Science - USA (2022-02-18)

between the area of the small Fermi surface
in the PIP 2 state at highjjneand the area of a
single Fermi pocket of the Sym 12 state, sug-
gestive of a continuous transition between
the two. This picture implicitly requires sub-
stantial reconstruction of the Fermi surfaces
relative to the single-particle band structure
through either intervalley coherence or the
development of nematic order. Alternatively,
the superconductor may arise from a partially
isospin-polarized phase that is distinct from
the PIP 2 phase, existing between the PIP 2 and
Sym 12 phases and breaking a different set of
spin, valley, or lattice symmetries.
Comparing transport measurements at zero
and finiteB‖(Fig. 4A) provides additional in-
formation. In the absence of a field, the Sym 12
and PIP 2 states are separated by a resistance
maximum (Fig. 4A), where the resistivity shows
strong nonlinearity. Specifically, the resistance
is constant up to a sharply defined threshold in
the applied current, where it abruptly changes
to a reduced value (Fig. 4B and fig. S11). This
behavior is reminiscent of phenomena ob-
served in charge density wave compounds
associated with electric field–induced depin-
ning ( 47 ). Both resistance peak and nonlinear-
ity are low-temperature phenomena, appearing
only belowT≈50 mK, as shown in Fig. 4B
and fig. S10. Like superconductivity at finite
B‖(Fig. 4, E and F), atB= 0, the threshold
current reaches a maximum between the
Sym 12 and PIP 2 states (Fig. 4, C and D). We
conclude that the observed nonlinear transport
is the signature of aB= 0, low-temperature
ground state distinct from the neighboring
PIP 2 phase.
The threshold observed atB=0iscon-
tinuously suppressed by applied magnetic
fields (Fig. 4, G and H) independent of the
field direction and disappears completely for
jjB>75 mT (fig. S9). This suggests that the
zero-field phase is spin unpolarized and that
the suppression of the nonlinearity is driven
by a spin polarization transition. As shown in
Fig. 4I, the boundary between the PIP 2 and
Sym 12 phases shifts to lowerjjne with in-
creasingB‖. Tracing the nonlinear transport
along the boundary between the Sym 12 and
PIP 2 phases (Fig. 4J) reveals that the suppres-
sion of the resistive state coincides precisely
with the onset of superconductivity. We con-
clude that the observed superconductivity
arises as soon as the Zeeman energy is suf-
ficient to spin polarize the electron system,
destroying theB= 0 phase and turning on the
superconducting ground state.
The current results introduce substantial
constraints to any universal theory of super-
conductivity in graphene systems—assuming
such a theory exists. In particular, the difference
in Fermi surface topology between the BBG,
RTG, and moiré systems in their respective
superconducting regimes suggests that Fermi

surface details are not central to the super-

conducting mechanism. By contrast, proximity

to an isospin ordered phase is a generic feature

of both moiré and crystalline graphene super-

conductors, suggestive of a fluctuation-mediated

or other purely repulsive mechanism.

However, our experiments do not yet rule

out a phonon-mediated mechanism, where

generic pairing only leads to observable super-

conductingTCin a narrow density range and

for a specific underlying isospin ordered phase.

With respect to resolving the mechanism of

superconductivity, the greatest effect of the

current work to be practical, the stability of

BBG allows exceptionally high-quality systems

to be made with high yield and reproducibility.

This should allow probes of the pairing sym-

metry, such as phase-sensitive measurements

in hybrid superconducting rings ( 48 ), which

may directly prove or disprove the spin-triplet

nature inferred in this work. Moreover, van

Hove singularities of the type explored here

and in RTGs are generic to all graphene mul-

tilayers, so we expect field effect–controlled

superconductivity to be a widespread phenom-

enon in graphene allotropes with sufficiently

low disorder.

REFERENCES AND NOTES

1. P. W. Anderson,J. Phys. Chem. Solids 11 , 26–30 (1959).


2. A. M. Clogston,Phys. Rev. Lett. 9 , 266–267 (1962).


3. B. S. Chandrasekhar,Appl. Phys. Lett. 1 ,7–8 (1962).


4. D. Aoki, K. Ishida, J. Flouquet,J. Phys. Soc. Jpn. 88 , 022001
   (2019).


5. F. Lévy, I. Sheikin, B. Grenier, A. D. Huxley,Science 309 ,
   1343 – 1346 (2005).


6. S. Ranet al.,Science 365 , 684–687 (2019).


7. Y. Caoet al.,Nature 556 , 43–50 (2018).


8. M. Yankowitzet al.,Nature 557 , 404–408 (2018).


9. Z. Haoet al.,Science 371 , 1133–1138 (2021).


10. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero,
    Nature 590 , 249–255 (2021).


11. H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, A. F. Young,
    Nature 598 , 434–438 (2021).


12. Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero,
    Nature 595 , 526–531 (2021).


13. L. Balents, C. R. Dean, D. K. Efetov, A. F. Young,Nat. Phys. 16 ,
    725 – 733 (2020).


14. K. S. Novoselovet al.,Nat. Phys. 2 , 177–180 (2006).


15. R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, A. Yacoby,
    Science 330 , 812–816 (2010).


16. A. S. Mayorovet al.,Science 333 , 860–863 (2011).


17. W. Baoet al.,Proc. Natl. Acad. Sci. U.S.A. 109 , 10802– 10805
    (2012).


18. F. Freitag, J. Trbovic, M. Weiss, C. Schönenberger,Phys. Rev.
    Lett. 108 , 076602 (2012).


19. J. Jung, A. H. MacDonald,Phys.Rev.B 89 , 035405
    (2014).


20. Y. Zhanget al.,Nature 459 , 820–823 (2009).


21. M. Koshino, E. McCann,Phys. Rev. B 80 , 165409
    (2009).


22. F. Zhang, B. Sahu, H. Min, A. H. MacDonald,Phys. Rev. B 82 ,
    035409 (2010).


23. A. A. Zibrovet al.,Nature 549 , 360–364 (2017).


24. See the supplementary materials.


25. J. P. Eisenstein, L. N. Pfeiffer, K. W. West,Phys. Rev. B 50 ,
    1760 – 1778 (1994).


26. E. V. Castro, N. M. R. Peres, T. Stauber, N. A. P. Silva,
    Phys. Rev. Lett. 100 , 186803 (2008).


27. H. Zhouet al.,Nature 598 , 429–433 (2021).


28. X. Xiet al.,Nat. Phys. 12 , 139–143 (2016).
    29. J. M. Luet al.,Science 350 , 1353–1357 (2015).
    30. D. Zhang, J. Falson,Nanotechnology 32 , 502003 (2021).
    31. H. Minet al.,Phys. Rev. B 74 , 165310 (2006).
    32. J. Sichauet al.,Phys. Rev. Lett. 122 , 046403 (2019).
    33. A. A. Abrikosov, L. P. Gor’kov,Zhur. EksptlÕ. i Teoret. Fiz. 39 ,
    4097498 (1960).
    34. M. Tinkham,Introduction to Superconductivity(McGraw-Hill,
    ed. 2, 1975).
    35. M. G. Vavilov, I. L. Aleiner,Phys. Rev. B 69 , 035303
    (2004).
    36. L. Wanget al.,Science 342 , 614–617 (2013).
    37. O. Vafek, J. M. Murray, V. Cvetkovic,Phys. Rev. Lett. 112 ,
    147002 (2014).
    38. T. Löthman, A. M. Black-Schaffer,Phys. Rev. B 96 , 064505
    (2017).
    39. R. Ojajärvi, T. Hyart, M. A. Silaev, T. T. Heikkilä,Phys. Rev. B
    98 , 054515 (2018).
    40. A. Ghazaryan, T. Holder, M. Serbyn, E. Berg,Phys. Rev. Lett.
    127 , 247001 (2021).
    41. S. Chatterjee, T. Wang, E. Berg, M. P. Zaletel, Inter-valley
    coherent order and isospin fluctuation mediated
    superconductivity in rhombohedral trilayer graphene.
    arXiv:2109.00002 [cond-mat.supr-con] (2021).
    42. Z. Dong, L. Levitov, Superconductivity in the vicinity of an
    isospin-polarized state in a cubic Dirac band. arXiv:2109.01133
    [cond-mat.supr-con] (2021).
    43. T. Cea, P. A. Pantaleón, V. T. Phong, F. Guinea, Superconductivity
    from Repulsive Interactions in Rhombohedral Trilayer
    Graphene: A Kohn-Luttinger-Like Mechanism. arXiv:2109.04345
    [cond-mat.mes-hall] (2021).
    44. A. Szabo, B. Roy, Parent (half)metal and emergent
    superconductivity in rhombohedral trilayer graphene.
    arXiv:2109.04466 [cond-mat.str-el] (2021).
    45. Y.-Z. You, A. Vishwanath, Kohn-Luttinger Superconductivity
    and Inter-Valley Coherence in Rhombohedral Trilayer
    Graphene. arXiv:2109.04669 [cond-mat.str-el] (2021).
    46. Y.-Z. Chou, F. Wu, J. D. Sau, S. Das Sarma,Phys. Rev. Lett. 127 ,
    187001 (2021).
    47. R. M. Fleming, C. C. Grimes,Phys. Rev. Lett. 42 , 1423– 1426
    (1979).
    48. C. C. Tsuei, J. R. Kirtley,Rev. Mod. Phys. 72 , 969– 1016
    (2000).
    49. H. Zhouet al., Isospin magnetism and spin-polarized
    superconductivity in Bernal bilayer graphene,Dryad(2021);
    https://doi.org/10.25349/D9G892.

ACKNOWLEDGMENTS

The authors acknowledge discussions with E. Berg, S. Das Sarma,

Y.-Z. Chou, A. Ghazaryan, L. Levitov, A. Macdonald, M. Serbyn,

C. Varma, and M. Zaletel.Funding:This project was primarily

funded by the Department of Energy under DE-SC0020043.

Support to purchase the cryogen-free dilution refrigerator was

provided by the Army Research Office under award W911NF-17-1-

323. A.F.Y. acknowledges the support of the Gordon and
     Betty Moore Foundation under award GBMF9471 and the
     Packard Foundation under award 2016-65145 for general group
     activities. K.W. and T.T. acknowledge support from the Elemental
     Strategy Initiative conducted by the MEXT, Japan (grant no.
     JPMXP0112101001), and JSPS KAKENHI (grant nos. 19H05790,
     20H00354, and 21H05233).Author contributions:H.Z. and
     L.H. fabricated the devices. H.Z., L.H., Y.S., W.H., C.L.P., and F.Y.
     performed the measurements. H.Z., L.H., Y.S., W.H., C.L.P., F.Y.,
     and A.F.Y. analyzed the data. L.C. assembled the cryogenic
     instrumentation. T.T. and K.W. grew the hexagonal boron nitride
     crystals. H.Z. and A.F.Y. conceived the experiment and wrote
     the paper.Competing interests:The authors declare no
     competing interests.Data and materials availability:All data
     shown in the main text and the supplementary materials are
     available in the Dryad data repository ( 49 ).

SUPPLEMENTARY MATERIALS

science.org/doi/10.1126/science.abm8386

Materials and Methods

Figs. S1 to S11

References ( 50 – 52 )

18 October 2021; accepted 4 January 2022

Published online 13 January 2022

10.1126/science.abm8386

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