- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Intrinsic finite-energy Cooper pairing in j=3/2 superconductors

Masoud Bahari1,2,*, Song-Bo Zhang1,2, and Björn Trauzettel1,2

  • 1Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
  • 2Würzburg-Dresden Cluster of Excellence ct.qmat, Germany

  • *masoud.bahari@physik.uni-wuerzburg.de
PDF

Phys. Rev. Research 4, L012017 – Published 11 February, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L012017

Abstract

We show that Cooper pairing can occur intrinsically away from the Fermi surface in j=3/2 superconductors with strong spin-orbit coupling and equally curved bands in the normal state. In contrast to conventional pairing between spin-1/2 electrons, we derive that pairing can happen between interband electrons having different magnetic quantum numbers, for instance, mj=1/2 and mj=3/2. Such superconducting correlations manifest themselves by a pair of indirect gaplike structures at finite excitation energies. An observable signature of this exotic pairing is the emergence of a pair of symmetric superconducting coherence peaks in the density of states at finite energies. Moreover, the angular-momentum-resolved density of states in the presence of a perturbative Zeeman field reflects the mj composition of the Cooper pairs. We argue that such finite-energy pairing is a generic feature of j=3/2 superconductors, both in the presence and absence of inversion symmetry.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (75)

  1. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
  2. J. P. Carbotte, Properties of boson-exchange superconductors, Rev. Mod. Phys. 62, 1027 (1990).
  3. M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991).
  4. T.-L. Ho and S. Yip, Pairing of Fermions with Arbitrary Spin, Phys. Rev. Lett. 82, 247 (1999).
  5. W. Yang, Y. Li, and C. Wu, Topological Septet Pairing with Spin-32 Fermions: High-Partial-Wave Channel Counterpart of the He3−B Phase, Phys. Rev. Lett. 117, 075301 (2016).
  6. P. M. R. Brydon, L. Wang, M. Weinert, and D. F. Agterberg, Pairing of j=3/2 Fermions in Half-Heusler Superconductors, Phys. Rev. Lett. 116, 177001 (2016).
  7. V. Kozii, J. W. F. Venderbos, and L. Fu, Three-dimensional Majorana fermions in chiral superconductors, Sci. Adv. 2, 12 (2016).
  8. L. Savary, J. Ruhman, J. W. F. Venderbos, L. Fu, and P. A. Lee, Superconductivity in three-dimensional spin-orbit coupled semimetals, Phys. Rev. B 96, 214514 (2017).
  9. J. W. F. Venderbos, L. Savary, J. Ruhman, P. A. Lee, and L. Fu, Pairing States of Spin-32 Fermions: Symmetry-Enforced Topological Gap Functions, Phys. Rev. X 8, 011029 (2018).
  10. P. Dutta, F. Parhizgar, and A. M. Black-Schaffer, Superconductivity in spin-3/2 systems: Symmetry classification, odd-frequency pairs, and Bogoliubov Fermi surfaces, Phys. Rev. Research 3, 033255 (2021).
  11. C. Fang, B. A. Bernevig, and M. J. Gilbert, Tri-Dirac surface modes in topological superconductors, Phys. Rev. B 91, 165421 (2015).
  12. I. Boettcher and I. F. Herbut, Superconducting quantum criticality in three-dimensional Luttinger semimetals, Phys. Rev. B 93, 205138 (2016).
  13. D. F. Agterberg, P. M. R. Brydon, and C. Timm, Bogoliubov Fermi Surfaces in Superconductors with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 118, 127001 (2017).
  14. C. Timm, A. P. Schnyder, D. F. Agterberg, and P. M. R. Brydon, Inflated nodes and surface states in superconducting half-Heusler compounds, Phys. Rev. B 96, 094526 (2017).
  15. J. Yu and C.-X. Liu, Singlet-quintet mixing in spin-orbit coupled superconductors with j=32 fermions, Phys. Rev. B 98, 104514 (2018).
  16. T. Kawakami, T. Okamura, S. Kobayashi, and M. Sato, Topological Crystalline Materials of J=3/2 Electrons: Antiperovskites, Dirac Points, and High Winding Topological Superconductivity, Phys. Rev. X 8, 041026 (2018).
  17. S. Kobayashi, A. Yamakage, Y. Tanaka, and M. Sato, Majorana Multipole Response of Topological Superconductors, Phys. Rev. Lett. 123, 097002 (2019).
  18. B. Roy, Sayed Ali Akbar Ghorashi, M. S. Foster, and A. H. Nevidomskyy, Topological superconductivity of spin-3/2 carriers in a three-dimensional doped Luttinger semimetal, Phys. Rev. B 99, 054505 (2019).
  19. H. Menke, C. Timm, and P. M. R. Brydon, Bogoliubov Fermi surfaces stabilized by spin-orbit coupling, Phys. Rev. B 100, 224505 (2019).
  20. S.-T. Tamura, S. Iimura, and S. Hoshino, Electronic multipoles and multiplet pairs induced by Pomeranchuk and Cooper instabilities of Bogoliubov Fermi surfaces, Phys. Rev. B 102, 024505 (2020).
  21. C. Timm, P. M. R. Brydon, and D. F. Agterberg, Distortional weak-coupling instability of Bogoliubov Fermi surfaces, Phys. Rev. B 103, 024521 (2021).
  22. A. L. Szabó, R. Moessner, and B. Roy, Interacting spin-32 fermions in a Luttinger semimetal: Competing phases and their selection in the global phase diagram, Phys. Rev. B 103, 165139 (2021).
  23. G. Goll, M. Marz, A. Hamann, T. Tomanic, K. Grube, T. Yoshino, and T. Takabatake, Thermodynamic and transport properties of the non-centrosymmetric superconductor LaBiPt, Phys. B: Condens. Matter 403, 1065 (2008).
  24. H. Lin, L. A. Wray, Y. Xia, S. Xu, S. Jia, R. J. Cava, A. Bansil, and M. Z. Hasan, Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena, Nat. Mater. 9, 546 (2010).
  25. S. Chadov, X. Qi, J. Kübler, G. H. Fecher, C. Felser, and S. C. Zhang, Tunable multifunctional topological insulators in ternary Heusler compounds, Nat. Mater. 9, 541 (2010).
  26. N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, and J. Paglione, Superconductivity in the topological semimetal YPtBi, Phys. Rev. B 84, 220504(R) (2011).
  27. T. V. Bay, T. Naka, Y. K. Huang, and A. de Visser, Superconductivity in noncentrosymmetric YPtBi under pressure, Phys. Rev. B 86, 064515 (2012).
  28. F. F. Tafti, T. Fujii, A. Juneau-Fecteau, S. R. de Cotret, N. Doiron-Leyraud, A. Asamitsu, and L. Taillefer, Superconductivity in the noncentrosymmetric half-Heusler compound LuPtBi: A candidate for topological superconductivity, Phys. Rev. B 87, 184504 (2013).
  29. Y. Pan, A. M. Nikitin, T. V. Bay, Y. K. Huang, C. Paulsen, B. H. Yan, and A. de Visser, Superconductivity and magnetic order in the noncentrosymmetric half-Heusler compound ErPdBi, Europhys. Lett. 104, 27001 (2013).
  30. G. Xu, W. Wang, X. Zhang, Y. Du, E. Liu, S. Wang, G. Wu, Z. Liu, and X. X. Zhang, Weak antilocalization effect and noncentrosymmetric superconductivity in a topologically nontrivial semimetal LuPdBi, Sci. Rep. 4, 5709 (2014).
  31. Y. Nakajima, R. Hu, K. Kirshenbaum, A. Hughes, P. Syers, X. Wang, K. Wang, R. Wang, S. R. Saha, D. Pratt, J. W. Lynn, and J. Paglione, Topological RPdBi half-Heusler semimetals: A new family of noncentrosymmetric magnetic superconductors, Sci. Adv. 1, e1500242 (2015).
  32. A. M. Nikitin, Y. Pan, X. Mao, R. Jehee, G. K. Araizi, Y. K. Huang, C. Paulsen, S. C. Wu, B. H. Yan, and A. de Visser, Magnetic and superconducting phase diagram of the half-Heusler topological semimetal HoPdBi, J. Phys.: Condens. Matter 27, 275701 (2015).
  33. Z. K. Liu, L. X. Yang, S.-C. Wu, C. Shekhar, J. Jiang, H. F. Yang, Y. Zhang, S.-K. Mo, Z. Hussain, B. Yan, C. Felser, and Y. L. Chen, Observation of unusual topological surface states in half-Heusler compounds LnPtBi (Ln=Lu, Y), Nat. Commun. 7, 12924 (2016).
  34. M. Meinert, Unconventional Superconductivity in YPtBi and Related Topological Semimetals, Phys. Rev. Lett. 116, 137001 (2016).
  35. D. Shrivastava and S. P. Sanyal, Electronic structure and superconducting behaviour of LuPtBi half-Heusler compound: A first principle study, Physica C 544, 22 (2018).
  36. H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak, P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M. R. Brydon, D. F. Agterberg, M. A. Tanatar, R. Prozorov, and J. Paglione, Beyond triplet: Unconventional superconductivity in a spin-3/2 topological semimetal, Sci. Adv. 4, 4 (2018).
  37. H. Xiao, T. Hu, W. Liu, Y. L. Zhu, P. G. Li, G. Mu, J. Su, K. Li, and Z. Q. Mao, Superconductivity in the half-Heusler compound TbPdBi, Phys. Rev. B 97, 224511 (2018).
  38. S. M. A. Radmanesh, C. Martin, Y. Zhu, X. Yin, H. Xiao, Z. Q. Mao, and L. Spinu, Evidence for unconventional superconductivity in half-Heusler YPdBi and TbPdBi compounds revealed by London penetration depth measurements, Phys. Rev. B 98, 241111(R) (2018).
  39. R. Majumder and M. M. Hossain, First-principles study of structural, electronic, elastic, thermodynamic and optical properties of topological superconductor LuPtBi, Comput. Condens. Matter 21, e00402 (2019).
  40. M. M. Hosen, G. Dhakal, K. Dimitri, H. Choi, F. Kabir, C. Sims, O. Pavlosiuk, P. Wiśniewski, T. Durakiewicz, J.-X. Zhu, D. Kaczorowski, and M. Neupane, Observation of Dirac state in half-Heusler material YPtBi, Sci. Rep. 10, 12343 (2020).
  41. V. Bhardwaj, A. Bhattacharya, S. Srivastava, V. V. Khovaylo, J. Sannigrahi, N. Banerjee, B. K. Mani, and R. Chatterjee, Strain driven emergence of topological non-triviality in YPdBi thin films, Sci. Rep. 11, 7535 (2021).
  42. K. Ishihara, T. Takenaka, Y. Miao, Y. Mizukami, K. Hashimoto, M. Yamashita, M. Konczykowski, R. Masuki, M. Hirayama, T. Nomoto, R. Arita, O. Pavlosiuk, P. Wisniewski, D. Kaczorowski, and T. Shibauchi, Tuning the Parity Mixing of Singlet-Septet Pairing in a Half-Heusler Superconductor, Phys. Rev. X 11, 041048 (2021).
  43. W. Al-Sawai, H. Lin, R. S. Markiewicz, L. A. Wray, Y. Xia, S.-Y. Xu, M. Z. Hasan, and A. Bansil, Topological electronic structure in half-Heusler topological insulators, Phys. Rev. B 82, 125208 (2010).
  44. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, Commentary: The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  45. C. Shi, X. Xi, Z. Hou, X. Zhang, G. Xu, E. Liu, W. Wang, W. Wang, J. Chen, and G. Wu, NMR investigation of atomic and electronic structures of half-Heusler topologically nontrivial semimetals, Phys. Status Solidi B 252, 357 (2015).
  46. For R∈{Y,Dy,Tb,Sm} in RPdBi, the downward curving mj split Γ8 bands lie close to the Fermi surface. This can be clearly seen for YPdBi in Refs. [24, 25, 41, 43, 44, 45] and for R∈{Dy,Tb,Sm} in Ref. [44].
  47. A. Moreo, M. Daghofer, A. Nicholson, and E. Dagotto, Interband pairing in multiorbital systems, Phys. Rev. B 80, 104507 (2009).
  48. G. Tang, C. Bruder, and W. Belzig, Magnetic Field-Induced “Mirage” Gap in an Ising Superconductor, Phys. Rev. Lett. 126, 237001 (2021).
  49. J. M. Luttinger and W. Kohn, Motion of electrons and holes in perturbed periodic fields, Phys. Rev. 97, 869 (1955).
  50. G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures, Phys. Rev. 100, 580 (1955).
  51. M. Tinkham, Group Theory and Quantum Mechanics, Dover Books on Chemistry and Earth Sciences (Dover, New York, 2003).
  52. M. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer, Berlin, 2007).
  53. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L012017 for the details on the folding down approach, power series expansion, band basis formalism, calculation of angular-momentum-resolved density of states, detection of the mj composition of Cooper pairs, spectrum of j=3/2 pairing at FEE, and constructing pairing Hamiltonians in cubic point group symmetry, which includes Refs. [6, 8, 9, 15, 20, 42, 51, 52, 54, 64].
  54. P. Löwdin, A note on the quantum-mechanical perturbation theory, J. Chem. Phys. 19, 1396 (1951).
  55. The analogous model for the (−+) subspace can be derived by substituting (+)↔(−) in Eq. (5).
  56. Due to our choice of the pairing state, the interband gaplike structure δ˚(k) is proportional to the identity matrix.
  57. The band crossings of the FEE appear for the C2, C2′, and C3 axes at E=±4|μβ/(4α+5β)|, E=±2|μ/(4α+5β)|β2+3γ2, and E=±4|γμ/(4α+5β)|, respectively. These crossings constitute the prerequisite for the GLSs.
  58. In the case of low-energy pairing of intraband electrons, the usual even (odd)-parity pseudospin singlet (triplet) as well as the d-vector representation of the odd-parity states [3, 75] can be deduced from Δ̂effνν′(−k)=−[Δ̂effνν′(k)]T with ν=ν′∈{+,−}.
  59. We expand the electron-electron interaction through the orthogonal basis set of cubic point group symmetry. In this regard, the expanded two-body interaction includes many terms, each distinguished by the basis label of cubic symmetry and Cooper pair quantum numbers. We investigate each term individually.
  60. For cubic anisotropy in the normal-state γ≠β, our results are still valid.
  61. The pairing channel nomenclature is based on the J quantum number of Cooper pairs due to the presence of strong spin-orbit coupling.
  62. There are three (eight) equivalent directions for the proper twofold (threefold) rotation C2(3) in cubic point group symmetry. The equivalent directions for the C2 rotation are [1,0,0], [0,1,0], and [0,0,1]. For the C3 rotation, the equivalent directions are [1,1,1], [1¯,1,1], [1,1¯,1], [1,1,1¯], [1¯,1¯,1], [1,1¯,1¯], [1¯,1,1¯], and [1¯,1¯,1¯], where 1¯≡−1.
  63. g (u) stands for even (odd) parity depending on the symmetry of the Cooper pair quantum numbers.
  64. H. Mäkelä and K.-A. Suominen, Inert States of Spin-S Systems, Phys. Rev. Lett. 99, 190408 (2007).
  65. W. Yang, T. Xiang, and C. Wu, Majorana surface modes of nodal topological pairings in spin-32 semimetals, Phys. Rev. B 96, 144514 (2017).
  66. In Fig. 2, the twofold degeneracy along the [0,0,1]* direction is not lifted in the presence of the inversion breaking ASOC term. This degeneracy is protected by mirror reflection symmetry and can be witnessed in time-reversal symmetric Td crystals [65].
  67. E. Bauer and M. Sigrist, Non-Centrosymmetric Superconductors: Introduction and Overview, Lecture Notes in Physics (Springer, Berlin, 2012), Vol. 847.
  68. P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Superconductivity without Inversion Symmetry: MnSi versus CePt3Si, Phys. Rev. Lett. 92, 097001 (2004).
  69. T. Dvir, F. Massee, L. Attias, M. Khodas, M. Aprili, C. H. L. Quay, and H. Steinberg, Spectroscopy of bulk and few-layer superconducting NbSe2 with van der Waals tunnel junctions, Nat. Commun. 9, 598 (2018).
  70. Y. Okada, Y. Ando, R. Shimizu, E. Minamitani, S. Shiraki, S. Watanabe, and T. Hitosugi, Scanning tunnelling spectroscopy of superconductivity on surfaces of LiTi2O4(111) thin films, Nat. Commun. 8, 15975 (2017).
  71. R. Kumar, A. Vasdev, S. Das, S. Howlader, K. S. Jat, P. Neha, S. Patnaik, and G. Sheet, The pressure-enhanced superconducting phase of Srx−Bi2Se3 probed by hard point contact spectroscopy, Sci. Rep. 11, 4090 (2021).
  72. D. Costanzo, H. Zhang, B. A. Reddy, H. Berger, and A. F. Morpurgo, Tunnelling spectroscopy of gate-induced superconductivity in MoS2, Nat. Nanotechnol. 13, 483 (2018).
  73. M. Hashimoto, I. M. Vishik, R.-H. He, T. P. Devereaux, and Z.-X. Shen, Energy gaps in high-transition-temperature cuprate superconductors, Nat. Phys. 10, 483 (2014).
  74. S. Kanasugi and Y. Yanase, Anapole superconductivity from PT-symmetric mixed-parity interband pairing, arXiv:2107.07096.
  75. R. Balian and N. R. Werthamer, Superconductivity with pairs in a relative p wave, Phys. Rev. 131, 1553 (1963).

Outline

Information

Phys. Rev. Research 4, L012017– Published 11 February, 2022
  • Received 3 August 2021
  • Accepted 19 January 2022
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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation