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.

Pauli-limit violation and re-entrant superconductivity in moiré graphene


Moiré quantum matter has emerged as a materials platform in which correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics1,2,3,4,5. However, direct evidence of unconventional pairing remains to be experimentally demonstrated. Here we show that magic-angle twisted trilayer graphene exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large (2–3 times) violation of the Pauli limit for conventional spin-singlet superconductors6,7. This is an unexpected observation for a system that is not predicted to have strong spin–orbit coupling. The Pauli-limit violation is observed over the entire superconducting phase, which indicates that it is not related to a possible pseudogap phase with large superconducting amplitude pairing. Notably, we observe re-entrant superconductivity at large magnetic fields, which is present over a narrower range of carrier densities and displacement fields. These findings suggest that the superconductivity in magic-angle twisted trilayer graphene is likely to be driven by a mechanism that results in non-spin-singlet Cooper pairs, and that the external magnetic field can cause transitions between phases with potentially different order parameters. Our results demonstrate the richness of moiré superconductivity and could lead to the design of next-generation exotic quantum matter.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Superconductivity in MATTG at high in-plane magnetic fields.
Fig. 2: Large Pauli-limit violation in MATTG.
Fig. 3: Re-entrant superconductivity.
Fig. 4: Field-induced transition between superconducting phases in MATTG.


  1. 1.

    Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    ADS  CAS  Google Scholar 

  2. 2.

    Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    ADS  CAS  Google Scholar 

  3. 3.

    Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    ADS  CAS  Google Scholar 

  4. 4.

    Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249–255 (2021).

    ADS  CAS  Google Scholar 

  5. 5.

    Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).

    ADS  CAS  Google Scholar 

  6. 6.

    Chandrasekhar, B. S. A note on the maximum critical field of high‐field superconductors. Appl. Phys. Lett. 1, 7–8 (1962).

    ADS  CAS  Google Scholar 

  7. 7.

    Clogston, A. M. Upper limit for the critical field in hard superconductors. Phys. Rev. Lett. 9, 266–267 (1962).

    ADS  Google Scholar 

  8. 8.

    Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    ADS  CAS  Google Scholar 

  9. 9.

    Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).

    ADS  CAS  Google Scholar 

  10. 10.

    Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

    ADS  CAS  Google Scholar 

  11. 11.

    Chen, G. et al. Evidence of a gate-tunable Mott insulator in a trilayer graphene moiré superlattice. Nat. Phys. 15, 237 (2019).

    CAS  Google Scholar 

  12. 12.

    Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

    ADS  CAS  Google Scholar 

  13. 13.

    Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    ADS  CAS  Google Scholar 

  14. 14.

    Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    CAS  Google Scholar 

  15. 15.

    Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    ADS  CAS  Google Scholar 

  16. 16.

    Strand, J. D. et al. The transition between real and complex superconducting order parameter phases in UPt3. Science 328, 1368–1369 (2010).

    ADS  CAS  Google Scholar 

  17. 17.

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

    ADS  CAS  Google Scholar 

  18. 18.

    Leggett, A. J. A theoretical description of the new phases of liquid 3He. Rev. Mod. Phys. 47, 331–414 (1975).

    ADS  CAS  Google Scholar 

  19. 19.

    Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Uspekhi 44, 131–136 (2001).

    ADS  Google Scholar 

  20. 20.

    Khalaf, E., Ledwith, P. & Vishwanath, A. Symmetry constraints on superconductivity in twisted bilayer graphene: Fractional vortices, 4e condensates or non-unitary pairing. Preprint at (2020).

  21. 21.

    Christos, M., Sachdev, S. & Scheurer, M. S. Superconductivity, correlated insulators, and Wess–Zumino–Witten terms in twisted bilayer graphene. Proc. Natl Acad. Sci. USA 117, 29543–29554 (2020).

    ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  22. 22.

    Cea, T. & Guinea, F. Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene. Preprint at (2021).

  23. 23.

    Khalaf, E., Kruchkov, A. J., Tarnopolsky, G. & Vishwanath, A. Magic angle hierarchy in twisted graphene multilayers. Phys. Rev. B 100, 085109 (2019).

    ADS  CAS  Google Scholar 

  24. 24.

    Carr, S. et al. Ultraheavy and ultrarelativistic Dirac quasiparticles in sandwiched graphenes. Nano Lett. 20, 3030–3038 (2020).

    ADS  CAS  Google Scholar 

  25. 25.

    Lei, C., Linhart, L., Qin, W., Libisch, F. & MacDonald, A. H. Mirror symmetry breaking and stacking-shift dependence in twisted trilayer graphene. Preprint at (2020).

  26. 26.

    Călugăru, D. et al. Twisted symmetric trilayer graphene: single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field. Phys. Rev. B 103, 195411 (2021).

    ADS  Google Scholar 

  27. 27.

    Cao, Y. et al. Nematicity and competing orders in superconducting magic-angle graphene. Science 372, 264–271 (2021).

    ADS  CAS  Google Scholar 

  28. 28.

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

    ADS  Google Scholar 

  29. 29.

    Larkin, A. I. & Ovchinnikov, Y. N. Nonuniform state of superconductors. Sov. Phys. JETP 20, 762–770 (1965).

    Google Scholar 

  30. 30.

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

    Google Scholar 

  31. 31.

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

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  32. 32.

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

    CAS  Google Scholar 

  33. 33.

    Xi, X. et al. Ising pairing in superconducting NbSe 2 atomic layers. Nat. Phys. 12, 139–143 (2016).

    CAS  Google Scholar 

  34. 34.

    Sichau, J. et al. Resonance microwave measurements of an intrinsic spin–orbit coupling gap in graphene: a possible indication of a topological state. Phys. Rev. Lett. 122, 046403 (2019).

    ADS  CAS  Google Scholar 

  35. 35.

    Banszerus, L. et al. Observation of the spin–orbit gap in bilayer graphene by one-dimensional ballistic transport. Phys. Rev. Lett. 124, 177701 (2020).

    ADS  CAS  Google Scholar 

  36. 36.

    Avsar, A. et al. Colloquium: Spintronics in graphene and other two-dimensional materials. Rev. Mod. Phys. 92, 021003 (2020).

    ADS  CAS  Google Scholar 

  37. 37.

    Bauer, E. & Sigrist, M. (eds) Non-Centrosymmetric Superconductors: Introduction and Overview (Springer, 2012).

  38. 38.

    Uji, S. et al. Magnetic-field-induced superconductivity in a two-dimensional organic conductor. Nature 410, 908–910 (2001).

    ADS  CAS  Google Scholar 

  39. 39.

    Balicas, L. et al. Superconductivity in an organic insulator at very high magnetic fields. Phys. Rev. Lett. 87, 067002 (2001).

    ADS  CAS  Google Scholar 

  40. 40.

    Schemm, E. R., Gannon, W. J., Wishne, C. M., Halperin, W. P. & Kapitulnik, A. Observation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3. Science 345, 190–193 (2014).

    ADS  CAS  Google Scholar 

  41. 41.

    Aoki, D., Ishida, K. & Flouquet, J. Review of U-based ferromagnetic superconductors: comparison between UGe2, URhGe, and UCoGe. J. Phys. Soc. Jpn 88, 022001 (2019).

    ADS  Google Scholar 

  42. 42.

    Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

    ADS  CAS  Google Scholar 

  43. 43.

    Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020).

    ADS  CAS  Google Scholar 

  44. 44.

    Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021).

    CAS  Google Scholar 

  45. 45.

    Bultinck, N. et al. Ground state and hidden symmetry of magic-angle graphene at even integer filling. Phys. Rev. X 10, 031034 (2020).

    CAS  Google Scholar 

  46. 46.

    Park, J. M. Replication data for: Pauli limit violation and reentrant superconductivity in moiré graphene. Harvard Dataverse (2021).

  47. 47.

    Xu, C. & Balents, L. Topological superconductivity in twisted multilayer graphene. Phys. Rev. Lett. 121, 087001 (2018).

    ADS  CAS  Google Scholar 

  48. 48.

    Uykur, E., Li, W., Kuntscher, C. A. & Dressel, M. Optical signatures of energy gap in correlated Dirac fermions. npj Quantum Mater. 4, 19 (2019).

    ADS  Google Scholar 

  49. 49.

    Scheurer, M. S. & Samajdar, R. Pairing in graphene-based moiré superlattices. Phys. Rev. Res. 2, 033062 (2020).

    CAS  Google Scholar 

  50. 50.

    Frigeri, P. A., Agterberg, D. F. & Sigrist, M. Spin susceptibility in superconductors without inversion symmetry. New J. Phys. 6, 115 (2004).

    ADS  Google Scholar 

  51. 51.

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

    ADS  CAS  Google Scholar 

  52. 52.

    Goh, S. K. et al. Anomalous upper critical field in CeCoIn5/YbCoIn5 superlattices with a Rashba-type heavy Fermion interface. Phys. Rev. Lett. 109, 157006 (2012).

    ADS  CAS  Google Scholar 

  53. 53.

    Werthamer, N. R., Helfand, E. & Hohenberg, P. C. Temperature and purity dependence of the superconducting critical field, Hc2. III. Electron spin and spin−orbit effects. Phys. Rev. 147, 295–302 (1966).

    ADS  CAS  Google Scholar 

  54. 54.

    Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).

    ADS  Google Scholar 

  55. 55.

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

    ADS  Google Scholar 

  56. 56.

    Lyon, T. J. et al. Probing electron spin resonance in monolayer graphene. Phys. Rev. Lett. 119, 066802 (2017).

    ADS  CAS  Google Scholar 

  57. 57.

    Nicholas, R. J., Haug, R. J., Klitzing, K. & Weimann, G. Exchange enhancement of the spin splitting in a GaAs–GaxAl1−xAs heterojunction. Phys. Rev. B 37, 1294–1302 (1988).

    ADS  CAS  Google Scholar 

  58. 58.

    Tutuc, E., Melinte, S. & Shayegan, M. Spin polarization and g factor of a dilute GaAs two-dimensional electron system. Phys. Rev. Lett. 88, 036805 (2002).

    ADS  CAS  Google Scholar 

  59. 59.

    Xu, S. et al. Odd-integer quantum hall states and giant spin susceptibility in p-type few-layer WSe2. Phys. Rev. Lett. 118, 067702 (2017).

    ADS  Google Scholar 

  60. 60.

    Rodan-Legrain, D. et al. Highly tunable junctions and non-local Josephson effect in magic-angle graphene tunnelling devices. Nat. Nanotechnol. (2021).

  61. 61.

    Fatemi, V. et al. Electrically tunable low-density superconductivity in a monolayer topological insulator. Science 362, 926–929 (2018).

    ADS  MathSciNet  CAS  Google Scholar 

  62. 62.

    Saito, Y., Itahashi, Y. M., Nojima, T. & Iwasa, Y. Dynamical vortex phase diagram of two-dimensional superconductivity in gated MoS2. Phys. Rev. Mater. 4, 074003 (2020).

    CAS  Google Scholar 

Download references


We thank L. Fu, F. Guinea, S. Kivelson, P. Lee, A. MacDonald, M. Sigrist, S. Todadri and A. Vishwanath for discussions. This work has been primarily supported by the US Department of Energy (DOE), Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under Award DE-SC0001819 (J.M.P.). Help with transport measurements and data analysis were supported by the National Science Foundation (DMR-1809802), and the STC Center for Integrated Quantum Materials (NSF grant no. DMR-1231319) (Y.C.). P.J.-H. acknowledges support from the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9643. P.J.-H. acknowledges partial support by the Fundación Ramon Areces and the CIFAR Quantum Materials program. The development of new nanofabrication and characterization techniques enabling this work has been supported by the US DOE Office of Science, BES, under award DE‐SC0019300. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant number JPMXP0112101001, JSPS KAKENHI grant numbers JP20H00354 and the CREST(JPMJCR15F3), JST. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the National Science Foundation (DMR-0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS-0335765).

Author information




J.M.P. and Y.C. fabricated the samples and performed transport measurements. K.W. and T.T. provided hBN samples. J.M.P., Y.C. and P.J.-H. performed data analysis, discussed the results and wrote the manuscript with input from all co-authors.

Corresponding authors

Correspondence to Yuan Cao or Jeong Min Park or Pablo Jarillo-Herrero.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Folkert de Vries, Yi-Ting Hsu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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 Pauli-limit violation for electron doping.

a, BT phase diagram at the stated density in the +2 − δ superconducting dome. The extracted Pauli-limit violation ratios using 10%, 20% and 30% of normal resistance as the threshold are 3.44, 2.98 and 2.83, respectively. b, B–T phase diagram at a density in the +2 + δ superconducting dome. The extracted Pauli-limit violation ratios using 10%, 20% and 30% of normal resistance as the threshold are 2.49, 2.37 and 2.65 respectively. The solid lines show the fit to the Ginzburg–Landau expression \(T\propto 1-\alpha {B}_{\parallel }^{2}\), and the colour tick marks at T = 0 show the corresponding Pauli limit, the same as in Fig. 2.

Extended Data Fig. 2 Pauli-limit violation in other devices.

a, BT phase diagram of device B with twist angle θ ≈ 1.44°. The extracted Pauli-limit violation ratios using 10%, 20% and 30% of the normal-state resistance as the threshold are 2.13, 2.00 and 2.00, respectively. b, BT phase diagram of device C with twist angle θ ≈ 1.4°. The extracted Pauli-limit violation ratios using 10%, 20% and 30% of the normal state resistance as the threshold are 2.29, 2.23 and 2.19, respectively. The solid lines show the fit to the Ginzburg–Landau expression \(T\propto 1-\alpha {B}_{\parallel }^{2}\), and the colour tick marks at T = 0 show the corresponding Pauli limit, the same as in Fig. 2.

Extended Data Fig. 3 Additional data on the high-field phases.

a, BD map of resistance at a lower temperature T = 0.3 K (see Fig. 4a for comparison). The filament-like transition between SC-I and SC-II is much less pronounced. b, c, Bidirectional sweeps in B at fixed D indicated by the white dashed line in a. The only change in measurement conditions between the two scans is a different arrangement of the BNC cables connecting to the lock-in amplifiers. Both scans are performed at 0.3 K. d, B–D map of resistance on the positive D side measured at T = 0.4 K (see Fig. 4a for comparison).

Extended Data Fig. 4 Extracted PVR as a function of displacement field at ν = −2.4.

Values of 10%, 20% and 30% normal-state resistance were used as the threshold.

Extended Data Fig. 5 Schematic of measurement setup and images of the main MATTG device from optical microscopy and atomic force microscopy.

The microscopy image shows that the core region of the device (inside the dashed rectangle) is clean and free of bubbles. The blue lines are the outlines of the Hall bar that were subsequently etched out.

Extended Data Fig. 6 Calibration of the perpendicular component using the X axis magnetic field Bx.

Calibration curves are shown for B = 0 T, 5 T and 10 T. The dashed lines indicate the calibrated zero perpendicular field condition at each B. The grey bar spans ±5 mT from the centre of the curves, showing that the minimum can be determined well within the bars. See Methods for more details.

Extended Data Fig. 7 Superconducting phases in a perpendicular magnetic field.

All measurements are taken at ν = −2.4, D/ε0 = −0.31 V nm−1. a, The suppression of SC-I and SC-II phases by a perpendicular field B at T = 0.4 K. The white dashed line denotes zero B. This rules out the possibility that the SC-II phase is due to imperfect sample alignment with the axis of B. be, Map of dVxx/dI versus I and B at four different in-plane fields, measured at T = 0.25 K.

Extended Data Fig. 8 Depairing energy for a spin-singlet inter-valley pairing state, calculated for a simple toy model.

The orange curve shows the total depairing energy averaged over the Fermi surface \({\bar{\varepsilon }}_{{\rm{depair}}}\), versus the valley depairing energy amplitude εV. Both quantities are normalized by the Zeeman depairing energy εZ. For comparison, the dashed lines show the cases when the Zeeman effect is omitted (blue dashed line) and when the valley depairing effect is omitted (purple dashed line). Regardless of εV/εZ, the total depairing effect is always stronger than the valley-only or the Zeeman-only case, which means that the critical magnetic field will be reduced from the Pauli limit (corresponding to the Zeeman-only case). Therefore it is unlikely that a spin-singlet inter-valley pairing state accounts for our experimental results.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cao, Y., Park, J.M., Watanabe, K. et al. Pauli-limit violation and re-entrant superconductivity in moiré graphene. Nature 595, 526–531 (2021).

Download citation


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