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Signatures of bosonic Landau levels in a finite-momentum superconductor

Abstract

Charged particles subjected to magnetic fields form Landau levels (LLs). Originally studied in the context of electrons in metals1, fermionic LLs continue to attract interest as hosts of exotic electronic phenomena2,3. Bosonic LLs are also expected to realize novel quantum phenomena4,5, but, apart from recent advances in synthetic systems6,7, they remain relatively unexplored. Cooper pairs in superconductors—composite bosons formed by electrons—represent a potential condensed-matter platform for bosonic LLs. Under certain conditions, an applied magnetic field is expected to stabilize an unusual superconductor with finite-momentum Cooper pairs8,9 and exert control over bosonic LLs10,11,12,13. Here we report thermodynamic signatures, observed by torque magnetometry, of bosonic LL transitions in the layered superconductor Ba6Nb11S28. By applying an in-plane magnetic field, we observe an abrupt, partial suppression of diamagnetism below the upper critical magnetic field, which is suggestive of an emergent phase within the superconducting state. With increasing out-of-plane magnetic field, we observe a series of sharp modulations in the upper critical magnetic field that are indicative of distinct vortex states and with a structure that agrees with predictions for Cooper pair LL transitions in a finite-momentum superconductor10,11,12,13,14. By applying Onsager’s quantization rule15, we extract the momentum. Furthermore, study of the fermionic LLs shows evidence for a non-zero Berry phase. This suggests opportunities to study bosonic LLs, topological superconductivity, and their interplay via transport16, scattering17, scanning probe18 and exfoliation techniques19.

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Fig. 1: Bosonic LLs, finite-q superconductivity and torque magnetization of Ba6Nb11S28.
Fig. 2: Evidence of an emergent superconducting state.
Fig. 3: Signatures of bosonic LL transitions.
Fig. 4: Berry phase of fermionic LLs.

Data availability

The data presented in this article are available from the Harvard Dataverse at https://doi.org/10.7910/DVN/PLWWKA.

Code availability

The codes used for the density functional theory and analytical calculations in this study are available from the corresponding author upon reasonable request.

References

  1. 1.

    Shoenberg, D. The de Haas–Van Alphen Effect. Phil. Trans. R. Soc. A 245, 1–57 (1952).

    ADS  Google Scholar 

  2. 2.

    Bartolomei, H. et al. Fractional statistics in anyon collisions. Science 368, 173–177 (2020).

    ADS  MathSciNet  CAS  PubMed  Article  PubMed Central  Google Scholar 

  3. 3.

    Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J. Direct observation of anyonic braiding statistics. Nat. Phys. 16, 931–936 (2020).

    CAS  Article  Google Scholar 

  4. 4.

    Cooper, N. R., Wilkin, N. K. & Gunn, J. M. F. Quantum phases of vortices in rotating Bose–Einstein condensates. Phys. Rev. Lett. 87, 120405 (2001).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  5. 5.

    Senthil, T. & Levin, M. Integer quantum Hall effect for bosons. Phys. Rev. Lett. 110, 046801 (2013).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  6. 6.

    Schine, N., Ryou, A., Gromov, A., Sommer, A. & Simon, J. Synthetic Landau levels for photons. Nature 534, 671–675 (2016).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  7. 7.

    Fletcher, R. J. et al. Geometric squeezing into the lowest Landau level. Science 372, 1318–1322 (2021).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  8. 8.

    Fulde, P. & Ferrell, R. A. Superconductivity in a strong spin-exchange field. Phys. Rev. 135, A550–A563 (1964).

    ADS  Article  Google Scholar 

  9. 9.

    Larkin, A. I. & Ovchinnikov, Y. I. Inhomogeneous state of superconductors. Sov. Phys. JETP 20, 762–769 (1965).

    MathSciNet  Google Scholar 

  10. 10.

    Bulaevskii, L. N. Inhomogeneous state and the anisotropy of the upper critical field in layered superconductors with Josephson layer interaction. Sov. Phys. JETP 38, 634–639 (1974).

    ADS  Google Scholar 

  11. 11.

    Shimahara, H. & Rainer, D. Crossover from vortex states to the Fulde–Ferrell–Larkin–Ovchinnikov state in two-dimensional s- and d-wave superconductors. J. Phys. Soc. Jpn 66, 3591–3599 (1997).

    ADS  CAS  Article  Google Scholar 

  12. 12.

    Klein, U. Two-dimensional superconductor in a tilted magnetic field: states with finite Cooper-pair momentum. Phys. Rev. B 69, 134518 (2004).

    ADS  Article  CAS  Google Scholar 

  13. 13.

    Yang, K. & MacDonald, A. H. Vortex-lattice structure of Fulde–Ferrell–Larkin–Ovchinnikov superconductors. Phys. Rev. B 70, 094512 (2004).

    ADS  Article  CAS  Google Scholar 

  14. 14.

    Bulaevskii, L. N. Magnetic properties of layered superconductors with weak interaction between the layers. Sov. Phys. JETP 37, 1133–1136 (1973).

    ADS  Google Scholar 

  15. 15.

    Onsager, L. Interpretation of the de Haas–van Alphen effect. Phil. Mag. 43, 1006–1008 (1952).

    Article  Google Scholar 

  16. 16.

    Kasahara, Y. et al. Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  17. 17.

    Huxley, A. et al. Realignment of the flux-line lattice by a change in the symmetry of superconductivity in UPt3. Nature 406, 160–164 (2000).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  18. 18.

    Wang, D. et al. Evidence for Majorana bound states in an iron-based superconductor. Science 362, 333–335 (2018).

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  19. 19.

    Yang, K. & Agterberg, D. F. Josephson effect in Fulde–Ferrell–Larkin–Ovchinnikov superconductors. Phys. Rev. Lett. 84, 4970–4973 (2000).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  20. 20.

    Wen, X.-G. Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995).

    ADS  Article  Google Scholar 

  21. 21.

    Lilly, M. P., Cooper, K. B., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Evidence for an anisotropic state of two-dimensional electrons in high Landau levels. Phys. Rev. Lett. 82, 394–397 (1999).

    ADS  CAS  Article  Google Scholar 

  22. 22.

    Amet, F. et al. Supercurrent in the quantum Hall regime. Science 352, 966–969 (2016).

    ADS  MathSciNet  CAS  PubMed  MATH  Article  PubMed Central  Google Scholar 

  23. 23.

    Klemm, R. A. Layered Superconductors (Oxford Univ. Press, 2012).

  24. 24.

    Burkhardt, H. & Rainer, D. Fulde–Ferrell–Larkin–Ovchinnikov state in layered superconductors. Ann. Phys. 506, 181–194 (1994).

    Article  Google Scholar 

  25. 25.

    Dao, V. H., Denisov, D., Buzdin, A. & Brison, J.-P. Role of crystal anisotropy on the vortex state in the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase. Phys. Rev. B 87, 174509 (2013).

    ADS  Article  CAS  Google Scholar 

  26. 26.

    Matsuda, Y. & Shimahara, H. Fulde–Ferrell–Larkin–Ovchinnikov state in heavy fermion superconductors. J. Phys. Soc. Jpn 76, 051005 (2007).

    ADS  Article  CAS  Google Scholar 

  27. 27.

    Wosnitza, J. FFLO states in layered organic superconductors. Ann. Phys. 530, 1700282 (2018).

    Article  CAS  Google Scholar 

  28. 28.

    Devarakonda, A. et al. Clean 2D superconductivity in a bulk van der Waals superlattice. Science 370, 231–236 (2020).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  29. 29.

    Tinkham, M. Introduction to Superconductivity (Dover Publications, 2004).

  30. 30.

    Gruenberg, L. W. & Gunther, L. Fulde–Ferrell effect in type-II superconductors. Phys. Rev. Lett. 16, 996–998 (1966).

    ADS  CAS  Article  Google Scholar 

  31. 31.

    Klemm, R. A., Luther, A. & Beasley, M. R. Theory of the upper critical field in layered superconductors. Phys. Rev. B 12, 877–891 (1975).

    ADS  Article  Google Scholar 

  32. 32.

    Lu, J. M. et al. Evidence for two-dimensional Ising superconductivity in gated MoS2. Science 350, 1353–1357 (2015).

    ADS  MathSciNet  CAS  PubMed  MATH  Article  PubMed Central  Google Scholar 

  33. 33.

    Uji, S. et al. Quantum vortex melting and phase diagram in the layered organic superconductor κ-(BEDT-TTF)2Cu(NCS)2. Phys. Rev. B 97, 024505 (2018).

    ADS  Article  Google Scholar 

  34. 34.

    Modler, R. “Anomalous peak effect”—Is it indicative of a generalized Fulde–Ferrell–Larkin–Ovchinnikov state? Czech. J. Phys. 46, 3123–3130 (1996).

    CAS  Article  Google Scholar 

  35. 35.

    Kawamata, S., Itoh, N., Okuda, K., Mochiku, T. & Kadowaki, K. Observation of anisotropic pinning effect in Bi2Sr2CaCu2O8+δ single crystals. Physica C 195, 103–108 (1992).

    ADS  CAS  Article  Google Scholar 

  36. 36.

    Abrikosov, A. A. Nobel Lecture: Type-II superconductors and the vortex lattice. Rev. Mod. Phys. 76, 975–979 (2004).

    ADS  CAS  Article  Google Scholar 

  37. 37.

    Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 2009).

  38. 38.

    Smidman, M., Salamon, M. B., Yuan, H. Q. & Agterberg, D. F. Superconductivity and spin–orbit coupling in non-centrosymmetric materials: a review. Rep. Prog. Phys. 80, 036501 (2017).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  39. 39.

    Mikitik, G. P. & Sharlai, Y. V. Manifestation of Berry’s phase in metal physics. Phys. Rev. Lett. 82, 2147–2150 (1999).

    ADS  CAS  Article  Google Scholar 

  40. 40.

    Falicov, L. M. & Sievert, P. R. Magnetoresistance and magnetic breakdown. Phys. Rev. Lett. 12, 558–561 (1964).

    ADS  Article  Google Scholar 

  41. 41.

    Schneider, J. M., Piot, B. A., Sheikin, I. & Maude, D. K. Using the de Haas–van Alphen effect to map out the closed three-dimensional Fermi surface of natural graphite. Phys. Rev. Lett. 108, 117401 (2012).

    ADS  CAS  PubMed  Article  Google Scholar 

  42. 42.

    Manchon, A., Koo, H. C., Nitta, J., Frolov, S. M. & Duine, R. A. New perspectives for Rashba spin–orbit coupling. Nat. Mater. 14, 871–882 (2015).

    ADS  CAS  PubMed  Article  Google Scholar 

  43. 43.

    Zhang, P. et al. Observation of topological superconductivity on the surface of an iron-based superconductor. Science 360, 182–186 (2018).

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  44. 44.

    Potter, A. C. & Lee, P. A. Engineering a p + ip superconductor: comparison of topological insulator and Rashba spin–orbit-coupled materials. Phys. Rev. B 83, 184520 (2011).

    ADS  Article  CAS  Google Scholar 

  45. 45.

    Hu, L. H., Liu, C. X. & Zhang, F. C. Topological Larkin–Ovchinnikov phase and Majorana zero mode chain in bilayer superconducting topological insulator films. Commun. Phys. 2, 25 (2019).

  46. 46.

    Wang, Y. et al. Field-enhanced diamagnetism in the pseudogap state of the cuprate Bi2Sr2CaCu2O8+δ superconductor in an intense magnetic field. Phys. Rev. Lett. 95, 247002 (2005).

    ADS  PubMed  Article  CAS  Google Scholar 

  47. 47.

    Bergemann, C. et al. Superconducting magnetization above the irreversibility line in Tl2Ba2CuO6+δ. Phys. Rev. B 57, 14387–14396 (1998).

    ADS  CAS  Article  Google Scholar 

  48. 48.

    Li, L. et al. Diamagnetism and Cooper pairing above Tc in cuprates. Phys. Rev. B 81, 054510 (2010).

    ADS  Article  CAS  Google Scholar 

  49. 49.

    Sugiura, S. et al. Fulde–Ferrell–Larkin–Ovchinnikov and vortex phases in a layered organic superconductor. npj Quantum Mater. 4, 7 (2019).

    ADS  Article  Google Scholar 

  50. 50.

    Campbell, A. M., Evetts, J. E. & Dew-hughes, D. The behaviour of type II superconductors. Phil. Mag. 10, 333–338 (1964).

    ADS  Article  Google Scholar 

  51. 51.

    Finnemore, D. K., Stromberg, T. F. & Swenson, C. A. Superconducting properties of high-purity niobium. Phys. Rev. 149, 231–243 (1966).

    ADS  CAS  Article  Google Scholar 

  52. 52.

    Altshuler, E. & Johansen, T. H. Colloquium: Experiments in vortex avalanches. Rev. Mod. Phys. 76, 471–487 (2004).

    ADS  CAS  Article  Google Scholar 

  53. 53.

    Swanson, A. G. et al. Flux jumps, critical fields, and de Haas–van Alphen effect in κ-(BEDT-TTF)2Cu(NCS)2. Solid State Commun. 73, 353–356 (1990).

    ADS  CAS  Article  Google Scholar 

  54. 54.

    Tenya, K. et al. Anomalous pinning behavior in Sr2RuO4. Physica B 403, 1101–1103 (2008).

    ADS  CAS  Article  Google Scholar 

  55. 55.

    Li, L., Checkelsky, J. G., Komiya, S., Ando, Y. & Ong, N. P. Low-temperature vortex liquid in La2-xSrxCuO4. Nat. Phys. 3, 311–314 (2007).

    CAS  Article  Google Scholar 

  56. 56.

    Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 1995).

  57. 57.

    Takahashi, M., Mizushima, T. & Machida, K. Multiband effects on Fulde–Ferrell–Larkin–Ovchinnikov states of Pauli-limited superconductors. Phys. Rev. B 89, 064505 (2014).

    ADS  Article  CAS  Google Scholar 

  58. 58.

    Campbell, A. M. & Evetts, J. E. Flux vortices and transport currents in type II superconductors. Adv. Phys. 21, 199–428 (1972).

    ADS  CAS  Article  Google Scholar 

  59. 59.

    Roy, S. B. & Chaddah, P. Anomalous superconducting properties in CeRu2: effects of magnetic and nonmagnetic substitutions. Phys. Rev. B 55, 11100–11102 (1997).

    ADS  CAS  Article  Google Scholar 

  60. 60.

    Lortz, R. et al. Calorimetric evidence for a Fulde–Ferrell–Larkin–Ovchinnikov superconducting state in the layered organic superconductor κ-(BEDT-TTF)2Cu(NCS)2. Phys. Rev. Lett. 99, 187002 (2007).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  61. 61.

    Farrell, D. E., Rice, J. P. & Ginsberg, D. M. Experimental evidence for flux-lattice melting. Phys. Rev. Lett. 67, 1165–1168 (1991).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  62. 62.

    Beck, R. G., Farrell, D. E., Rice, J. P., Ginsberg, D. M. & Kogan, V. G. Melting of the Abrikosov flux lattice in anisotropic superconductors. Phys. Rev. Lett. 68, 1594–1596 (1992).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  63. 63.

    Ooi, S., Shibauchi, T., Okuda, N. & Tamegai, T. Novel angular scaling of vortex phase transitions in Bi2Sr2CaCu2O8+y. Phys. Rev. Lett. 82, 4308–4311 (1999).

    ADS  CAS  Article  Google Scholar 

  64. 64.

    Uji, S. et al. Orbital effect on FFLO phase and energy dissipation due to vortex dynamics in magnetic-field-induced superconductor λ-(BETS)2FeCl4. J. Phys. Soc. Jpn 82, 034715 (2013).

    ADS  Article  CAS  Google Scholar 

  65. 65.

    Martínez, J. C. et al. Magnetic anisotropy of a Bi2Sr2CaCu2Ox single crystal. Phys. Rev. Lett. 69, 2276–2279 (1992).

    ADS  PubMed  Article  PubMed Central  Google Scholar 

  66. 66.

    Feinberg, D. & Villard, C. Intrinsic pinning and lock-in transition of flux lines in layered type-II superconductors. Phys. Rev. Lett. 65, 919–922 (1990).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  67. 67.

    Buzdin, A. I. & Kachkachi, H. Generalized Ginzburg–Landau theory for nonuniform FFLO superconductors. Phys. Lett. A 225, 341–348 (1997).

    ADS  CAS  Article  Google Scholar 

  68. 68.

    Houzet, M. & Buzdin, A. Influence of the paramagnetic effect on the vortex lattice in 2D superconductors. Europhys. Lett. 50, 375–381 (2000).

    ADS  CAS  Article  Google Scholar 

  69. 69.

    Denisov, D., Buzdin, A. & Shimahara, H. Types of Fulde–Ferrell–Larkin–Ovchinnikov states induced by anisotropy effects. Phys. Rev. B 79, 064506 (2009).

    ADS  Article  CAS  Google Scholar 

  70. 70.

    Shoenberg, D. Magnetization of a two-dimensional electron gas. J. Low Temp. Phys. 56, 417–440 (1984).

    ADS  Article  Google Scholar 

  71. 71.

    Yoshida, T., Sigrist, M. & Yanase, Y. Complex-stripe phases induced by staggered Rashba spin–orbit coupling. J. Phys. Soc. Jpn 82, 074714 (2013).

    ADS  Article  CAS  Google Scholar 

  72. 72.

    Zhou, T. & Ting, C. S. Phase diagram and local tunneling spectroscopy of the Fulde–Ferrell–Larkin–Ovchinnikov states of a two-dimensional square-lattice d-wave superconductor. Phys. Rev. B 80, 224515 (2009).

    ADS  Article  CAS  Google Scholar 

  73. 73.

    Yuan, N. F. Q. & Fu, L. Topological metals and finite-momentum superconductors. Proc. Natl Acad. Sci. USA 118, e2019063118 (2021).

    MathSciNet  CAS  PubMed  PubMed Central  Article  Google Scholar 

  74. 74.

    Agterberg, D. F. & Kaur, R. P. Magnetic-field-induced helical and stripe phases in Rashba superconductors. Phys. Rev. B 75, 064511 (2007).

    ADS  Article  CAS  Google Scholar 

  75. 75.

    Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

    CAS  Article  Google Scholar 

  76. 76.

    Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    ADS  CAS  Article  Google Scholar 

  77. 77.

    Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

    ADS  Article  Google Scholar 

  78. 78.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    ADS  CAS  Article  Google Scholar 

  79. 79.

    Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188–5192 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  80. 80.

    Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).

    ADS  CAS  Article  Google Scholar 

  81. 81.

    Mostofi, A. A. et al. wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).

    ADS  CAS  MATH  Article  Google Scholar 

  82. 82.

    Mostofi, A. A. et al. An updated version of wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 185, 2309–2310 (2014).

    ADS  CAS  MATH  Article  Google Scholar 

  83. 83.

    Liu, G.-B., Shan, W.-Y., Yao, Y., Yao, W. & Xiao, D. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides. Phys. Rev. B 88, 085433 (2013).

    ADS  Article  CAS  Google Scholar 

  84. 84.

    Fang, S. et al. Ab initio tight-binding Hamiltonian for transition metal dichalcogenides. Phys. Rev. B 92, 205108 (2015).

    ADS  Article  CAS  Google Scholar 

  85. 85.

    Xi, X. et al. Ising pairing in superconducting NbSe2 atomic layers. Nat. Phys. 12, 139–143 (2016).

    CAS  Article  Google Scholar 

  86. 86.

    Saito, Y. et al. Superconductivity protected by spin–valley locking in ion-gated MoS2. Nat. Phys. 12, 144–149 (2016).

    CAS  Article  Google Scholar 

  87. 87.

    Xiao, D., Chang, M. C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  CAS  MATH  Article  Google Scholar 

  88. 88.

    Fukui, T., Hatsugai, Y. & Suzuki, H. Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances. J. Phys. Soc. Jpn 74, 1674–1677 (2005).

    ADS  CAS  Article  Google Scholar 

  89. 89.

    Aroyo, M. I. et al. Crystallography online: Bilbao Crystallographic Server. Bulg. Chem. Commun. 43, 183–197 (2011).

    CAS  Google Scholar 

  90. 90.

    Aroyo, M. I. et al. Bilbao Crystallographic Server: I. Databases and crystallographic computing programs. Z. Kristallogr. Cryst. Mater. 221, 15–27 (2006).

    CAS  Article  Google Scholar 

  91. 91.

    Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. Acta Crystallogr. A 62, 115–128 (2006).

    ADS  PubMed  MATH  Article  CAS  PubMed Central  Google Scholar 

  92. 92.

    Culcer, D., MacDonald, A. & Niu, Q. Anomalous Hall effect in paramagnetic two-dimensional systems. Phys. Rev. B 68, 045327 (2003).

    ADS  Article  CAS  Google Scholar 

  93. 93.

    Wright, A. R. & McKenzie, R. H. Quantum oscillations and Berry’s phase in topological insulator surface states with broken particle–hole symmetry. Phys. Rev. B 87, 085411 (2013).

    ADS  Article  CAS  Google Scholar 

  94. 94.

    Alexandradinata, A., Wang, C., Duan, W. & Glazman, L. Revealing the topology of Fermi-surface wave functions from magnetic quantum oscillations. Phys. Rev. X 8, 011027 (2018).

    CAS  Google Scholar 

  95. 95.

    Kaganov, M. I. & Slutskin, A. A. Coherent magnetic breakdown. Phys. Rep. 98, 189–271 (1983).

    ADS  CAS  Article  Google Scholar 

  96. 96.

    Harrison, N. et al. Magnetic breakdown and quantum interference in the quasi-two-dimensional superconductor κ-(BEDT-TTF)2Cu(NCS)2 in high magnetic fields. J. Phys. Condens. Matter 8, 5415–5435 (1996).

    ADS  CAS  Article  Google Scholar 

  97. 97.

    Gvozdikov, V. M. & Taut, M. Magnetic quantum oscillations of electrons on a two-dimensional lattice: Numerical simulations and the magnetic breakdown approach. Phys. Rev. B 75, 155436 (2007).

    ADS  Article  CAS  Google Scholar 

  98. 98.

    Fuchs, J. N., Piéchon, F., Goerbig, M. O. & Montambaux, G. Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models. Eur. Phys. J. B 77, 351–362 (2010).

    ADS  CAS  Article  Google Scholar 

  99. 99.

    Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  100. 100.

    Analytis, J. G. et al. Two-dimensional surface state in the quantum limit of a topological insulator. Nat. Phys. 6, 960–964 (2010).

    CAS  Article  Google Scholar 

  101. 101.

    Xiong, J. et al. High-field Shubnikov–de Haas oscillations in the topological insulator Bi2Te2Se. Phys. Rev. B 86, 045314 (2012).

    ADS  Article  CAS  Google Scholar 

  102. 102.

    Das, B. et al. Evidence for spin splitting in InxGa1−xAs/In0.52Al0.48As hetetorstructures as B → 0. Phys. Rev. B 39, 1411–1414 (1989).

    ADS  CAS  Article  Google Scholar 

  103. 103.

    Nakamura, H., Koga, T. & Kimura, T. Experimental evidence of cubic Rashba effect in an inversion-symmetric oxide. Phys. Rev. Lett. 108, 206601 (2012).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  104. 104.

    Winkler, R. Spin–Orbit Coupling Effects in Two-dimensional Electron and Hole Systems (Springer, 2003).

  105. 105.

    Forsythe, C. et al. Band structure engineering of 2D materials using patterned dielectric superlattices. Nat. Nanotechnol. 13, 566–571 (2018).

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  106. 106.

    Alicea, J. Majorana fermions in a tunable semiconductor device. Phys. Rev. B 81, 125318 (2010).

    ADS  Article  CAS  Google Scholar 

  107. 107.

    Nakosai, S., Tanaka, Y. & Nagaosa, N. Topological superconductivity in bilayer Rashba system. Phys. Rev. Lett. 108, 147003 (2012).

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

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Acknowledgements

We are grateful to S. K. Yip, M. Shayegan and M. T. Randeria for discussions. This research is funded in part by the Gordon and Betty Moore Foundation through grant GBMF9070 to J.G.C. (instrumentation development), the Office of Naval Research (ONR) under award N00014-21-1-2591 (advanced characterization), the US Department of Energy (DOE) Office of Science, Basic Energy Sciences, under award DE-SC0019300 (material synthesis) and award DE-SC0022028 (structure analysis), the STC Center for Integrated Quantum Materials, NSF grant DMR-1231319 (E.K.) and the DOE Office of Basic Energy Sciences under award DE-SC0018945 (L.F.). S.F. acknowledges support from the Rutgers Center for Materials Theory Distinguished Postdoctoral Fellowship. Computations were performed on the Cannon cluster supported by the FAS Division of Science Research Computing Group (FASRC) at Harvard University. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement DMR-1157490, the State of Florida, and DOE.

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A.D. and J.G.C. conceived the project. A.D. grew the single crystals, characterized the materials, and performed the measurements with T.S., J.Z., D.G. and M.K. supporting. A.D. performed the analytical calculations and S.F. and A.D. the electronic structure calculations with L.F. and E.K. supporting. A.D. and J.G.C. wrote the manuscript with contributions and discussions from all authors. J.G.C. supervised the project.

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Correspondence to J. G. Checkelsky.

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Extended data figures and tables

Extended Data Fig. 1 Torque magnetometry.

a Non-collinear response of Μ to Η results in τ which can be measured using torque magnetometry techniques. b Capacitive torque magnetometer (sample S1). c Resistive torque magnetometer (sample S2). Scale bars are 1 mm. d δC(Η) at θ = 89.7º and Τ = 0.3 K for sample S1. (inset) The experimental geometry for S1. δC < 0 is consistent with diamagnetic response of \(M\). e Comparison of ρxx(T) and Mτ(T) measured with μ0H = 0.4 mT for sample S1. f Comparison of ρxx(H) and Mτ(H) at θ = 89.7º and T = 0.3 K for sample S1. g \(M(H)\) loop for θ = 0º at T = 0.4 K measured by SQUID. h Mτ(H) loop for θ = 46º at T = 0.3 K measured by torque magnetometry (S1). (inset) τ(H) from which we obtain Mτ(H) in the main panel. i δ quantum oscillation frequency F(θH) (orange points) with fit to equation (2) (dashed line). (inset) θ where Mτ(H) oscillates around zero corresponds to θ = 90º.

Extended Data Fig. 2 Survey of H-T-θ phase space (sample S1).

a 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) (see Fig. 2a) at θ = 89.7º and various fixed temperatures, vertically offset for clarity. The peak-like structure marking H is traced by a dashed guide to the eye. b Up- (solid) and down-sweep (dashed) Mτ(H) measured at T = 0.31 K at various angles between 80º and 90º, vertically offset for clarity. c Up- (solid) and down-sweep (dashed) Mτ(H) measured at θ = 89.2º at various fixed temperatures, vertically offset for clarity. d 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) at θ = 89.2º (shown in c) for various fixed temperatures, vertically offset for clarity. The peak-like structure marking H is absent here. e T-H phase diagram at θ = 89.2º showing Hc2 and Hc1.

Extended Data Fig. 3 H-T phase diagram (sample S2).

a Up- (solid) and down-sweep (dashed) Mτ(H) measured at θ = 88.9º for various temperatures, vertically offset for clarity. b 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) at θ = 88.9º (shown in a) for fixed temperatures down to T = 33 mK, vertically offset for clarity. The peak-like structure at H is traced by a dashed guide to the eye. c T-H phase diagram at θ = 88.9º showing Hc2, H, and Hc1. Black circle marks Tc.

Extended Data Fig. 4 Mτ(H) at fixed T and θ (sample S1).

a Up- (solid) and down-sweeps (dashed) of Mτ(H) for T = 395 mK at equally spaced, fixed θ between 89.30º and 89.95º, vertically offset for clarity. b 𝜕Μτ(H)/𝜕H of up-sweep in (a), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ). c Up- (solid) and down-sweeps (dashed) of Mτ(H) for T = 460 mK at equally spaced, fixed θ between 89.75º and 89.95º, vertically offset for clarity. d 𝜕Μτ(H)/𝜕H of up-sweep in (c), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ).

Extended Data Fig. 5 Low temperature Mτ(H) at fixed θ.

a Up- and b down-sweeps of Mτ(H) for T = 20 mK at equally spaced, fixed θ between 88.45º and 89.95º, vertically offset for clarity. Guide to the eye (grey dashed line) in (b) traces angular evolution of Hʹ(θ) c 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) in (a), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ) (inset) Up- and down-sweep Mτ(H) (solid and dashed, respectively) at fixed angles near 90º for a third sample, S3.

Extended Data Fig. 6 Ginzburg-Landau modeling.

a GL free energy δF(qx, qy) in the finite-q pairing state of an isotropic superconductor. Finite-q pairing with |q| = q0 minimizes δF (red contour). b (black line) Radial cut of δF(q) for H = 0. (orange points) For H ≠ 0, discrete LLs are formed. The LL with (2n+1)½/𝓁 closest to q0 is the optimal solution. c Superconductivity opens a gap ΔSC around the Fermi energy EF (orange). d Magnetic field causes Zeeman splitting of a spin-degenerate band (dashed) into spin-up (blue) and spin-down (red) bands. e Zeeman split bands create two Fermi surface contours in k-space separated by q.

Extended Data Fig. 7 Electronic structure modeling.

a The Fermi surface contours for monolayer H-NbS2. The supercell Brillouin zones and corresponding M points are shown as white lines and open blue circles, respectively. b Electronic structure resulting from 3 × 3 zone-folding. This is formed by overlapping the nine cells marked by white lines in (a). c Fermi surface contours from first-principles calculations capturing monolayer H-NbS2 and the spacer layer 3 × 3 perturbation. This resembles (b) but with additional band gaps and renormalization due to the spacer layer perturbation. d Electronic structure around K, Kʹ from first-principles with the δ pocket corresponding to fδ identified. e Electronic structure around M from first-principles showing the inner α1 and outer α2 pockets. f Bloch sphere showing spin-texture along the δ pocket. (inset) Expanded view near the north pole of the Bloch sphere. Blue arrow shows sense of spin evolution for clockwise motion along the δ pocket, see white arrow in (d). g Bloch sphere showing spin-texture along the α1 (blue) and α2 (red) pockets. Blue (red) arrow shows sense of spin evolution for clockwise motion along the α12) pocket, see white arrows in (e). h Coordinate system used for M point kp model. i Bloch sphere showing the segments \({{\mathscr{C}}}_{a}\) (red) and \({{\mathscr{C}}}_{b}\) (blue) traced by the spin-1/2 eigenvector \({\rm{|}}\varPsi {\rm{\rangle }}\) which together form a closed  contour \({\mathscr{C}}\). j Berry curvature ΩB(kx,ky). For ΔM ≠ 0, ΩB integrated over the area Ak defined by the FS contour yields φB ≠ π.

Extended Data Fig. 8 de Haas-van Alphen oscillations.

a \({M}_{\tau }^{{osc}}\)(H) at T = 0.3 K for various angles. The oscillations are aligned across 75º when plotted versus H indicating the two-dimensionality of the FSs. b \({M}_{\tau }^{{osc}}(H)\) at θ = 18º for various temperatures. c \({\widetilde{A}}_{{FFT}}(T)\) for the α and δ oscillations (points) with fits to \({R}_{T}^{i}\) (Methods) to extract the effective mass. d \({M}_{\tau }^{{osc}}(1/H)\) measured at T = 20 mK for θ = 8.2º (blue). The result of applying a sliding window centered at various equally spaced 1/H0 are shown as gray and orange traces, vertically offset for clarity. e FFT of un-windowed (blue) and windowed (gray) \({M}_{\tau }^{\mathrm{osc}}(1/H)\) in (d), vertically offset for clarity. The peak at \({f}_{{{\rm{\alpha }}}^{\star }}\) (grey dashed guide) appears for 1/ μ0H0 < 0.21 T−1, orange trace in (d) and (e).

Extended Data Fig. 9 Phase analysis.

a Mτ(H) measured at T = 0.3 K and θ = 80º showing both the Meissner effect (left) and dHvA quantum oscillation (right). b \({M}_{\tau }^{\mathrm{osc}}\) (1/H) at T = 0.3 K (T = 20 mK) for sample(s) S1 (S4-S5) at angles near θ = 16º. c-f Phase-shift function K(f, φB)around the δ (left) and α (right) oscillations for samples S1, S3, S4, and S5 using \({M}_{\tau }^{\mathrm{osc}}\) shown in (b). The red contours are drawn at 98% of the local maxima. K(f, φB) for S2 is shown in Fig. 4b.

Extended Data Fig. 10 Calculated Landau level spectrum.

a \(M\)-point pockets resulting from the kp model with the experimentally determined SOC parameters. The inner (outer) contour in red (blue) corresponds to the measured \({f}_{{{\rm{\alpha }}}_{1}}\) (\({f}_{{{\rm{\alpha }}}_{2}}\)) frequency. The intersecting dashed green circles are the contours in the presence of mirror symmetry. b Calculated density of states in the presence of H showing oscillations due to Landau quantization of the electronic structure in (a). (inset) FFT power spectrum shows three modes corresponding to \({f}_{{{\rm{\alpha }}}_{1}}\) (red), \({f}_{{\alpha }_{2}}\) (blue), and the breakdown frequency \({f}_{{{\rm{\alpha }}}^{\star }}\) (green). c The breakdown contribution at \({f}_{{{\rm{\alpha }}}^{\star }}\) in the FFT power spectrum vanishes at μ0Hmb ≈ 6 T. d FFT power spectrum at various values of the chemical potential EF showing a Landau fan. By comparing with the observed pocket sizes, we can estimate EF.

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Devarakonda, A., Suzuki, T., Fang, S. et al. Signatures of bosonic Landau levels in a finite-momentum superconductor. Nature 599, 51–56 (2021). https://doi.org/10.1038/s41586-021-03915-3

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