High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure

A key question regarding the unconventional superconductivity of Sr2RuO4 remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat-capacity signature of any second transition of a few percent of that of the primary superconducting transition. The normalized jump in heat capacity, ΔC/C, grows smoothly as a function of uniaxial pressure, favoring order parameters which are allowed to maximize in the same part of the Brillouin zone as the well-studied van Hove singularity. Thanks to the high precision of our measurements, these findings place stringent constraints on theories of the superconductivity of Sr2RuO4.

Obtaining a full understanding of the superconductivity of Sr2RuO4 is a core challenge for condensed-matter physics. Since soon after its discovery over a quarter of a century ago (1), the superconducting order parameter of Sr2RuO4 has been known to be unconventional (2, 3) and to condense from a well-understood and fairly simple quasi-two-dimensional Fermi liquid metallic state (4–7). Given the profound advances in theoretical techniques in recent decades a full understanding of its superconductivity is an important, and attainable, challenge for the field. The form of the wave-vector-dependent susceptibility of Sr2RuO4 leads to the prediction of a rich superconducting phase diagram in weak-coupling calculations which aim to perform a bias-free estimate of the condensation energies of different order parameters. A notable feature of the results is how close a number of different odd- and even-parity solutions are seen to be in energy (8–10). On the one hand, this emphasizes the potential of Sr2RuO4 as a test-bed material on which to refine the predictive capabilities of modern theories of unconventional superconductivity (11). On the other hand, realizing this potential will likely first require a conclusive experimental determination of which of the many possible order parameters wins out in the real material. This is a particularly exciting stage of the quest to complete this empirical determination, for reasons that we will now outline.For over 20 y the large majority of attention was paid to odd-parity order parameter candidates for Sr2RuO4 (12), because of NMR measurements of spin susceptibility in the superconducting state that seemed to be inconsistent with any even-parity order parameter (13). However, thanks to the discovery of a systematic error in the original NMR measurements (14, 15), that situation has now been more or less completely reversed. Taking into account the most recent measurements of the magnetic field dependence of the spin susceptibility (16), it seems clear that the order parameter must be even-parity or at least dominated by an even-parity component. The spin susceptibility results would be most easily describable in terms of a single-component, likely d-wave, order parameter, but recent thermodynamic evidence from ultrasound experiments is most straightforwardly interpreted in terms of an order parameter with two degenerate components (17, 18). Such order parameters do not of necessity break time-reversal symmetry, but they can, if the two degenerates have the appropriate phase relationship. In the context it is significant that long-standing muon-spin relaxation (μSR) (19, 20) and magneto-optic Kerr rotation measurements (21) have indicated time-reversal symmetry breaking in the superconducting state.To investigate any order parameter with two degenerate components, whether or not it breaks time-reversal symmetry, uniaxial pressure is a powerful probe because it can split the degeneracy, creating a split superconducting phase transition (22). In a significant experimental advance, the muon-spin relaxation experiments have recently been extended to high uniaxial pressures (23). In line with naive expectation, the temperature at which time-reversal symmetry is broken (TTRSB) splits from the main superconducting transition (Tc), with TTRSB remaining nearly pressure-independent while Tc increases under the application of the pressure. However, there has been a long-standing question about whether the Kerr and muon signals correspond to bulk thermodynamic transitions, so it is highly desirable to compare the new muon-spin relaxation data with those from a bulk thermodynamic probe. In this context, it is natural to look at heat capacity, because it has an intrinsic sensitivity to transitions within the superconducting state, as is well known from work on UPt3 (24, 25).