Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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


Strontium ruthenate (Sr2RuO4) continues to present an important test of our understanding of unconventional superconductivity, because while its normal-state electronic structure is known with precision, its superconductivity remains unexplained. There is evidence that its order parameter is chiral, but reconciling this with recent observations of the spin part of the pairing requires an order parameter that is either finely tuned or implies a new form of pairing. Therefore, a definitive resolution of whether the superconductivity of Sr2RuO4 is chiral is important for the study of superconductivity. Here we report the measurement of zero-field muon spin relaxation—a probe sensitive to weak magnetism—on samples under uniaxial stresses. We observe stress-induced splitting between the onset temperatures of superconductivity and time-reversal symmetry breaking—consistent with the qualitative expectations for a chiral order parameter—and argue that this observation cannot be explained by conventional magnetism. In addition, we report the appearance of bulk magnetic order under higher uniaxial stress, above the critical pressure at which a Lifshitz transition occurs in Sr2RuO4.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Hypothesis and setup.
Fig. 2: Results on unstressed Sr2RuO4.
Fig. 3: Stress-induced splitting between Tc and TTRSB.
Fig. 4: Magnetic order.
Fig. 5: Experimental phase diagram.

Data availability

The data shown in Figs. 25 are available as the source data. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


  1. Maeno, Y. et al. Superconductivity in a layered perovskite without copper. Nature 372, 532–534 (1994).

    Article  ADS  Google Scholar 

  2. Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing. Rev. Mod. Phys. 75, 657–712 (2003).

    Article  ADS  Google Scholar 

  3. Mackenzie, A. P., Scaffidi, T., Hicks, C. W. & Maeno, Y. Even odder after twenty-three years: the superconducting order parameter puzzle of Sr2RuO4. npj Quantum Mater. 2, 40 (2017).

    Article  ADS  Google Scholar 

  4. Maeno, Y., Kittaka, S., Nomura, T., Yonezawa, S. & Ishida, K. Evaluation of spin-triplet superconductivity in Sr2RuO4. J. Phys. Soc. Jpn 81, 011009 (2012).

    Article  ADS  Google Scholar 

  5. Kallin, C. Chiral p-wave order in Sr2RuO4. Rep. Prog. Phys. 75, 042501 (2012).

    Article  ADS  Google Scholar 

  6. Luke, G. M. et al. Time-reversal symmetry-breaking superconductivity in Sr2RuO4. Nature 394, 558–561 (1998).

    Article  ADS  Google Scholar 

  7. Xia, J., Maeno, Y., Beyersdorf, P. T., Fejer, M. M. & Kapitulnik, A. High resolution polar Kerr effect measurements of Sr2RuO4: evidence for broken time-reversal symmetry in the superconducting state. Phys. Rev. Lett. 97, 167002 (2006).

    Article  ADS  Google Scholar 

  8. Nakamura, T. et al. Essential configuration of Pb/Ru/Sr2RuO4 junctions exhibiting anomalous superconducting interference. J. Phys. Soc. Jpn 81, 064708 (2012).

    Article  ADS  Google Scholar 

  9. Anwar, M. A. et al. Anomalous switching in Nb/Ru/Sr2RuO4 topological junctions by chiral domain wall motion. Sci. Rep. 3, 2480 (2013).

    Article  Google Scholar 

  10. Ishida, K. et al. Spin-triplet superconductivity in Sr2RuO4 identified by 17O Knight shift. Nature 396, 658–660 (1998).

    Article  ADS  Google Scholar 

  11. Duffy, J. A. et al. Polarized-neutron scattering study of the Cooper-pair moment in Sr2RuO4. Phys. Rev. Lett. 85, 5412 (2000).

    Article  ADS  Google Scholar 

  12. Pustogow, A. et al. Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance. Nature 574, 72–75 (2019).

    Article  ADS  Google Scholar 

  13. Ishida, K., Manago, M. & Maeno, Y. Reduction of the 17O Knight shift in the superconducting state and the heat-up effect by NMR pulses on Sr2RuO4. J. Phys. Soc. Jpn 89, 034712 (2020).

    Article  ADS  Google Scholar 

  14. Petsch, A. N. et al. Reduction of the spin susceptibility in the superconducting state of Sr2RuO4 observed by polarized neutron scattering. Phys. Rev. Lett. 125, 217004 (2020).

    Article  ADS  Google Scholar 

  15. Suh, H. G. et al. Stabilizing even-parity chiral superconductivity in Sr2RuO4. Phys. Rev. Res. 2, 032023(R) (2019).

    Article  Google Scholar 

  16. Sharma, R. et al. Momentum-resolved superconducting energy gaps of Sr2RuO4 from quasiparticle interference imaging. Proc. Natl Acad. Sci. USA 117, 5222–5227 (2020).

    Article  ADS  Google Scholar 

  17. Shiroka, T. et al. μSR studies of superconductivity in eutectically grown mixed ruthenates. Phys. Rev. B 85, 134527 (2012).

    Article  ADS  Google Scholar 

  18. Luke, G. M. et al. Unconventional superconductivity in Sr2RuO4. Physica B 289–290, 373–376 (2000).

    Article  ADS  Google Scholar 

  19. Higemoto, W., Koda, A., Kadono, R., Yoshida, Y. & Onuki, Y. Investigation of spontaneous magnetic field in spin-triplet superconductor Sr2RuO4. JPS Conf. Proc. 2, 010202 (2014).

    Google Scholar 

  20. Hicks, C. W. et al. Limits on superconductivity-related magnetization in Sr2RuO4 and PrOs4Sb12 from scanning SQUID microscopy. Phys. Rev. B 81, 214501 (2010).

    Article  ADS  Google Scholar 

  21. Kashiwaya, S. et al. Time-reversal invariant superconductivity of Sr2RuO4 revealed by Josephson effects. Phys. Rev. B 100, 094530 (2019).

    Article  ADS  Google Scholar 

  22. Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).

    Article  ADS  Google Scholar 

  23. Li, Y.-S. et al. High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure. Proc. Natl Acad. Sci. USA 118, e2020492118 (2021).

    Article  Google Scholar 

  24. Hicks, C. W. et al. Strong increase of Tc of Sr2RuO4 under both tensile and compressive strain. Science 344, 283–285 (2014).

    Article  ADS  Google Scholar 

  25. Watson, C. A., Gibbs, A. S., Mackenzie, A. P., Hicks, C. W. & Moler, K. A. Micron-scale measurements of low anisotropic strain response of local Tc in Sr2RuO4. Phys. Rev. B 98, 094521 (2018).

    Article  ADS  Google Scholar 

  26. Mackenzie, A. P. et al. Extremely strong dependence of superconductivity on disorder in Sr2RuO4. Phys. Rev. Lett. 80, 161–164 (1998).

    Article  ADS  Google Scholar 

  27. Maeno, Y. et al. Enhancement of superconductivity of Sr2RuO4 to 3 K by embedded metallic microdomains. Phys. Rev. Lett. 81, 3765–3768 (1998).

    Article  ADS  Google Scholar 

  28. Steppke, A. et al. Strong peak in Tc of Sr2RuO4 under uniaxial pressure. Science 355, eaaf9398 (2017).

    Article  Google Scholar 

  29. Barber, M. E. et al. Role of correlations in determining the Van Hove strain in Sr2RuO4. Phys. Rev. B 100, 245139 (2019).

    Article  ADS  Google Scholar 

  30. Sunko, V. et al. Direct observation of a uniaxial stress-driven Lifshitz transition in Sr2RuO4. npj Quantum Mater. 4, 46 (2019).

    Article  ADS  Google Scholar 

  31. Steffens, P. et al. Spin fluctuations in Sr2RuO4 from polarized neutron scattering: implications for superconductivity. Phys. Rev. Lett. 122, 047004 (2019).

    Article  ADS  Google Scholar 

  32. Cobo, S., Ahn, F., Eremin, I. & Akbari, A. Anisotropic spin fluctuations in Sr2RuO4: role of spin-orbit coupling and induced strain. Phys. Rev. B 94, 224507 (2016).

    Article  ADS  Google Scholar 

  33. Minakata, M. & Maeno, Y. Magnetic ordering in Sr2RuO4 induced by nonmagnetic impurities. Phys. Rev. B 63, 180504 (2001).

    Article  ADS  Google Scholar 

  34. Braden, M. et al. Incommensurate magnetic ordering in Sr2Ru1–xTixO4. Phys. Rev. Lett. 88, 197002 (2002).

    Article  ADS  Google Scholar 

  35. Carlo, J. P. et al. New magnetic phase diagram of (Sr,Ca)2RuO4. Nat. Mater. 11, 323–328 (2012).

    Article  ADS  Google Scholar 

  36. Liu, Y.-C., Zhang, F.-C., Rice, T. M. & Wang, Q.-H. Theory of the evolution of superconductivity in Sr2RuO4 under anisotropic strain. npj Quantum Mater. 2, 12 (2017).

    Article  ADS  Google Scholar 

  37. Brodsky, D. O. et al. Strain and vector magnetic field tuning of the anomalous phase in Sr3Ru2O7. Sci. Adv. 3, e1501804 (2017).

    Article  ADS  Google Scholar 

  38. Spehling, J. et al. Magnetic order and spin dynamics in the proximity of a ferromagnetic quantum critical point: a μSR study of YbNi4P2. Phys. Rev. B 85, 140406(R) (2012).

    Article  ADS  Google Scholar 

  39. Lausberg, S. et al. Avoided ferromagnetic quantum critical point: unusual short-range ordered state in CeFePO. Phys. Rev. Lett. 109, 216402 (2012).

    Article  ADS  Google Scholar 

  40. Wu, W. D. et al. Muon spin relaxation studies of magnetic order in Y1–xUxPd3 and UPd4. Phys. Rev. Lett. 72, 3722–3725 (1994).

    Article  ADS  Google Scholar 

  41. Luke, G. M. et al. Muon spin relaxation in UPt3. Phys. Rev. Lett. 71, 1466–1469 (1993).

    Article  ADS  Google Scholar 

  42. Dalmas de Réotier, P. et al. Absence of zero field muon spin relaxation induced by superconductivity in the B phase of UPt3. Phys. Lett. A 205, 239–243 (1995).

    Article  ADS  Google Scholar 

  43. Grinenko, V. et al. Superconductivity with broken time-reversal symmetry in ion-irradiated Ba0.27K0.73Fe2As2 single crystals. Phys. Rev. B 95, 214511 (2017).

    Article  ADS  Google Scholar 

  44. Grinenko, V. et al. Superconductivity with broken time-reversal symmetry inside a superconducting s-wave state. Nat. Phys. 16, 789–794 (2020).

    Article  Google Scholar 

  45. Zhang, J. et al. Broken time-reversal symmetry in superconducting Pr1−xLaxPt4Ge12. Phys. Rev. B 100, 024508 (2019).

    Article  ADS  Google Scholar 

  46. Schemm, E. R., Gannon, W. J., Wishne, C. M., Halperin, W. P. & Kapitulnik, A. Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3. Science 345, 190–193 (2014).

    Article  ADS  Google Scholar 

  47. Strand, J. D. et al. The transition between real and complex superconducting order parameter phases in UPt3. Science 328, 1368–1369 (2010).

    Article  ADS  Google Scholar 

  48. Avers, K. E. et al. Broken time-reversal symmetry in the topological superconductor UPt3. Nat. Phys. 16, 531–535 (2020).

    Article  Google Scholar 

  49. Ghosh, S. et al. Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4. Nat. Phys. (2020).

  50. Benhabib, S. et al. Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4. Nat. Phys. (2020).

  51. Fischer, M. H. & Berg, E. Fluctuation and strain effects in a chiral p-wave superconductor. Phys. Rev. B 93, 054501 (2016).

    Article  ADS  Google Scholar 

  52. Yu, Y. & Raghu, S. Effect of strain inhomogeneity on a chiral p-wave superconductor. Phys. Rev. B 100, 094517 (2019).

    Article  ADS  Google Scholar 

  53. Puetter, C. M. & Kee, H.-Y. Identifying spin-triplet pairing in spin-orbit coupled multi-band superconductors. Europhys. Lett. 98, 27010 (2012).

    Article  ADS  Google Scholar 

  54. Ramires, A. & Sigrist, M. Superconducting order parameter of Sr2RuO4: a microscopic perspective. Phys. Rev. B 100, 104501 (2019).

    Article  ADS  Google Scholar 

  55. Rømer, A. T., Scherer, D. D., Eremin, I. M., Hirschfeld, P. J. & Andersen, B. M. Knight shift and leading superconducting instability from spin fluctuations in Sr2RuO4. Phys. Rev. Lett. 123, 247001 (2019).

    Article  ADS  Google Scholar 

  56. Kivelson, S. A., C., Y. A., Ramshaw, B. J. & Thomale, R. A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4. npj Quantum Mat. 5, 43 (2020).

    Article  ADS  Google Scholar 

  57. Rømer, A. T. et al. Theory of strain-induced magnetic order and splitting of Tc and TTRSB in Sr2RuO4. Phys. Rev. B 102, 054506 (2020).

    Article  ADS  Google Scholar 

  58. Bobowski, J. S. et al. Improved single-crystal growth of Sr2RuO4. Condens. Matter 4, 6 (2019).

    Article  Google Scholar 

  59. Ghosh, S. et al. Piezoelectric-driven uniaxial pressure cell for muon spin relaxation and neutron scattering experiments. Rev. Sci. Instrum. 91, 103902 (2020).

    Article  ADS  Google Scholar 

  60. Maisuradze, A., Khasanov, R., Shengelaya, A. & Keller, H. Comparison of different methods for analyzing μSR line shapes in the vortex state of type-II superconductors. J. Phys. Condens. Matter 21, 075701 (2009).

    Article  ADS  Google Scholar 

  61. Suter, A. & Wojek, B. M. Musrfit: a free platform-independent framework for μSR data analysis. Phys. Proc. 30, 69–73 (2012).

    Article  ADS  Google Scholar 

  62. Clawson, C. W. et al. Low-temperature mobility of positive muons in copper. Phys. Rev. Lett. 51, 114–117 (1983).

    Article  ADS  Google Scholar 

  63. Brandt, E. H. Properties of the ideal Ginzburg-Landau vortex lattice. Phys. Rev. B 68, 054506 (2003).

    Article  ADS  Google Scholar 

  64. Grinenko, V. et al. Low-temperature breakdown of antiferromagnetic quantum critical behavior in FeSe. Phys. Rev. B 97, 201102(R) (2018).

    Article  ADS  Google Scholar 

Download references


This work has been financially supported by the Deutsche Forschungsgemeinschaft (GR 4667/1, GRK 1621 and SFB 1143 projects C02 and C09) and the Max Planck Society. Y.M., T.M. and J.S.B. acknowledge the financial support of JSPS Kakenhi (JP15H5852, JP15K21717 and JP17H06136) and the JSPS Core-to-Core Program. N.K. acknowledges the financial support from JSPS Kakenhi (no. JP18K04715) and JST-Mirai Program (no. JPMJMI18A3). A.N. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 701647. This work was partially performed at the Swiss Muon Source (SμS), PSI, Villigen. We acknowledge fruitful discussions with A. Amato, B. Andersen, E. Babaev, S. Blundell, A. Charnukha, D. Efremov, I. Eremin, C. Kallin, A. Ramires, B. Ramshaw, A. Rømer, T. Scaffidi, M. Sigrist, C. Timm and S. Yonezawa. We also acknowledge H.-S. Xu for his contribution in the crystal growth as well as T. Shiroka and C. Wang for technical support. We thank A. Gilman, P. P. Orth and R. M. Fernandes for results from the Landau theory of a two-component order parameter.

Author information

Authors and Affiliations



V.G., S.G., R.S., J.-C.O., A.N., D.D., Z.G., H.L. and H.-H.K. performed the μSR measurements. V.G. performed the heat capacity measurements. V.G., S.G. and H.L. analysed the raw μSR data. S.G., A.N., M.E., F.B. and C.W.H. built and characterized the stress apparatus. M.E.B. and J.P. characterized the samples. N.K., D.A.S., J.S.B. and T.M. grew the samples. V.G., Y.M., A.P.M., H.L., C.W.H. and H.-H.K. supervised the research. V.G., C.W.H. and H.-H.K. wrote the paper. C.W.H. designed the stress apparatus. H.-H.K. initiated this study. All the authors discussed the results and implications, and they commented on the manuscript.

Corresponding authors

Correspondence to Vadim Grinenko, Clifford W. Hicks or Hans-Henning Klauss.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review informationNature Physics thanks Morten Eskildsen, Jeff Sonier 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 The sample setup.

a, Photographs of Sample F, showing the locations of the ac susceptibility coil, the hematite masks, and the force-sensing strain gauges. b, A view from the muon direction. The beam diameter is 1 cm. (c) A schematic of the sample holder, in which the moving portion is coloured blue. This holder slots into a force generator, described in Ref. 59. d, Force-displacement data for Samples D-F. For Sample D, a chip broke from one of the two pieces of Sr2RuO4 at a load of 500 N. For Samples E and F, the force-displacement relationships are straight lines, showing that the sample did not fracture under the load. Note that in Sample F the data fall on two distinct lines. This may have originated in a slip in the apparatus that altered the calibration of the displacement sensor; it is not of the form expected for damage to the sample.

Extended Data Fig. 2 TF data.

(a-b) Transverse-field μSR time spectra above and below Tc, with a muon spin polarization 45 with respect to the sample c axis and Bc. (c-d) Fourier transforms of the time spectra shown in panels (a) and (b). The fit to determine the sample contribution is done in the time domain.

Extended Data Fig. 3 Testing different backgrounds.

a, A(t) resulting from the single histogram analysis of Sample D at − 0.43 GPa under different assumptions about the background: non-relaxing, and with 55% of the background muon spins taken to relax following the relaxation function of copper. b, Results from the single histogram analysis for the relaxation rate λ taking different portions of copper in the background. c, Data in panel (b) with temperature independent constants subtracted. Assuming different backgrounds is seen to have essentially no effect on temperature-dependent relative changes in λ extracted from analysis. d, Fourier transforms of the transverse-field μSR time spectra of Sample A2 above and below Tc. e, Comparison of the temperature dependence of the zero field muon spin relaxation rate (left scale) and the specific heat data (right scale) for Samples A and A2.

Extended Data Fig. 4 Control measurement.

a, Photograph of the ‘sample’ for these measurements: a hematite mask is put in place of Sr2RuO4. Therefore the signal originates from muons stopping in hematite and in the non-magnetic background (cryostat walls, sample holder frame, and etc.). b, Strongly damped oscillations of the muon spin polarisation can be seen in the first 50 ns. There is no notable change in their form between 1.6 and 0.4 K. c, Weak transverse-field measurements, showing the signal from muons that did not implant into the hematite. d, Zero-field measurements: the polarisation of the muons implanting outside the hematite decays on a μs time scale. Fits exclude the first 50 ns, shown in panel (b). e, The zero-field muon spin relaxation rate has no significant temperature dependence.

Extended Data Fig. 5 Further characterization of the samples.

Additional sample characterisation data for samples a-e. Penetration depths were determined through transverse-field μSR measurements. Heat capacity data for Samples c,d, and e were recorded from portions of the samples that were removed from the holder after the μSR measurement; for Sample a the measurement was performed on a portion of the sample before the μSR measurement. Note that for Sample c, we show heat capacity data from Sample e, which was extracted from Sample c. Susceptibility data were recorded in situ using the susceptibility coils shown in Extended Data Fig. 1. (g-h) Additional data on Samples E and F at, respectively, -0.70 and -0.79 GPa. (h-i) In situ ac susceptibility raw data at different uniaxial stresses for Samples e and f.

Extended Data Fig. 6 Testing different models to describe magnetic order.

Zero-field asymmetry A(t) in the magnetic state for Sample F at -1.05 GPa, fitted with a Bessel function describing incommensurate spin density wave order, and a damped cosine describing ferromagnetism or commensurate magnetic order. The damped cosine fit performs noticably worse at early times, and also fails to capture the third oscillation at t ≈ 3.5μs. Note that in the single histogram analysis a functional form for A(t) must be assumed to fit the dark count rate, which results in different experimental A(t) for the two forms, even though both are derived from a single data set.

Extended Data Fig. 7 Knight shift.

a, Temperature dependence of the muon Knight shift for Sample A2 in Bab = 8 T. The Knight shift is defined as Kab = [BSRO − BAg]/BAg, where BSRO and BAg are the magnetic fields at the muon stopping site in Sr2RuO4 and the Ag sample holder, respectively. The inset shows a Fourier transform of the high transversal field time spectra at T = 0.12 K. The frequencies for the Ag and Sr2RuO4 are well-resolved. b, Temperature dependence of the muon spin relaxation rate for Sample A2 and the Ag sample holder in Bab = 8 T.

Supplementary information

Supplementary Information

Further discussion of possible magnetic mechanisms and a Ginzburg–Landau analysis.

Source data

Source Data Fig. 2

μSR and specific heat data at zero stress shown in Fig. 2.

Source Data Fig. 3

μSR and a.c. susceptibility data at different uniaxial pressures shown in Fig. 3.

Source Data Fig. 4

μSR and a.c. susceptibility data at P = −1.05 GPa shown in Fig. 4.

Source Data Fig. 5

Superconducting Tc, TTRSB and magnetic transition TN temperatures at different values of the uniaxial pressure shown in Fig. 5.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grinenko, V., Ghosh, S., Sarkar, R. et al. Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress. Nat. Phys. 17, 748–754 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing