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Evidence for Dirac flat band superconductivity enabled by quantum geometry

Abstract

In a flat band superconductor, the charge carriers’ group velocity vF is extremely slow. Superconductivity therein is particularly intriguing, being related to the long-standing mysteries of high-temperature superconductors1 and heavy-fermion systems2. Yet the emergence of superconductivity in flat bands would appear paradoxical, as a small vF in the conventional Bardeen–Cooper–Schrieffer theory implies vanishing coherence length, superfluid stiffness and critical current. Here, using twisted bilayer graphene3,4,5,6,7, we explore the profound effect of vanishingly small velocity in a superconducting Dirac flat band system8,9,10,11,12,13. Using Schwinger-limited non-linear transport studies14,15, we demonstrate an extremely slow normal state drift velocity vn ≈ 1,000 m s–1 for filling fraction ν between −1/2 and −3/4 of the moiré superlattice. In the superconducting state, the same velocity limit constitutes a new limiting mechanism for the critical current, analogous to a relativistic superfluid16. Importantly, our measurement of superfluid stiffness, which controls the superconductor’s electrodynamic response, shows that it is not dominated by the kinetic energy but instead by the interaction-driven superconducting gap, consistent with recent theories on a quantum geometric contribution8,9,10,11,12. We find evidence for small Cooper pairs, characteristic of the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation crossover17,18,19, with an unprecedented ratio of the superconducting transition temperature to the Fermi temperature exceeding unity and discuss how this arises for ultra-strong coupling superconductivity in ultra-flat Dirac bands.

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Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.  Source data are provided with this paper.

Code availability

The code that supports the findings of this study is available from the corresponding authors upon reasonable request.

References

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

2. Stewart, G. R. Unconventional superconductivity. Adv. Phys. 66, 75–196 (2017).

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

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

5. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

6. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

7. Lau, C. N., Bockrath, M. W., Mak, K. F. & Zhang, F. Reproducibility in the fabrication and physics of moiré materials. Nature 602, 41–50 (2022).

8. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

9. Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 123, 237002 (2019).

10. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

11. Julku, A., Peltonen, T. J., Liang, L., Heikkilä, T. T. & Törmä, P. Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene. Phys. Rev. B 101, 060505 (2020).

12. Verma, N., Hazra, T. & Randeria, M. Optical spectral weight, phase stiffness, and Tc bounds for trivial and topological flat band superconductors. Proc. Nat. Acad. Sci. 118, e2106744118 (2021).

13. Herzog-Arbeitman, J., Peri, V., Schindler, F., Huber, S. D. & Bernevig, B. A. Superfluid weight bounds from symmetry and quantum geometry in flat band. Phys. Rev. Lett. 128, 087002 (2022)

14. Berdyugin, A. I. et al. Out-of-equilibrium criticalities in graphene superlattices. Science 375, 430–433 (2022).

15. Schwinger, J. On gauge invariance and vacuum polarization. Phys. Rev. 82, 664–679 (1951).

16. Nishida, Y. & Abuki, H. BCS-BEC crossover in a relativistic superfluid and its significance to quark matter. Phys. Rev. D 72, 096004 (2005).

17. Chen, Q., Stajic, J., Tan, S. & Levin, K. BCS–BEC crossover: from high temperature superconductors to ultracold superfluids. Phys. Rep. 412, 1–88 (2005).

18. Randeria, M. & Taylor, E. Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein Condensation and the Unitary Fermi Gas. Annu. Rev. Condens. Matter Phys. 5, 209–232 (2014).

19. Nakagawa, Y. et al. Gate-controlled BCS-BEC crossover in a two-dimensional superconductor. Science 372, 190–195 (2021).

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

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

22. Allor, D., Cohen, T. D. & McGady, D. A. Schwinger mechanism and graphene. Phys. Rev. D 78, 096009 (2008).

23. Polshyn, H. et al. Large linear-in-temperature resistivity in twisted bilayer graphene. Nat. Phys. 15, 1011–1016 (2019).

24. 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).

25. Berezinsky, V. L. Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems. Sov. Phys. JETP 32, 493–500 (1971).

26. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C: Solid State Phys. 6, 1181–1203 (1973).

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

28. Codecido, E. et al. Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle. Sci. Adv. 5, eaaw9770 (2019).

29. Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of two-dimensional superfluids. Phys. Rev. Lett. 39, 1201 (1977).

30. Tinkham, M. Introduction to Superconductivity 2nd edn (McGraw-Hill, 1996).

31. Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).

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

33. Sensarma, R., Randeria, M. & Ho, T.-L. Vortices in superfluid Fermi gases through the BEC to BCS crossover. Phys. Rev. Lett. 96, 090403 (2006).

34. Hazra, T., Verma, N. & Randeria, M. Bounds on the superconducting transition temperature: applications to twisted bilayer graphene and cold atoms. Phys. Rev. X 9, 031049 (2019).

35. Ahn, J., Park, S. & Yang, B.-J. Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).

36. Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).

37. Song, Z. et al. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

38. Ma, C. et al. Moiré band topology in twisted bilayer graphene. Nano Lett. 20, 6076–6083 (2020).

39. Fortin-Deschênes, M. et al. Uncovering Topological Edge States in Twisted Bilayer Graphene. Nano Lett. 22, 6186–6193 (2022).

40. Khalaf, E., Chatterjee, S., Bultinck, N., Zaletel, M. P. & Vishwanath, A. Charged skyrmions and topological origin of superconductivity in magic-angle graphene. Sci. Adv. 7, eabf5299 (2021).

41. Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

42. Uemura, Y. J. et al. Basic similarities among cuprate, bismuthate, organic, Chevrel-phase, and heavy-fermion superconductors shown by penetration-depth measurements. Phys. Rev. Lett. 68, 2712–2712 (1992).

43. Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Independent superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. 16, 926–930 (2020).

44. Saito, Y. et al. Isospin Pomeranchuk effect in twisted bilayer graphene. Nature 592, 220–224 (2021).

45. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021).

46. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).

47. Zhang, K., Zhang, Y., Fu, L. & Kim, E.-A. Fractional correlated insulating states at n±1/3 filled magic angle twisted bilayer graphene. Commun. Phys. 5, 250 (2022)

48. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Nat. Acad. Sci. 108, 12233 (2011).

49. Koshino, M. et al. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

50. Dóra, B. & Moessner, R. Nonlinear electric transport in graphene: quantum quench dynamics and the Schwinger mechanism. Phys. Rev. B 81, 165431 (2010).

51. Sainz-Cruz, H., Cea, T., Pantaleón, P. A. & Guinea, F. High transmission in twisted bilayer graphene with angle disorder. Phys. Rev. B 104, 075144 (2021).

52. Beenakker, C. W. J. & van Houten, H. in Solid State Physics Vol. 44 (eds Ehrenreich, H. & Turnbull, D.) 1–228 (Academic Press, 1991).

Acknowledgements

We thank P. Stepanov for advice on device fabrication. The experiments are supported by DOE BES Division under grant number DE-SC0020187. M.R. and the nanofabrication facility were supported by NSF Materials Research Science and Engineering Center Grant DMR-2011876. T.X., P.C. and F.Z. were supported by the Army Research Office under grant number W911NF-18-1-0416 and by the National Science Foundation under grant numbers DMR-1945351 through the CAREER programme, DMR-1921581 through the DMREF programme and DMR-2105139 through the CMP programme. T.X., P.C. and F.Z. acknowledge the Texas Advanced Computing Center for providing resources that have contributed to the research results reported in this work. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan (grant number JPMXP0112101001) and JSPS KAKENHI (grant numbers 19H05790, 20H00354 and 21H05233).

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Authors

Contributions

H.T., C.N.L. and M.W.B. conceived the project. H. T., X.G., Y.Z. and S.C. fabricated samples. H.T. X.G. and Y.Z. performed transport measurements. T.X. and P.C. performed theoretical calculations under the supervision of F.Z. K.W. and T.T. provided hBN crystals. H.T., C.N.L. and M.W.B. analysed the data. M.W.B., C.N.L., M.R. and F.Z. interpreted the data and co-wrote the manuscript. All authors discussed and commented on the manuscript.

Corresponding authors

Correspondence to Chun Ning Lau or Marc W. Bockrath.

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Extended data figures and tables

Extended Data Fig. 1 Rxx (Vbg, B) of the device at T = 0.3 K.

The numbers on the right indicate the filling factors of the peaks (where the full-filling of the minibands correspond to filling factors $$\widetilde{\nu }$$ = ±4). Using the convention that the full-filling of the minibands correspond to filling factor $$\widetilde{\nu }$$ = ±4, we identify a number of peaks at fractional fillings of $$\widetilde{\nu }$$ = −0.5, 1.5 and N± 1/3, where N is an integer.

Extended Data Fig. 2 Hall resistance and inferred charge densities.

Left: Symmetrized Rxy versus density. Right: Measured Hall density compared to inferred density from capacitance; the red line has a unit slope to show the agreement between the two.

Extended Data Fig. 3 Non-linear transport at B = 0 and higher temperatures.

ab dV/dI versus J and ñ at T = 5 K, and dV/dI in kΩ versus J at ñ = −1 (blue), −2 (green) and −3 (red) × 1011 cm−2, respectively. cd dV/dI in kΩ versus J and T at ñ ~ −2.8 and −1.7 × 1011 cm−2, respectively. The dV/dI peaks disappear at higher temperatures, which is consistent with an ultra-small Fermi energy of ~1 meV.

Extended Data Fig. 4 The shape of Fermi surface in the lab frame for various rescaled drift velocity $${\boldsymbol{\beta }}={{\boldsymbol{v}}}_{{\bf{n}}}/{{\boldsymbol{v}}}_{{\bf{F}}}$$.

Plot of the Fermi surface for various β versus x and y momentum components px and py.

Extended Data Fig. 5 Theoretical modelling of Fermi velocity vF and the critical drift velocity vn.

a $${v}_{{\rm{F}}}$$ and $${v}_{{\rm{n}}}$$ in units of the Fermi velocity at the Dirac point $${v}_{{\rm{DP}}}$$, as well as $${v}_{{\rm{n}}}/{v}_{{\rm{F}}}$$, versus electron density $${n}_{e}$$. b the effective masses at Fermi energy, $$\hbar {k}_{{\rm{F}}}/{v}_{{\rm{F}}}$$ in theory and $$\hbar {k}_{{\rm{F}}}/{v}_{{\rm{n}}}$$ in measurements, in units of the bare electron mass $${m}_{e}$$ as functions of the electron density $${n}_{e}$$.

Extended Data Fig. 6 Comparison between velocity measured from quantum Hall effect, Shubnikov–de Hass oscillations and non-linear transport measurements near charge neutrality.

a dV/dI versus density n and bias current I for device D2 with θ = 1.06º. Peaks due to the Schwinger effect are indicated by the red dashed lines. b Rxx versus n at T = 30, 25, 20, 18, 12, 10, 7, 5 and 2.02 K, respectively (blue to black). Inset: Activation plot of Rxx measured in the quantum Hall νq = 4 valley indicated by the arrow in the main panel taken at B = 4 T. cd Same as ab but for device D3. Inset in c: Zoom-in of same data in main panel with background subtracted. Colour scale: black: −1 kΩ; white: 3 kΩ. From blue to black, temperatures in d are T = 10, 6, 4, 2.5, 1.8, 1.2, 0.8, 0.4, 0.1 and 0.03 K. e Plot of vQH versus vNLT for D2 and D3. The dotted line indicates vQH = vNLT.

Extended Data Fig. 7 Non-linear transport data near charge neutrality and half-filling for device D4.

a dV/dI (n, I) near charge neutrality. Velocity obtained from slope of features near zero density such as shown by the red dashed line, yielding vNL = ~ 1.7 × 104 m s–1; averaging the slopes of features over four quadrants yields vNL = ~ 1.5 × 104 m s–1. b dV/dI data near half-filling. Features indicated by red dashed lines follow nearly equal slopes, yielding vNL = 2.3 × 103 m s–1.

Extended Data Fig. 8 Comparison of ac and dc measurements.

a R(n) at B = 0, T = 0.3 K and zero bias, measured using ac lock-in techniques with a large dynamic range. The superconducting region displays a “residual” resistance of ~20–30 Ω. b DC voltage-current curve at ñ = −1.65 × 1011 cm−2, B = 0 and T = 0.3 K. The blue line is a line fit to the zero-bias region, which has a slope of −0.2 ± 1.4 Ω.

Extended Data Fig. 9

Transport data over extended range. Nonlinear transport data dV/dI (J,ν) in kΩ over a large density range at B = 0 and T = 0.3 K.

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Tian, H., Gao, X., Zhang, Y. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023). https://doi.org/10.1038/s41586-022-05576-2

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