflict with both triplet pairing and time-reversal symmetry breaking at Tc and, thus, are unlikely. Another possibility is that sketched in Fig. 3A: that Sr2RuO4 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.

Fig. 3 Electronic structure under strain.

(A) A possible phase diagram with a weak cusp and strong underlying response to strain. (B and C) Calculated Fermi surfaces (labeled α, β, and γ) at zero z-axis momentum under 0.5% compression (black) and tension (blue). The dashed lines are, for reference, the two-dimensional zone boundaries of the RuO2 sheets. (B) [100] strain (along Embedded Image); (C) [110] strain (along Embedded Image). In the lower right (shaded) portion of (C) only, the surfaces have been distorted slightly to make the zone boundaries match, and differences between the surfaces have been exaggerated by a factor of 5.

To gain further insight, we used the Wien2k package (32) to calculate the electronic structure in the presence of strain; the Fermi surfaces for ε100 and ε110 = ±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 Embedded Image than Embedded Image strains, a result supported by ultrasound data: Above Tc, sound waves that generate xx and yy strains are damped much more than waves generating xy strains (33).

The shifting Fermi surfaces will alter Tc. With px ± ipy order, a strong cusp is expected when the longitudinal and transverse responses to strain are very different. In Sr2RuO4, this occurs most clearly on the {100} sections of the γ sheet, near the van Hove points: Under Embedded Image 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 dTc/dε100 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 (3436), in which case a weaker cusp is expected, which could have been obscured by a strong underlying response. Finally, px ± ipy 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 Embedded Image than Embedded Image strains. But [110] strain changes the dispersions on the (110) and Embedded Image sections of the γ sheet almost identically, so the cusp might be tiny. We estimate that dTc/dε110 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 px ± ipy order. The challenge for degenerate p-wave models is to explain simultaneously the apparent weakness of the cusp and the large nonlinear response to Embedded Image 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 Sr2RuO4: The characteristic energy scale of a 0.5% change in strain is large—roughly that percentage of the bandwidth. For Sr2RuO4, 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

Materials and Methods

Supplementary Text

Figs. S1 to S7

References (4347)

References and Notes

  1. In addition to px ± ipy, ground states of unstrained Sr2RuO4 that would yield cusps in Tc at ε = 0 include dxz ± idyz (22) and the nonchiral orders px and px + py.
  2. 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.
  3. More details on our methods and calculations can be found in the supplementary materials on Science Online.
  4. We use ε100 and ε110 to denote the applied strains, implying simultaneous transverse strains following the Poisson’s ratios, whereas εij are defined as usual.
  5. Angle-resolved photoemission spectroscopy under uniaxial stress has been performed [see (42)].
  6. 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.
  7. 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.
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