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

You-Sheng Li; Naoki Kikugawa; Dmitry A. Sokolov; Fabian Jerzembeck; Alexandra S. Gibbs; Yoshiteru Maeno; Clifford W. Hicks; Jörg Schmalian; Michael Nicklas; Andrew P. Mackenzie

doi:10.1073/pnas.2020492118

Edited by Zachary Fisk, University of California, Irvine, CA, and approved January 13, 2021 (received for review September 30, 2020)

Significance

Research on the unconventional superconductivity of Sr2RuO4 is undergoing a renaissance since recent spin susceptibility measurements ruling out the spin triplet order parameter which had been widely favored for over two decades. With ultrasound, Kerr rotation, and muon spin relaxation data all providing evidence for a two-component order parameter, it is vital that this possibility be investigated thermodynamically by studying the dependence of the heat-capacity anomaly on uniaxial pressure. Here, the relevant experimental results are combined with theoretical analysis that shows how strongly the data constrain theories of the order parameter. In particular, we do not observe any signs of transition splitting of two-order-parameter components. Sr2RuO4 thus offers a unique test bed for theories of unconventional superconductivity.

Abstract

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.

Footnotes

↵¹To whom correspondence may be addressed. Email: Michael.Nicklas{at}cpfs.mpg.de or andy.mackenzie{at}cpfs.mpg.de.

Author contributions: C.W.H., M.N., and A.P.M. designed research; Y.-S.L., N.K., D.A.S., F.J., A.S.G., Y.M., and C.W.H. performed research; Y.-S.L. and M.N. analyzed data; and J.S. and A.P.M. wrote the paper with contributions from the other authors.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2020492118/-/DCSupplemental.

Data Availability

Heat-capacity data that underpin the findings of this study are available from the University of St Andrews research portal: https://doi.org/10.17630/b5ff3cfd-bb2d-46c9-a240-7a7064562588.

Published under the PNAS license.

View Full Text

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, Direct observation of a uniaxial stress-driven Lifshitz transition in Sr2RuO4. npj Quant. Mat. 4, 2397–4648 (2019).
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1. I. Žutić,
2. I. Mazin
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1. H. G. Suh et al.
, Stabilizing even-parity chiral superconductivity in Sr2RuO4. Phys. Rev. Res. 2, 032023(R) (2020).
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Proceedings of the National Academy of Sciences: 118 (10)

Table of Contents

Submit

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

[2] Y. Maeno et al.

[3] ↵
A. P. Mackenzie,
Y. Maeno
, The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing. Rev. Mod. Phys. 75, 657–712 (2003).
OpenUrl CrossRef

[4] A. P. Mackenzie,

[5] Y. Maeno

[6] ↵
A. P. Mackenzie et al.
, Extremely strong dependence of superconductivity on disorder in Sr2RuO4. Phys. Rev. Lett. 80, 161–164 (1998).
OpenUrl CrossRef

[7] A. P. Mackenzie et al.

[8] ↵
A. P. Mackenzie et al.
, Quantum oscillations in the layered perovskite superconductor Sr2RuO4. Phys. Rev. Lett. 76, 3786–3789 (1996).
OpenUrl CrossRef PubMed

[9] A. P. Mackenzie et al.

[10] ↵
Y. Maeno et al.
, Two-dimensional Fermi liquid behavior of the superconductor Sr2RuO4. J. Phys. Soc. Jpn. 66, 1405–1408 (1997).
OpenUrl

[11] Y. Maeno et al.

[12] ↵
C. Bergemann,
S. R. Julian,
A. P. Mackenzie,
S. NishiZaki,
Y. Maeno
, Detailed topography of the Fermi surface of Sr2RuO4. Phys. Rev. Lett. 84, 2662–2665 (2000).
OpenUrl CrossRef PubMed

[13] C. Bergemann,

[14] S. R. Julian,

[15] A. P. Mackenzie,

[16] S. NishiZaki,

[17] Y. Maeno

[18] ↵
C. Bergemann,
A. P. Mackenzie,
S. R. Julian,
D. Forsythe,
E. Ohmichi
, Quasi-two-dimensional Fermi liquid properties of the unconventional superconductor Sr2RuO4. Adv. Phys. 52, 639–725 (2003).
OpenUrl CrossRef PubMed

[19] C. Bergemann,

[20] A. P. Mackenzie,

[21] S. R. Julian,

[22] D. Forsythe,

[23] E. Ohmichi

[24] ↵
S. Raghu,
A. Kapitulnik,
S. A. Kivelson
, Hidden quasi-one-dimensional superconductivity in Sr2RuO4. Phys. Rev. Lett. 105, 136401 (2010).
OpenUrl CrossRef PubMed

[25] S. Raghu,

[26] A. Kapitulnik,

[27] S. A. Kivelson

[28] ↵
T. Scaffidi,
J. C. Romers,
S. H. Simon
, Pairing symmetry and dominant band in Sr2RuO4. Phys. Rev. B 89, 220510(R) (2014).
OpenUrl

[29] T. Scaffidi,

[30] J. C. Romers,

[31] S. H. Simon

[32] ↵
H. S. Røising,
T. Scaffidi,
F. Flicker,
G. F. Lange,
S. H. Simon
, Superconducting order of Sr2RuO4 from a three-dimensional microscopic model. Phys. Rev. Res. 1, 033108 (2019).
OpenUrl

[33] H. S. Røising,

[34] T. Scaffidi,

[35] F. Flicker,

[36] G. F. Lange,

[37] S. H. Simon

[38] ↵
A. P. Mackenzie,
T. Scaffidi,
C. W. Hicks,
Maeno
, Even odder after twenty-three years: The superconducting order parameter puzzle of Sr2RuO4. npj Quant. Mater. 2, 40 (2017).
OpenUrl

[39] A. P. Mackenzie,

[40] T. Scaffidi,

[41] C. W. Hicks,

[42] Maeno

[43] ↵
Y. Maeno,
S. Kittaka,
T. Nomura,
S. Yonezawa,
K. Ishida
, Evaluation of spin-triplet superconductivity in Sr2RuO4. J. Phys. Soc. Jpn. 81, 11009 (2012).
OpenUrl

[44] Y. Maeno,

[45] S. Kittaka,

[46] T. Nomura,

[47] S. Yonezawa,

[48] K. Ishida

[49] ↵
B. Burganov et al.
, Strain control of fermiology and many-body interactions in two-dimensional ruthenates. Phys. Rev. Lett. 116, 197003 (2016).
OpenUrl CrossRef PubMed

[50] B. Burganov et al.

[51] ↵
A. Pustogow et al.
, Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance. Nature 574, 72–75 (2019).
OpenUrl

[52] A. Pustogow et al.

[53] ↵
K. Ishida,
M. Manago,
K. Kinjo,
Y. Maeno
, 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).
OpenUrl

[54] K. Ishida,

[55] M. Manago,

[56] K. Kinjo,

[57] Y. Maeno

[58] ↵
A. Chronister et al.
, Evidence for even parity unconventional superconductivity in Sr2RuO4. arXiv [Preprint] (2020). https://arxiv.org/abs/2007.13730 (Accessed 17 February 2021).

[59] A. Chronister et al.

[60] ↵
S. Benhabib et al.
, Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4. Nat. Phys. 17, 194–198 (2021).
OpenUrl

[61] S. Benhabib et al.

[62] ↵
S. Ghosh et al.
, Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4. Nat. Phys. 17, 99–204 (2021).
OpenUrl

[63] S. Ghosh et al.

[64] ↵
G. M. Luke et al.
, Time-reversal symmetry breaking superconductivity in Sr2RuO4. Nature 394, 558–561 (1998).
OpenUrl CrossRef

[65] G. M. Luke et al.

[66] ↵
G. M. Luke et al.
, Unconventional superconductivity in Sr2RuO4. Physica B 289–290, 373–376 (2000).
OpenUrl

[67] G. M. Luke et al.

[68] ↵
J. Xia,
Y. Maeno,
P. T. Beyersdorf,
M. M. Fejer,
A. Kapitulnik
, High resolution polar kerr effect measurements of Sr2RuO4: Evidence for broken time-reversal symmetry in the superconducting state. Phys. Rev. Lett. 97, 167002 (2006).
OpenUrl CrossRef PubMed

[69] J. Xia,

[70] Y. Maeno,

[71] P. T. Beyersdorf,

[72] M. M. Fejer,

[73] A. Kapitulnik

[74] ↵
M. Sigrist,
K. Ueda
, Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).
OpenUrl CrossRef

[75] M. Sigrist,

[76] K. Ueda

[77] ↵
V. Grinenko et al.
, Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress. Nat. Phys., doi:10.1038/s41567-021-01182-7 (2021).
OpenUrl CrossRef

[78] V. Grinenko et al.

[79] ↵
R. A. Fisher et al.
, Specific heat of UPt₃: Evidence for unconventional superconductivity. Phys. Rev. Lett. 62, 1411–1414 (1989).
OpenUrl CrossRef PubMed

[80] R. A. Fisher et al.

[81] ↵
K. Hasselbach,
L. Taillefer,
J. Flouquet
, Critical point in the superconducting phase diagram of UPt3. Phys. Rev. Lett. 63, 93–96 (1989).
OpenUrl CrossRef PubMed

[82] K. Hasselbach,

[83] L. Taillefer,

[84] J. Flouquet

[85] ↵
Y. S. Li,
R. Borth,
C. W. Hicks,
A. P. Mackenzie,
M. Nicklas
, Heat-capacity measurements under large uniaxial pressure using a piezo-driven device. Rev. Sci. Inst. 91, 103903 (2020).
OpenUrl

[86] Y. S. Li,

[87] R. Borth,

[88] C. W. Hicks,

[89] A. P. Mackenzie,

[90] M. Nicklas

[91] ↵
C. W. Hicks et al.
, Strong increase of Tc of Sr2RuO4 under both tensile and compressive strain. Science 344, 283–285 (2014).
OpenUrl Abstract/FREE Full Text

[92] C. W. Hicks et al.

[93] ↵
A. Steppke et al.
, Strong peak in Tc of Sr2RuO4 under uniaxial pressure. Science 355, eaaf9398 (2017).
OpenUrl Abstract/FREE Full Text

[94] A. Steppke et al.

[95] ↵
Y. T. Hsu et al.
, Manipulating superconductivity in ruthenates through Fermi surface engineering. Phys. Rev. B 94, 045118 (2016).
OpenUrl

[96] Y. T. Hsu et al.

[97] ↵
M. E. Barber et al.
, Role of correlations in determining the van Hove strain in Sr2RuO4. Phys. Rev. B 100, 245139 (2019).
OpenUrl

[98] M. E. Barber et al.

[99] ↵
V. Sunko et al.
, Direct observation of a uniaxial stress-driven Lifshitz transition in Sr2RuO4. npj Quant. Mat. 4, 2397–4648 (2019).
OpenUrl

[100] V. Sunko et al.

[101] ↵
I. Žutić,
I. Mazin
, Phase-sensitive tests of the pairing state symmetry in Sr2RuO4. Phys. Rev. Lett. 95, 217004 (2005).
OpenUrl CrossRef PubMed

[102] I. Žutić,

[103] I. Mazin

[104] ↵
H. G. Suh et al.
, Stabilizing even-parity chiral superconductivity in Sr2RuO4. Phys. Rev. Res. 2, 032023(R) (2020).
OpenUrl

[105] H. G. Suh et al.

[106] ↵
S. A. Kivelson,
A. C. Yuan,
B. J. Ramshaw,
R. Thomale
, A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4. npj Quant. Mater. 5, 43 (2020).
OpenUrl

[107] S. A. Kivelson,

[108] A. C. Yuan,

[109] B. J. Ramshaw,

[110] R. Thomale

[111] ↵
A. T. Rømer,
D. D. Scherer,
I. M. Eremin,
P. J. Hirschfeld,
B. M. Andersen
, Knight shift and leading superconducting instability from spin fluctuations in Sr2RuO4. Phys. Rev. Lett. 123, 247001 (2019).
OpenUrl

[112] A. T. Rømer,

[113] D. D. Scherer,

[114] I. M. Eremin,

[115] P. J. Hirschfeld,

[116] B. M. Andersen

[117] ↵
T. Scaffidi
, Degeneracy between even- and odd-parity superconductivity in the quasi-1D Hubbard model and implications for Sr2RuO4. arXiv [Preprint] (2020). https://arxiv.org/abs/2007.13769 (Accessed 17 February 2021).

[118] T. Scaffidi

[119] ↵
K. Matano,
M. Kriener,
K. Segawa,
Y. Ando,
G. Zheng
, Spin-rotation symmetry breaking in the superconducting state of CuxBi2Se3. Nat. Phys. 12, 852–854 (2016).
OpenUrl

[120] K. Matano,

[121] M. Kriener,

[122] K. Segawa,

[123] Y. Ando,

[124] G. Zheng

[125] ↵
S. Yonezawa et al.
, Thermodynamic evidence for nematic superconductivity in CuxBi2Se3. Nat. Phys. 13, 123–126 (2017).
OpenUrl

[126] S. Yonezawa et al.

[127] ↵
C. Cho et al.
, .Z3-vestigial nematic order due to superconducting fluctuations in the doped topological insulators NbxBi2Se3 and CuxBi2Se3. Nat. Commun. 11, 3056 (2020).
OpenUrl

[128] C. Cho et al.

[129] ↵
R. Willa,
M. Hecker,
R. M. Fernandes,
J. Schmalian
, Inhomogeneous time-reversal symmetry breaking in Sr2RuO4. arXiv [Preprint] (2020). https://arxiv.org/abs/2011.01941 (Accessed 17 February 2021).

[130] R. Willa,

[131] M. Hecker,

[132] R. M. Fernandes,

[133] J. Schmalian

[134] ↵
J. S. Bobowski et al.
, Improved single-crystal growth of Sr2RuO4. Condens. Matter 4, 6 (2019).
OpenUrl

[135] J. S. Bobowski et al.

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

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