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.

  • Article
  • Published:

Orbital Fulde–Ferrell–Larkin–Ovchinnikov state in an Ising superconductor


In superconductors possessing both time and inversion symmetries, the Zeeman effect of an external magnetic field can break the time-reversal symmetry, forming a conventional Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state characterized by Cooper pairings with finite momentum1,2. In superconductors lacking (local) inversion symmetry, the Zeeman effect may still act as the underlying mechanism of FFLO states by interacting with spin–orbit coupling (SOC). Specifically, the interplay between the Zeeman effect and Rashba SOC can lead to the formation of more accessible Rashba FFLO states that cover broader regions in the phase diagram3,4,5. However, when the Zeeman effect is suppressed because of spin locking in the presence of Ising-type SOC, the conventional FFLO scenarios are no longer effective. Instead, an unconventional FFLO state is formed by coupling the orbital effect of magnetic fields with SOC, providing an alternative mechanism in superconductors with broken inversion symmetries6,7,8. Here we report the discovery of such an orbital FFLO state in the multilayer Ising superconductor 2H-NbSe2. Transport measurements show that the translational and rotational symmetries are broken in the orbital FFLO state, providing the hallmark signatures of finite-momentum Cooper pairings. We establish the entire orbital FFLO phase diagram, consisting of a normal metal, a uniform Ising superconducting phase and a six-fold orbital FFLO state. This study highlights an alternative route to achieving finite-momentum superconductivity and provides a universal mechanism to preparing orbital FFLO states in similar materials with broken inversion symmetries.

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

Access options

Buy this article

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

Fig. 1: Superconductivity and possible pairing states in NbSe2 multilayers.
Fig. 2: Upper critical fields and vortex dynamics in the orbital FFLO state.
Fig. 3: Six-fold anisotropy in the orbital FFLO state (measured in the 17-nm flake).
Fig. 4: Phase diagram of the orbital FFLO state under parallel magnetic fields.

Similar content being viewed by others

Data availability

All relevant data shown are provided with this paper. Additional data that support the plots and other analyses in this work are available from the corresponding author upon request.


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

    Article  ADS  Google Scholar 

  2. Larkin, A. I. & Ovchinnikov, Y. N. Nonuniform state of superconductors. Sov. Phys. JETP 47, 1136–1146 (1964).

    Google Scholar 

  3. Barzykin, V. & Gor’kov, L. P. Inhomogeneous stripe phase revisited for surface superconductivity. Phys. Rev. Lett. 89, 227002 (2002).

    Article  ADS  PubMed  Google Scholar 

  4. Zheng, Z. et al. FFLO superfluids in 2D spin-orbit coupled Fermi gases. Sci. Rep. 4, 6535 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Sigrist, M. et al. Superconductors with staggered non-centrosymmetricity. J. Phys. Soc. Jpn 83, 061014 (2014).

    Article  ADS  Google Scholar 

  6. Liu, C.-X. Unconventional superconductivity in bilayer transition metal dichalcogenides. Phys. Rev. Lett. 118, 087001 (2017).

    Article  ADS  PubMed  Google Scholar 

  7. Nakamura, Y. & Yanase, Y. Odd-parity superconductivity in bilayer transition metal dichalcogenides. Phys. Rev. B 96, 054501 (2017).

    Article  ADS  Google Scholar 

  8. 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).

    Article  Google Scholar 

  9. Bauer, E. & Sigrist, M. Non-Centrosymmetric Superconductors: Introduction and Overview Vol. 847 (Springer Science & Business Media, 2012).

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

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

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

    Article  CAS  Google Scholar 

  12. Aslamazov, L. G. Influence of impurities on the existence of an inhomogeneous state in a ferromagnetic superconductor. Sov. Phys. JETP 28, 773–775 (1969).

    ADS  Google Scholar 

  13. Takada, S. Superconductivity in a molecular field. II: stability of Fulde-Ferrel phase. Prog. Theor. Phys. 43, 27–38 (1970).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  15. Shimahara, H. Fulde-Ferrell state in quasi-two-dimensional superconductors. Phys. Rev. B 50, 12760–12765 (1994).

    Article  ADS  CAS  Google Scholar 

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

    Article  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  17. Kumagai, K. et al. Fulde-Ferrell-Larkin-Ovchinnikov state in a perpendicular field of quasi-two-dimensional CeCoIn5. Phys. Rev. Lett. 97, 227002 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Kumagai, K., Shishido, H., Shibauchi, T. & Matsuda, Y. Evolution of paramagnetic quasiparticle excitations emerged in the high-field superconducting phase of CeCoIn5. Phys. Rev. Lett. 106, 137004 (2011).

    Article  ADS  CAS  PubMed  Google Scholar 

  19. 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).

    Article  ADS  CAS  PubMed  Google Scholar 

  20. Beyer, R., Bergk, B., Yasin, S., Schlueter, J. A. & Wosnitza, J. Angle-dependent evolution of the Fulde-Ferrell-Larkin-Ovchinnikov state in an organic superconductor. Phys. Rev. Lett. 109, 027003 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  21. Uji, S. et al. Vortex dynamics and the Fulde-Ferrell-Larkin-Ovchinnikov state in a magnetic-field-induced organic superconductor. Phys. Rev. Lett. 97, 157001 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Devarakonda, A. et al. Signatures of bosonic Landau levels in a finite-momentum superconductor. Nature 599, 51–56 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  23. Kinjo, K. et al. Superconducting spin smecticity evidencing the Fulde-Ferrell-Larkin-Ovchinnikov state in Sr2RuO4. Science 376, 397–400 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Yasuzuka, S. et al. Highly isotropic in-plane upper critical field in the anisotropic s-wave superconductor 2H-NbSe2. J. Supercond. Nov. Magn. 33, 953–958 (2020).

    Article  CAS  Google Scholar 

  25. Fletcher, J. D. et al. Penetration depth study of superconducting gap structure of 2H−NbSe2. Phys. Rev. Lett. 98, 057003 (2007).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Foner, S. & McNiff, E. J. Upper critical fields of layered superconducting NbSe2 at low temperature. Phys. Lett. A 45, 429–430 (1973).

    Article  ADS  CAS  Google Scholar 

  27. de la Barrera, S. C. et al. Tuning Ising superconductivity with layer and spin–orbit coupling in two-dimensional transition-metal dichalcogenides. Nat. Commun. 9, 1427 (2018).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  28. Prober, D. E., Schwall, R. E. & Beasley, M. R. Upper critical fields and reduced dimensionality of the superconducting layered compounds. Phys. Rev. B 21, 2717–2733 (1980).

    Article  ADS  CAS  Google Scholar 

  29. Bulaevskii, L., Buzdin, A. & Maley, M. Intrinsic pinning of vortices as a direct probe of the nonuniform Larkin-Ovchinnikov-Fulde-Ferrell state in layered superconductors. Phys. Rev. Lett. 90, 067003 (2003).

    Article  ADS  CAS  PubMed  Google Scholar 

  30. Croitoru, M. & Buzdin, A. In search of unambiguous evidence of the Fulde–Ferrell–Larkin–Ovchinnikov state in quasi-low dimensional superconductors. Condens. Matter 2, 30 (2017).

    Article  Google Scholar 

  31. Buzdin, A., Matsuda, Y. & Shibauchi, T. FFLO state in thin superconducting films. Europhys. Lett. 80, 67004 (2007).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  33. Bulaevskiǐ, L. 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 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  35. Kwok, W. et al. Direct observation of dissipative flux motion and pinning by twin boundaries in YBa2Cu3O7–δ single crystals. Phys. Rev. Lett. 64, 966–969 (1990).

    Article  ADS  CAS  PubMed  Google Scholar 

  36. Goldobin, E., Koelle, D., Kleiner, R. & Buzdin, A. Josephson junctions with second harmonic in the current-phase relation: properties of φ junctions. Phys. Rev. B 76, 224523 (2007).

  37. Hamill, A. et al. Two-fold symmetric superconductivity in few-layer NbSe2. Nat. Phys. 17, 949–954 (2021).

    Article  CAS  Google Scholar 

  38. Doh, H., Song, M. & Kee, H. Novel route to a finite center-of-mass momentum pairing state for superconductors: a current-driven Fulde-Ferrell-Larkin-Ovchinnikov state. Phys. Rev. Lett. 97, 257001 (2006).

    Article  ADS  PubMed  Google Scholar 

  39. Harper, F. E. & Tinkham, M. The mixed state in superconducting thin films. Phys. Rev. 172, 441–450 (1968).

    Article  ADS  Google Scholar 

  40. Cho, C. et al. Evidence for the Fulde–Ferrell–Larkin–Ovchinnikov state in bulk NbS2. Nat. Commun. 12, 3676 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  41. Rahn, D. J. et al. Gaps and kinks in the electronic structure of the superconductor 2H-NbSe2 from angle-resolved photoemission at 1 K. Phys. Rev. B 85, 224532 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  42. Kiss, T. et al. Charge-order-maximized momentum-dependent superconductivity. Nat. Phys. 3, 720–725 (2007).

    Article  CAS  Google Scholar 

  43. Dvir, T. et al. Spectroscopy of bulk and few-layer superconducting NbSe2 with van der Waals tunnel junctions. Nat. Commun. 9, 598 (2018).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  44. Gurevich, A. Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors. Phys. Rev. B 67, 184515 (2003).

    Article  ADS  Google Scholar 

  45. Talantsev, E. F. et al. On the origin of critical temperature enhancement in atomically thin superconductors. 2D Mater. 4, 025072 (2017).

    Article  Google Scholar 

  46. Takahashi, S. & Tachiki, M. New phase diagram in superconducting superlattices. Phys. Rev. B 34, 3162–3164 (1986).

    Article  ADS  CAS  Google Scholar 

  47. Gor’kov, L. P. & Rashba, E. I. Superconducting 2D system with lifted spin degeneracy: mixed singlet-triplet state. Phys. Rev. Lett. 87, 037004 (2001).

    Article  ADS  PubMed  Google Scholar 

  48. Frigeri, P. A., Agterberg, D. F., Koga, A. & Sigrist, M. Superconductivity without inversion symmetry: MnSi versus CePt3Si. Phys. Rev. Lett. 92, 097001 (2004).

    Article  ADS  CAS  PubMed  Google Scholar 

  49. Ran, S. et al. Nearly ferromagnetic spin-triplet superconductivity. Science 365, 684–687 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  52. Zheliuk, O. et al. Josephson coupled Ising pairing induced in suspended MoS2 bilayers by double-side ionic gating. Nat. Nanotechnol. 14, 1123–1128 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  53. He, W. Y. et al. Magnetic field driven nodal topological superconductivity in monolayer transition metal dichalcogenides. Commun. Phys. 1, 40 (2018).

  54. Shaffer, D., Kang, J., Burnell, F. J. & Fernandes, R. M. Crystalline nodal topological superconductivity and Bogolyubov Fermi surfaces in monolayer NbSe2. Phys. Rev. B 101, 224503 (2020).

    Article  ADS  CAS  Google Scholar 

  55. Cho, C. et al. Nodal and nematic superconducting phases in NbSe2 monolayers from competing superconducting channels. Phys. Rev. Lett. 129, 087002 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  56. Shimahara, H. Structure of the Fulde-Ferrell-Larkin-Ovchinnikov state in two-dimensional superconductors. J. Phys. Soc. Jpn 67, 736–739 (1998).

    Article  ADS  CAS  Google Scholar 

  57. Bowers, J. A. & Rajagopal, K. Crystallography of color superconductivity. Phys. Rev. D 66, 065002 (2002).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  59. Maloney, M. D., de la Cruz, F. & Cardona, M. Superconducting parameters and size effects of aluminum films and foils. Phys. Rev. B 5, 3558–3572 (1972).

    Article  ADS  Google Scholar 

  60. Kozuka, Y. et al. Two-dimensional normal-state quantum oscillations in a superconducting heterostructure. Nature 462, 487–490 (2009).

    Article  ADS  CAS  PubMed  Google Scholar 

  61. Wang, B. Y. et al. Isotropic Pauli-limited superconductivity in the infinite-layer nickelate Nd0.775Sr0.225NiO2. Nat. Phys. 17, 473–477 (2021).

    Article  CAS  Google Scholar 

  62. Takada, S. & Izuyama, T. Superconductivity in a molecular field. I. Prog. Theor. Phys. 41, 635–663 (1969).

    Article  ADS  Google Scholar 

Download references


We thank J. Zoestbergen for technical support. This publication is part of the project TOPCORE (with project number OCENW.GROOT.2019.048) of the research programme Open Competition ENW Groot, which is (partly) financed by the Dutch Research Council (NWO). P.W. acknowledges the research program ‘Materials for the Quantum Age’ (QuMat) for financial support. This program (registration number 024.005.006) is part of the Gravitation program financed by the Dutch Ministry of Education, Culture and Science (OCW). O.Z. acknowledges financial support from the CogniGron research center and the Ubbo Emmius Funds (University of Groningen). N.F.Q.Y. acknowledges the National Natural Science Foundation of China (grant number 12174021) for their financial support. The high-field measurement was supported by HFML-RU/NWO-I, a member of the European Magnetic Field Laboratory (EMFL). It is part of the research programme of the Netherlands Organisation for Scientific Research (NWO) funded by the National Roadmap for Large-Scale Research Facilities.

Author information

Authors and Affiliations



P.W., O.Z. and J.Y. conceived the research. P.W., O.Z., X.P., L.Z. and M.L. fabricated the devices. P.W. carried out the magnetotransport and anisotropy measurements. P.W., O.Z., X.P., S.W. and U.Z. carried out the high-field magnetotransport measurements. N.F.Q.Y. constructed the theoretical model of the orbital FFLO states. P.W. and J.Y. analysed the data. P.W., N.F.Q.Y. and J.Y. interpreted the data and prepared the manuscript with inputs from N.E.H. and T.T.M.P. All authors commented on the manuscript.

Corresponding author

Correspondence to Jianting Ye.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Yong Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Absence of the upturn in upper critical fields in a downgraded device.

a, The temperature dependence of sheet resistances for two flakes showing RRR = 12.6 and 28 for 11 and 17 nm thick flakes, respectively. b, The Bc2,|| measured for the 11 and 17 nm thick flakes. The 2D Ginzburg-Landau fittings, that is, the solid black curves, yield thickness dfit = 8 and 12 nm, respectively. Overall, the reduced thicknesses obtained from the GL fittings are due to the protection from the Ising SOC, which becomes more robust in thinner flakes59,60,61. Close to Tc0, the 11 nm flake shows a steeper temperature dependence of Bc2, consistent with its reduced thickness. Nevertheless, for the 17 nm flake with a larger RRR, an upturn in the Bc2 can be observed at B = 0.36BP, indicating the orbital FFLO state, which eventually enhances Bc2 to exceed that measured in the 11 nm flake. As a larger RRR indicates better sample quality, the contrasting behaviour in the temperature dependence of Bc2 suggests that the absence of the orbital FFLO phase in the thin flake might be caused by the downgraded quality, which suppresses the finite-momentum pairing via scattering13,62.

Extended Data Fig. 2 Illustration of 2-axis rotation of a 2D sample in an external magnetic field with an installation canting angle γ.

A 2D sample is mounted on a 2-axis rotational stage. The 2D surface of the sample (yellow plane) makes a canting angle γ with respect to one of the rotation planes of the stage (grey plane). To simplify the discussion and isolate the effect of canting angle γ, we assume that the stage can make precise rotations so that, as shown in Fig. 3b, we can always align the sample plane precisely parallel to the external B field. When this exact parallelism is aligned at a given φ, due to the canting angle γ, further rotation along the stage axis 1 or 2 can cause a correlation between θ and φ, which are labelled as different θ(φ) values.

Extended Data Fig. 3 Procedure for subtracting the canting angle γ.

a, Magnetoresistance R||(φ) in an in-plane B field B|| (for the 17 nm device). b, Variation of θ0 (as defined in Extended Data Fig. 2) as a function of φ when the B field is adjusted to be parallel to the sample plane. The solid black curve is fitted using Eqn. 2 when θ  = 0°, which yields a canting angle γ = 0.71°. c, The data are shown in Fig. 2f before correcting the effect of γ. The black line is the same fitting that is shown in panel b. d, After correcting for the canting angle, the magnetoresistance R(φ, θ) shows a two-fold anisotropy.

Extended Data Fig. 4 Measurement configuration of the 17 nm device.

a, Device orientation and the applied current direction. One crystalline direction of NbSe2, as indicated by the white dashed line, is defined as φ = 0. b, c, Transport measurements using two sets of electrode pairs on two sides of the Hall bar show a small shift (~5°). It is consistent with the small deviation of φ when changing the current direction in Fig. 3j.

Extended Data Fig. 5 Six-fold anisotropy of another multilayer NbSe2 flake.

The thickness of the flake is 22 nm. The anisotropy is measured at B|| = 8.9 T. a, The magnetoresistance R||(φ) in the coexisting state shows a six-fold anisotropy in the B|| field. b, The mapping of R(θ, φ) exhibits six-fold anisotropy when rotating θ close to 0.

Extended Data Fig. 6 Determination of the critical point where the upturn of Bc2(Jc) occurs.

a, An example of I-V measurements at different B|| fields at a fixed temperature. b, The critical current densities Jc were extracted from a. The critical current density is determined as the point where V/I is half the normal resistance RN at T = 10 K. c, The upturn in the BJc plot is determined by the kink in dB/dJc.

Extended Data Fig. 7 Critical current density as a function of temperature under zero magnetic field.

At B = 0 T, no upturn was observed in the temperature dependence of Jc, ruling out the two-gap scenario as the cause of the upturns.

Supplementary information

Supplementary Information

Supplementary Figs. 1–7, sections 1–5 and references.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wan, P., Zheliuk, O., Yuan, N.F.Q. et al. Orbital Fulde–Ferrell–Larkin–Ovchinnikov state in an Ising superconductor. Nature 619, 46–51 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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