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.
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The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
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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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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.
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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.
a–b 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. c–d 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. c–d Same as a–b 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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DOI: https://doi.org/10.1038/s41586-022-05576-2
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