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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.

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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.


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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, Jeong Min Park or Pablo Jarillo-Herrero.

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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.

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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.

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

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