Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene

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Abstract

The discovery of superconducting and insulating states in magic-angle twisted bilayer graphene (MATBG)1,2 has ignited considerable interest in understanding the nature of electronic interactions in this chemically pristine material. The transport properties of MATBG as a function of doping are similar to those of high-transition-temperature copper oxides and other unconventional superconductors1,2,3, which suggests that MATBG may be a highly interacting system. However, to our knowledge, there is no direct experimental evidence of strong many-body correlations in MATBG. Here we present high-resolution spectroscopic measurements, obtained using a scanning tunnelling microscope, that provide such evidence as a function of carrier density. MATBG displays unusual spectroscopic characteristics that can be attributed to electron–electron interactions over a wide range of doping levels, including those at which superconductivity emerges in this system. We show that our measurements cannot be explained with a mean-field approach for modelling electron–electron interactions in MATBG. The breakdown of a mean-field approach when applied to other correlated superconductors, such as copper oxides, has long inspired the study of the highly correlated Hubbard model3. We show that a phenomenological extended-Hubbard-model cluster calculation, which is motivated by the nearly localized nature of the relevant electronic states of MATBG, produces spectroscopic features that are similar to those that we observed experimentally. Our findings demonstrate the critical role of many-body correlations in understanding the properties of MATBG.

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Fig. 1: Non-interacting spectroscopic properties of MATBG.
Fig. 2: Breakdown of non-interacting description.
Fig. 3: Evidence for correlated insulating state at half-filling.
Fig. 4: Extended Hubbard model.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

Change history

  • 22 August 2019

    This article was amended to correct the Peer review information, which was originally incorrect.

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Acknowledgements

We acknowledge discussions with N. P. Ong, S. Wu, D. Wong, P. Sambo, L. Glazman, P. Phillips and A. MacDonald. We also acknowledge collaborations with E. Tutuc, K. Kim and Y. Wang during the initial stages of this project. This work has been primarily supported by the Gordon and Betty Moore Foundation as part of the EPiQS initiative (GBMF4530) and DOE-BES grant DE-FG02-07ER46419. Other support for the experimental work was provided by NSF-MRSEC programmes through the Princeton Center for Complex Materials DMR-142054, NSF-DMR-1608848, ExxonMobil through the Andlinger Center for Energy and the Environment at Princeton, and the Princeton Catalysis Initiative. B.J. acknowledges funding from the Alexander von Humboldt foundation through a Feodor Lynen postdoctoral fellowship. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, A3 Foresight by JSPS and the CREST (JPMJCR15F3), JST. B.L. acknowledges support from the Princeton Center for Theoretical Science at Princeton University. B.A.B. acknowledges support from the Department of Energy DE-SC0016239, Simons Investigator Award, the Packard Foundation, the Schmidt Fund for Innovative Research, NSF EAGER grant DMR-1643312, and NSF-MRSEC DMR-1420541.

Author information

Y.X., B.J., C.-L.C., X.L. and A.Y. designed, performed and analysed the STM experiments. X.L., Y.X. and B.J. fabricated the sample. X.L. carried out the finite element electrostatic simulation. B.L. and B.A.B. performed the theoretical calculations. K.W. and T.T. provided hBN crystals. All authors contributed to the writing of the manuscript.

Correspondence to Ali Yazdani.

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The authors declare no competing interests.

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Peer review information Nature thanks Miguel M. Ugeda and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Fig. 1 Non-interacting spectroscopic properties on another region.

a, STM topography showing the moiré superlattice with θ = 1.14°. b, Scanning tunnelling spectra measured on an AA site for slight electron doping (Vg = −10 V, Vset = −200 mV, Iset = 500 pA, Vmod = 0.5 mV). The blue and green arrows mark the step-like features. c, Band structure calculated using the continuum model including the effects of strain and relaxation. MMΛm is a non-high-symmetry direction along which the Dirac points (protected by C2zT symmetry) are located. The black dotted line indicates the Fermi level. The blue (green) dashed line corresponds to the VHS of the first conduction (valence) remote band. d, Corresponding \(\sqrt{{\rm{L}}{\rm{D}}{\rm{O}}{\rm{S}}}\) (offset by −28 meV) on an AA site. The blue and green arrows mark the VHS of the first conduction and valence remote bands.

Extended Data Fig. 2 Lifetime broadening.

a, b, Peak width of the VHS as a function of the VHS energy when both flat bands are filled (a) or emptied (b). The green (a) and yellow (b) curves are the fit using p(E) = p0 + λE2 with p0 = 5.25 meV, λ = 0.021 meV−2 for a and with p0 = 5.1 meV, λ = 0.00072 meV−2 for b. The black dotted line marks E = 17 meV.

Extended Data Fig. 3 Tip band bending effect

a, Schematic (not to scale) of the electrostatic simulation for the device geometry in our experiment. b, Density under and away from the tip as a function of gate voltage, calculated using the geometry in a with height z = 4 Å, radius r = 0.6 nm, a work function difference of 0.25 V and the band structure from Fig. 1e. c, Density and DOS at the Fermi level as a function of gate voltage under the tip with the same parameters as in b. d, e, Zero-bias conductance as a function of gate voltage with different set point conditions (Vset = +200 mV, Iset = 120 pA for d and Vset = −200 mV, Iset = 500 pA for e) showing different apparent gate efficiencies.

Extended Data Fig. 4 Additional example of the breakdown of the non-interacting description.

a, dI/dV spectra measured on an AA site as a function of sample bias (energy) and gate voltage on the region with θ = 1.14° (Vset = −200 mV, Iset = 500 pA, Vmod = 0.5 mV). b, Normalized dI/dV spectra at different gate voltages extracted from a. The curves are shifted vertically by 0.5 nS each for clarity. The black dotted line indicates the Fermi level.

Extended Data Fig. 5 Density-dependent spectroscopy on a non-magic-angle device.

a, dI/dV measured on an AA site as a function of sample bias and gate voltage on the region with θ = 1.72° (Vset = −100 mV, Iset = 180 pA, Vmod = 2 mV). be, Normalized dI/dV spectra at different gate voltages extracted from a. The curves are shifted vertically by 0.4 each for clarity. The black dotted line indicates the Fermi level. We note that the electron and hole dopings are symmetric, which can be achieved by different tip conditioning (see Methods for discussion).

Extended Data Fig. 6 Mean-field calculations.

a, b, Normalized dI/dV spectra obtained from a non-interacting model (a) and a Hartree–Fock calculation (b) for different filling factors. The curves are each shifted vertically by 0.3 each for clarity. c, Individual flavour filling as a function of total filling factor, indicating the presence of spontaneous spin or valley polarization near half-filling of the flat bands from the Hartree–Fock calculation.

Extended Data Fig. 7 Extended Hubbard model with six sites.

a, Schematic of a six-site cluster two-orbital Hubbard model with on-site energy \(\pm \epsilon \), hopping t, on-site Coulomb interaction U, near-neighbour interactions V0 (same orbital) and V1 (different orbitals). b, Local spectral weight computed from the exact diagonalization of a six-site cluster Hubbard model for different filling factors with \(\epsilon =9\), t = 0.75, U = 30, V0 = 5, V1 = 3.8 (see Methods). The curves beyond \(\pm {n}_{0}\) are obtained by assuming a constant DOS at the Fermi level from the remote bands. The curves are shifted vertically by 0.25 each for clarity. c, Band broadening as a function of the ratio of the on-site Coulomb interaction U to the non-interacting band width 6t, while maintaining V0 = U/6, V1 = U/7.9.

Extended Data Fig. 8 Extended Hubbard model on a honeycomb lattice.

a, Schematics of a 6-unit-cell lattice with one orbital per site, hopping t, nearest site interaction V0 (different sublattices) and next-nearest site interaction V1 (same lattices). b, Local spectral weight computed from the exact diagonalization of a 6-unit-cell lattice Hubbard model for different filling factors with \(\epsilon =8.5\), t = 0.75, V0 = 18.2, V1 = 4.4 (see Methods). The curves beyond \(\pm {n}_{0}\) are obtained by assuming a constant DOS at the Fermi level from the remote bands. The curves are shifted vertically by 0.25 each for clarity. c, Band broadening as a function of the ratio of the nearest site interaction V0 to the non-interacting band width 3t, while maintaining V0 = 4.2V1.

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