Maximized electron interactions at the magic angle in twisted bilayer graphene


The electronic properties of heterostructures of atomically thin van der Waals crystals can be modified substantially by moiré superlattice potentials from an interlayer twist between crystals1,2. Moiré tuning of the band structure has led to the recent discovery of superconductivity3,4 and correlated insulating phases5 in twisted bilayer graphene (TBG) near the ‘magic angle’ of twist of about 1.1 degrees, with a phase diagram reminiscent of high-transition-temperature superconductors. Here we directly map the atomic-scale structural and electronic properties of TBG near the magic angle using scanning tunnelling microscopy and spectroscopy. We observe two distinct van Hove singularities (VHSs) in the local density of states around the magic angle, with an energy separation of 57 millielectronvolts that drops to 40 millielectronvolts with high electron/hole doping. Unexpectedly, the VHS energy separation continues to decrease with decreasing twist angle, with a lowest value of 7 to 13 millielectronvolts at a magic angle of 0.79 degrees. More crucial to the correlated behaviour of this material, we find that at the magic angle, the ratio of the Coulomb interaction to the bandwidth of each individual VHS (U/t) is maximized, which is optimal for electronic Cooper pairing mechanisms. When doped near the half-moiré-band filling, a correlation-induced gap splits the conduction VHS with a maximum size of 6.5 millielectronvolts at 1.15 degrees, dropping to 4 millielectronvolts at 0.79 degrees. We capture the doping-dependent and angle-dependent spectroscopy results using a Hartree–Fock model, which allows us to extract the on-site and nearest-neighbour Coulomb interactions. This analysis yields a U/t of order unity indicating that magic-angle TBG is moderately correlated. In addition, scanning tunnelling spectroscopy maps reveal an energy- and doping-dependent three-fold rotational-symmetry breaking of the local density of states in TBG, with the strongest symmetry breaking near the Fermi level and further enhanced when doped to the correlated gap regime. This indicates the presence of a strong electronic nematic susceptibility or even nematic order in TBG in regions of the phase diagram where superconductivity is observed.

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Fig. 1: Atomic insights on TBG structure near the magic angle.
Fig. 2: LDOS and bandwidth of TBG at the magic angle.
Fig. 3: Doping dependence of magic-angle LDOS peaks.
Fig. 4: Wavefunctions and symmetry breaking in magic-angle TBG under doping.

Data availability

The data presented in this work is available upon reasonable request to A.N.P.

Code availability

Code for the analysis described in Methods section ‘Anisotropy quantification technique’ and other analyses presented in this paper are available upon reasonable request.


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We thank A. Millis, J. Schmalian, L. Fu, R. Fernandes and S. Todadri for discussions. This work is supported by the Programmable Quantum Materials (Pro-QM) programme at Columbia University, an Energy Frontier Research Center established by the Department of Energy (grant DE-SC0019443). Equipment support is provided by the Office of Naval Research (grant N00014-17-1-2967) and Air Force Office of Scientific Research (grant FA9550-16-1-0601). Support for sample fabrication at Columbia University is provided by the NSF MRSEC programme through Columbia in the Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634). Theoretical work was supported by the European Research Council (ERC-2015-AdG694097). The Flatiron Institute is a division of the Simons Foundation. L.X. acknowledges the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement number 709382 (MODHET). A.N.P. and A.R. acknowledge support from the Max Planck—New York City Center for Non-Equilibrium Quantum Phenomena. D.M.K. acknowledges funding from the Deutsche Forschungsgemeinschaft through the Emmy Noether programme (KA 3360/2-1). C.D. acknowledges support by the Army Research Office under W911NF-17-1-0323 and The David and Lucile Packard foundation.

Author information




A.K. performed STM measurements. L.J.M., M.Y. and S.C. fabricated samples for STM measurements. A.K. and L.J.M. performed experimental data analysis. K.W. and T.T. provided hBN crystals. D.M.K. and L.X. performed theoretical calculations. J.H., C.D., A.R. and A.N.P. advised. A.K. wrote the manuscript with assistance from all authors.

Corresponding authors

Correspondence to Angel Rubio or Abhay N. Pasupathy.

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: The authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 0.79° LDOS doping dependence.

a, STS LDOS as a function of doping at a 0.79° TBG AA site for a doping range of −0.95ns to 0.8ns (ns being full filling of the moiré band with four electrons or holes). Spectra were taken in a closed loop at 100-meV and 50-pA setpoints with a 0.5-meV oscillation. b, Experimental VHS separation versus doping (bottom axis) and theoretical mean-field VHS separation as a function of chemical potential (Edμ) relative to charge neutrality (top axis) for the 0.79° doping-dependent LDOS. c, LDOS comparison of the correlated gap at half-filling (−0.5ns) in 1.15° TBG and 0.79° TBG. d, Peak-to-peak gap size as a function of doping, offset to half-filling (0.5ns) for 1.15° and 0.79°, where x is additional carrier doping in carriers per square centimetre around half-filling. e, Comparison of 0.79°, 1.15° and 1.10° LDOS when doped near the Fermi level, which is 0 V in the plot. The doping level of each curve is indicated in the legend. Error bars in b and d are estimated from the sum of squares of the lock-in oscillation (0.5 meV for 1.15° and 1 meV for 0.79°) used to determine feature positions.

Extended Data Fig. 2 Asymmetry in valence VHS with doping.

Half-widths of the trailing edge and leading edge of the valence VHS as a function of doping in the 1.15° sample. Error bars are estimated from the sum of squares of the lock-in oscillation (0.5 meV) used, which determines the peak half-width position.

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Supplementary Information

Supplementary Notes 1-10 including Supplementary Figures 1-10 and Supplementary References.

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Kerelsky, A., McGilly, L.J., Kennes, D.M. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

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