- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Open Access

Evidence for vertical line nodes in Sr2RuO4 from nonlocal electrodynamics

J. F. Landaeta1,2,*, K. Semeniuk2, J. Aretz2, K. R. Shirer2, D. A. Sokolov2, N. Kikugawa3, Y. Maeno4, I. Bonalde5, J. Schmalian6,7 et al.

A. P. Mackenzie2,8 and E. Hassinger1,2,†

  • *Contact author: javier.landaeta@cpfs.mpg.de
  • †Contact author: elena.hassinger@tu-dresden.de
PDF

Phys. Rev. B 110, L100503 – Published 17 September, 2024

DOI: https://doi.org/10.1103/PhysRevB.110.L100503

Abstract

By determining the superconducting lower and upper critical fields Hc1(T) and Hc2(T), respectively, in a high-purity spherical Sr2RuO4 sample via ac-susceptibility measurements, we obtain the temperature dependence of the coherence length ξ and the penetration depth λ down to 0.04Tc. Given the high sample quality, the observed T2 dependence of λ at low temperatures cannot be explained in terms of impurity effects. Instead, we argue that the weak type-II superconductor Sr2RuO4 has to be treated in the nonlocal limit. By comparing our data with existing theory in that limit, the penetration depth in Sr2RuO4 agrees with a gap structure having vertical line nodes, while horizontal line nodes cannot account for the observation. The work highlights the potential benefits of purifying other unconventional superconductors in order to access the fascinating nonlocal regime in more materials and to determine their Cooper pair wave functions.

View figure in article

Physics Subject Headings (PhySH)

See Also

Superconducting penetration depth through a Van Hove singularity: Sr2RuO4 under uniaxial stress

Eli Mueller, Yusuke Iguchi, Fabian Jerzembeck, Jorge O. Rodriguez, Marisa Romanelli, Edgar Abarca-Morales, Anastasios Markou, Naoki Kikugawa, Dmitry A. Sokolov, Gwansuk Oh, Clifford W. Hicks, Andrew P. Mackenzie, Yoshiteru Maeno, Vidya Madhavan, and Kathryn A. Moler
Phys. Rev. B 110, L100502 (2024)

Nonlocal electrodynamics and the penetration depth of superconducting Sr2RuO4

Henrik S. Røising, Andreas Kreisel, and Brian M. Andersen
Phys. Rev. B 110, 094511 (2024)

Article Text

Supplemental Material

References (60)

  1. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Superconductivity in a layered perovskite without copper, Nature (London) 372, 532 (1994).
  2. A. P. Mackenzie, S. R. Julian, A. J. Diver, G. J. McMullan, M. P. Ray, G. G. Lonzarich, Y. Maeno, S. Nishizaki, and T. Fujita, Quantum oscillations in the layered perovskite superconductor Sr2RuO4, Phys. Rev. Lett. 76, 3786 (1996).
  3. A. P. Mackenzie and Y. Maeno, The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing, Rev. Mod. Phys. 75, 657 (2003).
  4. A. P. Mackenzie, A personal perspective on the unconventional superconductivity of Sr2RuO4, J. Supercond. Novel Magn. 33, 177 (2020).
  5. A. P. Mackenzie, T. Scaffidi, C. W. Hicks, and Y. Maeno, Even odder after twenty-three years: The superconducting order parameter puzzle of Sr2RuO4, npj Quantum Mater. 2, 40 (2017).
  6. Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, Evaluation of spin-triplet superconductivity in Sr2RuO4, J. Phys. Soc. Jpn. 81, 011009 (2012).
  7. C. Kallin, Chiral p-wave order in Sr2RuO4, Rep. Prog. Phys. 75, 042501 (2012).
  8. Y. Liu and Z.-Q. Mao, Unconventional superconductivity in Sr2RuO4, Phys. C: Supercond. Appl. 514, 339 (2015).
  9. A. J. Leggett and Y. Liu, Symmetry properties of superconducting order parameter in Sr2RuO4: A brief review, J. Supercond. Nov. Magn. 34, 1647 (2021).
  10. A. Pustogow, Y. Luo, A. Chronister, Y. S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-17 nuclear magnetic resonance, Nature (London) 574, 72 (2019).
  11. K. Ishida, M. Manago, K. Kinjo, and Y. Maeno, Reduction of the O17 Knight shift in the superconducting state and the heat-up effect by NMR pulses on Sr2RuO4, J. Phys. Soc. Jpn. 89, 034712 (2020).
  12. A. Chronister, A. Pustogow, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, C. W. Hicks, A. P. Mackenzie, E. D. Bauer, and S. E. Brown, Evidence for even parity unconventional superconductivity in Sr2RuO4, Proc. Natl. Acad. Sci. USA 118, e2025313118 (2021).
  13. G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Time-reversal symmetry-breaking superconductivity in Sr2RuO4, Nature (London) 394, 558 (1998).
  14. V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brückner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H.-H. Klauss, Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress, Nat. Phys. 17, 748 (2021).
  15. Y.-S. Li, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, A. S. Gibbs, Y. Maeno, C. W. Hicks, J. Schmalian, M. Nicklas, and A. P. Mackenzie, High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure, Proc. Natl. Acad. Sci. USA 118, e2020492118 (2021).
  16. R. Willa, M. Hecker, R. M. Fernandes, and J. Schmalian, Inhomogeneous time-reversal symmetry breaking in Sr2RuO4, Phys. Rev. B 104, 024511 (2021).
  17. C. Lupien, W. A. MacFarlane, C. Proust, L. Taillefer, Z. Q. Mao, and Y. Maeno, Ultrasound attenuation in Sr2RuO4: An angle-resolved study of the superconducting gap function, Phys. Rev. Lett. 86, 5986 (2001).
  18. I. Bonalde, B. D. Yanoff, M. B. Salamon, D. J. Van Harlingen, E. M. E. Chia, Z. Q. Mao, and Y. Maeno, Temperature dependence of the penetration depth in Sr2RuO4: Evidence for nodes in the gap function, Phys. Rev. Lett. 85, 4775 (2000).
  19. K. Deguchi, Z. Q. Mao, and Y. Maeno, Determination of the superconducting gap structure in all bands of the spin-triplet superconductor Sr2RuO4, J. Phys. Soc. Jpn. 73, 1313 (2004).
  20. S. Kittaka, S. Nakamura, T. Sakakibara, N. Kikugawa, T. Terashima, S. Uji, D. A. Sokolov, A. P. Mackenzie, K. Irie, Y. Tsutsumi, K. Suzuki, and K. Machida, Searching for gap zeros in Sr2RuO4 via field-angle-dependent specific-heat measurement, J. Phys. Soc. Jpn. 87, 093703 (2018).
  21. K. Izawa, H. Takahashi, H. Yamaguchi, Y. Matsuda, M. Suzuki, T. Sasaki, T. Fukase, Y. Yoshida, R. Settai, and Y. Onuki, Superconducting gap structure of spin-triplet superconductor Sr2RuO4 studied by thermal conductivity, Phys. Rev. Lett. 86, 2653 (2001).
  22. M. A. Tanatar, S. Nagai, Z. Q. Mao, Y. Maeno, and T. Ishiguro, Thermal conductivity of superconducting Sr2RuO4 in oriented magnetic fields, Phys. Rev. B 63, 064505 (2001).
  23. M. Suzuki, M. A. Tanatar, Z. Q. Mao, Y. Maeno, and T. Ishiguro, Quasi-particle density in Sr2RuO4 probed by means of the phonon thermal conductivity, J. Phys.: Condens. Matter 14, 7371 (2002).
  24. E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. René de Cotret, G. Grissonnanche, M. S. Anwar, Y. Maeno, N. Doiron-Leyraud, and L. Taillefer, Vertical line nodes in the superconducting gap structure of Sr2RuO4, Phys. Rev. X 7, 011032 (2017).
  25. K. Iida, M. Kofu, K. Suzuki, N. Murai, S. Ohira-Kawamura, R. Kajimoto, Y. Inamura, M. Ishikado, S. Hasegawa, T. Masuda, Y. Yoshida, K. Kakurai, K. Machida, and S. Lee, Horizontal line nodes in Sr2RuO4 proved by spin resonance, J. Phys. Soc. Jpn. 89, 053702 (2020).
  26. K. Machida, Spin singlet pairing in Sr2RuO4 with horizontal nodes: Present status and future prospect, JPS Conf. Proc. 30, 011038 (2020).
  27. A. Ramires and M. Sigrist, Superconducting order parameter of Sr2RuO4: A microscopic perspective, Phys. Rev. B 100, 104501 (2019).
  28. H. G. Suh, H. Menke, P. M. R. Brydon, C. Timm, A. Ramires, and D. F. Agterberg, Stabilizing even-parity chiral superconductivity in Sr2RuO4, Phys. Rev. Res. 2, 032023(R) (2020).
  29. G. Palle, C. Hicks, R. Valentí, Z. Hu, Y.-S. Li, A. Rost, M. Nicklas, A. P. Mackenzie, and J. Schmalian, Constraints on the superconducting state of Sr2RuO4 from elastocaloric measurements, Phys. Rev. B 108, 094516 (2023).
  30. H. S. Røising, T. Scaffidi, F. Flicker, G. F. Lange, and S. H. Simon, Superconducting order of Sr2RuO4 from a three-dimensional microscopic model, Phys. Rev. Res. 1, 033108 (2019).
  31. A. T. Rømer, A. Kreisel, M. A. Müller, P. J. Hirschfeld, I. M. Eremin, and B. M. Andersen, Theory of strain-induced magnetic order and splitting of Tc and TTRSB in Sr2RuO4, Phys. Rev. B 102, 054506 (2020).
  32. S. A. Kivelson, A. C. Yuan, B. Ramshaw, and R. Thomale, A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4, npj Quantum Mater. 5, 43 (2020).
  33. I. Kosztin and A. J. Leggett, Nonlocal effects on the magnetic penetration depth in d-wave superconductors, Phys. Rev. Lett. 79, 135 (1997).
  34. E. E. M. Chia, D. J. Van Harlingen, M. B. Salamon, B. D. Yanoff, I. Bonalde, and J. L. Sarrao, Nonlocality and strong coupling in the heavy fermion superconductor CeCoIn5: A penetration depth study, Phys. Rev. B 67, 014527 (2003).
  35. A. Carrington, I. J. Bonalde, R. Prozorov, R. W. Giannetta, A. M. Kini, J. Schlueter, H. H. Wang, U. Geiser, and J. M. Williams, Low-temperature penetration depth of κ−(ET)2Cu[N(CN)2]Br and κ−(ET)2Cu(NCS)2, Phys. Rev. Lett. 83, 4172 (1999).
  36. P. J. Hirschfeld and N. Goldenfeld, Effect of strong scattering on the low-temperature penetration depth of a d-wave superconductor, Phys. Rev. B 48, 4219 (1993).
  37. H. Kusunose and M. Sigrist, The penetration depth in Sr2RuO4: Evidence for orbital-dependent superconductivity, Europhys. Lett. 60, 281 (2002).
  38. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.110.L100503 for detailed information about the experimental methods, sample preparation, determination of critical field values, lower critical field and demagnetization factors, thermodynamic critical field Hc, coherence length and penetration depth from the critical fields, and formalism of nonlocal electrodynamics, which also includes Refs. [56, 57, 58, 59, 60].
  39. A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, Extremely strong dependence of superconductivity on disorder in Sr2RuO4, Phys. Rev. Lett. 80, 161 (1998).
  40. K. Yoshida, Y. Maeno, S. Nishizaki, and T. Fujita, Anisotropic flux pinning in a layered superconductor Sr2RuO4, J. Phys. Soc. Jpn. 65, 2220 (1996).
  41. F. Jerzembeck, H. S. Røising, A. Steppke, H. Rosner, D. A. Sokolov, N. Kikugawa, T. Scaffidi, S. H. Simon, A. P. Mackenzie, and C. W. Hicks, The superconductivity of Sr2RuO4 under c-axis uniaxial stress, Nat. Commun. 13, 4596 (2022).
  42. M. Tinkham, Introduction to Superconductivity, 2nd ed. (Dover Publications, Mineola, New York, 2004).
  43. S. Tsuchiya, M. Matsuno, R. Ishiguro, H. Kashiwaya, S. Kashiwaya, S. Nomura, H. Takayanagi, and Y. Maeno, Magnetization of a mesoscopic superconducting Sr2RuO4 plate on micro-dc-SQUIDs, J. Phys. Soc. Jpn. 83, 094715 (2014).
  44. E. H. Brandt, Properties of the ideal Ginzburg-Landau vortex lattice, Phys. Rev. B 68, 054506 (2003).
  45. C.-R. Hu, Numerical constants for isolated vortices in superconductors, Phys. Rev. B 6, 1756 (1972).
  46. R. Khasanov, A. Ramires, V. Grinenko, I. Shipulin, N. Kikugawa, D. A. Sokolov, J. A. Krieger, T. J. Hicken, Y. Maeno, H. Luetkens, and Z. Guguchia, In-plane magnetic penetration depth in Sr2RuO4: Muon-spin rotation and relaxation study, Phys. Rev. Lett. 131, 236001 (2023).
  47. S. NishiZaki, Y. Maeno, and Z. Mao, Changes in the superconducting state of Sr2RuO4 under magnetic fields probed by specific heat, J. Phys. Soc. Jpn. 69, 572 (2000).
  48. P. J. Baker, R. J. Ormeno, C. E. Gough, Z. Q. Mao, S. Nishizaki, and Y. Maeno, Microwave surface impedance measurements of Sr2RuO4: The effect of impurities, Phys. Rev. B 80, 115126 (2009).
  49. R. J. Ormeno, M. A. Hein, T. L. Barraclough, A. Sibley, C. E. Gough, Z. Q. Mao, S. Nishizaki, and Y. Maeno, Electrodynamic response of Sr2RuO4, Phys. Rev. B 74, 092504 (2006).
  50. M. Suzuki, M. A. Tanatar, N. Kikugawa, Z. Q. Mao, Y. Maeno, and T. Ishiguro, Universal heat transport in Sr2RuO4, Phys. Rev. Lett. 88, 227004 (2002).
  51. D. Einzel, P. J. Hirschfeld, F. Gross, B. S. Chandrasekhar, K. Andres, H. R. Ott, J. Beuers, Z. Fisk, and J. L. Smith, Magnetic field penetration depth in the heavy-electron superconductor UBe13, Phys. Rev. Lett. 56, 2513 (1986).
  52. K. Ishida, H. Murakawa, H. Mukuda, Y. Kitaoka, Z. Mao, and Y. Maeno, NMR and NQR studies on superconducting Sr2RuO4, J. Phys. Chem. Solids 69, 3108 (2008).
  53. A. B. Pippard and W. L. Bragg, An experimental and theoretical study of the relation between magnetic field and current in a superconductor, Proc. R. Soc. A 216, 547 (1953).
  54. E. Mueller, Y. Iguchi, F. Jerzembeck, J. O. Rodriguez, M. Romanelli, E. Abarca-Morales, A. Markou, N. Kikugawa, D. A. Sokolov, G. Oh, C. W. Hicks, A. P. Mackenzie, Y. Maeno, V. Madhavan, and K. A. Moler, companion paper, Superconducting penetration depth through a Van Hove singularity: Sr2RuO4 under uniaxial stress, Phys. Rev. B 110, L100502 (2024).
  55. H. S. Røising, A. Kreisel, and B. M. Andersen, companion paper, Nonlocal electrodynamics and the penetration depth of superconducting Sr2RuO4, Phys. Rev. B 110, 094511 (2024).
  56. J. F. Landaeta, A. M. León, S. Zwickel, T. Lühmann, M. Brando, C. Geibel, E.-O. Eljaouhari, H. Rosner, G. Zwicknagl, E. Hassinger, and S. Khim, Conventional type-II superconductivity in locally noncentrosymmetric LaRh2As2 single crystals, Phys. Rev. B 106, 014506 (2022).
  57. R. Liang, D. A. Bonn, W. N. Hardy, and D. Broun, Lower critical field and superfluid density of highly underdoped YBa2Cu3O6+x single crystals, Phys. Rev. Lett. 94, 117001 (2005).
  58. R. Prozorov and V. G. Kogan, Effective demagnetizing factors of diamagnetic samples of various shapes, Phys. Rev. Appl. 10, 014030 (2018).
  59. Y. Maeno, Z. Mao, S. Nishizaki, and T. Akima, Superconducting state of Sr2RuO4 under magnetic fields, Phys. B: Condens. Matter 280, 285 (2000).
  60. D. Scalapino, The case for dx2−y2 pairing in the cuprate superconductors, Phys. Rep. 250, 329 (1995).

Outline

Information

Phys. Rev. B 110, L100503– Published 17 September, 2024
  • Received 22 December 2023
  • Revised 10 May 2024
  • Accepted 7 August 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation