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# Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4

## Abstract

Sr2RuO4 has stood as the leading candidate for a spin-triplet superconductor for 26 years1. However, recent NMR experiments have cast doubt on this candidacy2,3 and it is difficult to find a theory of superconductivity that is consistent with all experiments. The order parameter symmetry for this material therefore remains an open question. Symmetry-based experiments are needed that can rule out broad classes of possible superconducting order parameters. Here, we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr2RuO4 through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus c66, which implies that the superconducting order parameter has two components. A two-component p-wave order parameter, such as px + ipy, naturally satisfies this requirement. As this order parameter appears to have been precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely $$\{{d}_{xz},{d}_{yz}\}$$ and $$\{{d}_{{x}^{2}-{y}^{2}},{g}_{xy({x}^{2}-{y}^{2})}\}$$, are now the prime candidates for the order parameter of Sr2RuO4.

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## Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We acknowledge helpful discussions with K. Modic, S. Kivelson, I. Mazin, D. Agterberg, R. Thomale, P. Hirschfeld, R. Fernandes, I. Paul, C. Proust and L. Taillefer. B.J.R. and S.G. are grateful for help with the experimental design from E. Smith and J. Parpia, and from the machine shop staff of the Laboratory of Atomic and Solid State Physics at Cornell University. B.J.R and S.G. acknowledge support for building the experiment, collecting and analysing the data, and writing the manuscript from the Office of Basic Energy Sciences of the United States Department of Energy under award no. DE-SC0020143. B.J.R. and S.G. acknowledge support from the Cornell Center for Materials Research with funding from the Materials Research Science and Engineering Centers program of the National Science Foundation (cooperative agreement no. DMR-1719875). A.S. acknowledges support from the National High Magnetic Field Laboratory, which is supported by the National Science Foundation (cooperative agreement no. DMR-1644779) and the State of Florida. N.K. acknowledges support from KAKENHI (Grants-in-Aid for Scientific Research, grant no. JP18K04715) of the Japan Society for the Promotion of Science.

## Author information

Authors

### Contributions

S.G. and B.J.R. designed the experiment. F.J., D.A.S., N.K., M.B., C.W.H. and A.P.M. prepared the crystal and performed characterization measurements. S.G. acquired and analysed the ultrasound data. S.G., A.S., C.W.H. and B.J.R. wrote the manuscript with input from all co-authors.

### Corresponding author

Correspondence to B. J. Ramshaw.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Johnpierre Paglione and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 RUS frequency data.

Temperature evolution of 18 resonance frequencies of Sr2RuO4 through Tc, with panels (a) and (b) each showing 9 frequencies. Plots are vertically shifted for visual clarity.

### Extended Data Fig. 2 Characterization of the Sr2RuO4 rod.

(a) Specific heat and (b) susceptibility measurements of the upper critical field, measured on different parts of the same rod from which the sample for RUS experiment was obtained. Tc varies by about 200 mK between different parts of the rod.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–3, Table 1 and Discussion.

## Source data

### Source Data Fig. 2

Ultrasound spectra raw data.

### Source Data Fig. 3

Temperature dependence of RUS frequencies and elastic moduli.

### Source Data Extended Data Fig. 1

RUS frequencies data through superconducting transition.

### Source Data Extended Data Fig. 2

Specific heat and critical field data on Sr2RuO4.

## Rights and permissions

Reprints and Permissions

Ghosh, S., Shekhter, A., Jerzembeck, F. et al. Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4. Nat. Phys. 17, 199–204 (2021). https://doi.org/10.1038/s41567-020-1032-4

• Accepted:

• Published:

• Issue Date:

• ### The symmetry of superconducting Sr2RuO4

• Daniel F. Agterberg

Nature Physics (2021)

• ### Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress

• , Shreenanda Ghosh
• , Rajib Sarkar
• , Jean-Christophe Orain
• , Artem Nikitin
• , Matthias Elender
• , Debarchan Das
• , Zurab Guguchia
• , Felix Brückner
• , Mark E. Barber
• , Joonbum Park
• , Naoki Kikugawa
• , Dmitry A. Sokolov
• , Jake S. Bobowski
• , Takuto Miyoshi
• , Yoshiteru Maeno
• , Andrew P. Mackenzie
• , Hubertus Luetkens
• , Clifford W. Hicks
•  & Hans-Henning Klauss

Nature Physics (2021)

• ### Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4

• S. Benhabib
• , C. Lupien
• , I. Paul
• , L. Berges
• , M. Dion
• , M. Nardone
• , A. Zitouni
• , Z. Q. Mao
• , Y. Maeno
• , A. Georges
• , L. Taillefer
•  & C. Proust

Nature Physics (2021)

• ### A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4

• Steven Allan Kivelson
• , Andrew Chang Yuan