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Cascade of electronic transitions in magic-angle twisted bilayer graphene

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Abstract

Magic-angle twisted bilayer graphene exhibits a variety of electronic states, including correlated insulators1,2,3, superconductors2,3,4 and topological phases3,5,6. Understanding the microscopic mechanisms responsible for these phases requires determination of the interplay between electron–electron interactions and quantum degeneracy (the latter is due to spin and valley degrees of freedom). Signatures of strong electron–electron correlations have been observed at partial fillings of the flat electronic bands in recent spectroscopic measurements7,8,9,10, and transport experiments have shown changes in the Landau level degeneracy at fillings corresponding to an integer number of electrons per moiré unit cell2,3,4. However, the interplay between interaction effects and the degeneracy of the system is currently unclear. Here we report a cascade of transitions in the spectroscopic properties of magic-angle twisted bilayer graphene as a function of electron filling, determined using high-resolution scanning tunnelling microscopy. We find distinct changes in the chemical potential and a rearrangement of the low-energy excitations at each integer filling of the moiré flat bands. These spectroscopic features are a direct consequence of Coulomb interactions, which split the degenerate flat bands into Hubbard sub-bands. We find these interactions, the strength of which we can extract experimentally, to be surprisingly sensitive to the presence of a perpendicular magnetic field, which strongly modifies the spectroscopic transitions. The cascade of transitions that we report here characterizes the correlated high-temperature parent phase11,12 from which various insulating and superconducting ground-state phases emerge at low temperatures in magic-angle twisted bilayer graphene.

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Fig. 1: Cascade of transitions in spectroscopy at an AA site.
Fig. 2: Theoretical model with on-site Hubbard repulsion U.
Fig. 3: Chemical potential observed through remote band oscillations.

Data availability

The data that support the findings of this study are available at https://doi.org/10.5281/zenodo.3653524 or from the corresponding author upon reasonable request.

Change history

  • 08 July 2020

    The online publication date in the printed version of this article was listed incorrectly as 10 June 2020; the date was correct online

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Acknowledgements

We thank P. Jarillo-Herrero, E. Berg, A. Stern, A. H. MacDonald, B. Jäck, X. Liu, C.-L. Chiu, N. P. Ong, S. Wu, S. Todadri and S. Kahn for useful discussions. This work was primarily supported by the Gordon and Betty Moore Foundation as part of the EPiQS initiative (grant GBMF4530) and by DOE-BES grant DE-FG02-07ER46419. Other support for the experimental work was provided by NSF-MRSEC through the Princeton Center for Complex Materials grants DMR-1420541, NSF-DMR-1608848 and NSF-DMR-1904442, by ExxonMobil through the Andlinger Center for Energy and the Environment at Princeton, and by the Princeton Catalysis Initiative. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, from A3 Foresight by JSPS and from CREST (grant JPMJCR15F3), JST. B.L. acknowledges support from the Princeton Center for Theoretical Science at Princeton University. B.A.B. acknowledges support from Department of Energy grant DE-SC0016239, a Simons Investigator Award, the Packard Foundation, the Schmidt Fund for Innovative Research, NSF EAGER grant DMR-1643312, and NSF-MRSEC DMR-1420541. A.Y. acknowledges the hospitality of Trinity College and the Cavendish Laboratory in Cambridge, UK, during the preparation of this manuscript, which was also funded in part by a QuantEmX grant from the Institute for Complex Adaptive Matter and the Gordon and Betty Moore Foundation (grant GBMF5305).

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Contributions

D.W., K.P.N., M.O. and A.Y. designed the research strategy, carried out STM/STS measurements, and performed the data analysis. D.W., M.O. and K.P.N fabricated samples. S.J., D.W., K.P.N., M.O. and A.Y. constructed the STM. K.W. and T.T. synthesized the h-BN crystals. B.L. and B.A.B. performed the theoretical calculations. All authors discussed the results and contributed to the writing of the manuscript.

Corresponding author

Correspondence to Ali Yazdani.

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The authors declare no competing interests.

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Peer review information Nature thanks Klaus Ensslin, Mathias Scheurer and Miguel Ugeda for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 dI/dV curves on an AA site.

a, Optical micrograph of the MATBG/h-BN device from which data in main text Fig. 1 were acquired (device A). The white dashed line encloses the MATBG area. b, STM topographic image of 1.06° MATBG (Vb = −500 mV, I = 10 pA). c, dI/dV spectra acquired at the centre of an AA site, where both flat bands are completely filled (red), the valence flat band is filled but the conduction flat band is empty (purple), and both flat bands are completely empty (blue). These data were acquired on the same AA site as that shown in main text Fig. 1c. Sequential curves are vertically offset by 20 nS for clarity. d, Same as c but for more gate voltages (Vg = 50 V to −50 V). Sequential curves are vertically offset by 2.5 nS for clarity. A feature due to inelastic tunnelling is sometimes seen at Vb = ±60 mV but has no influence on observations closer to EF than36 ±60 mV. The dI/dV values for spectra with negative ν appear to be generically larger than for positive ν. This asymmetry is caused in part by the constant-current feedback condition in STS measurements. As ν decreases, the flat bands are emptied and contribute less to the tunnelling current at the set point; the tip moves closer to the sample to compensate for the loss in current, amplifying the value of dI/dV as ν decreases. We cannot however rule out the possibility that intrinsic electron–hole asymmetry also plays a role in the difference in dI/dV between positive and negative ν. e, dEH/dVg as a function of gate voltage (rolling average over a ΔVg = 0.65 V window) obtained by extracting the energy EH of the peak in dI/dV for the cascade features shown in main text Fig. 1. Upward pointing triangles identify discontinuous transitions between each cascade feature in the conduction flat band (red), around charge neutrality (purple), and in the valence flat band (blue). f, dI/dV(Vb = 0 V, Vg) on the AA site. Equally spaced shaded bars are drawn over dips in the zero-bias conductance, which allow us to identify each integer filling ν of the flat bands. Zero-bias conductance peaks are marked by downward pointing triangles, which can be attributed to each of the discontinuities in a.

Extended Data Fig. 2 Cascade features at other angles near the magic angle.

The cascade features that we report in this paper were reproduced 8 times in device A and 9 times in device B (all angles between 0.97° and 1.17°). Each observation can be regarded as independent because (i) the STM tip is different for each measurement and (ii) the local twist angle is different for each measurement owing to large spatial inhomogeneity in each device. A small moiré between graphene and h-BN (varying around 8°) was often seen in device A, while no graphene/h-BN moiré was seen in device B. Panels ad along with main text Fig. 1c, d show our first five observations of the cascade features in device A. a, The left panel shows dI/dV(Vb, Vg) measured at the centre of an AA site, while the right panel shows the same quantity divided by I/Vb. The twist angle is 1.01° with 0.2% strain (initial tunnelling parameters: Vb = −300 mV, I = 0.3 nA, 551.7 Hz sinusoidal modulation of Vrms = 1 mV). b, Same as a for a twist angle of 0.97° and 0% strain (initial tunnelling parameters: Vb = −80 mV, I = 1.0 nA, 4.117 kHz sinusoidal modulation of Vrms = 0.3 mV). c, Same as a for a twist angle of 1.03° and 0.2% strain (initial tunnelling parameters: Vb = −100 mV, I = 0.5 nA, 4.121 kHz sinusoidal modulation of Vrms = 1.0 mV). d, Same as a for a twist angle of 1.07° and 0.1% strain (initial tunnelling parameters: Vb = −80 mV, I = 1.0 nA, 4.121 kHz sinusoidal modulation of Vrms = 1.0 mV). Twist angles are measured within ±0.01°.

Extended Data Fig. 3 Exact diagonalization.

a, Theoretical spectral weight for different values of the filling ν calculated by exact diagonalization of the two-flavour, two-band Hubbard model consisting of 4 sites, interband separation 2E0 = 16 meV, nearest-neighbour hopping t = 0.6 meV, and on-site Coulomb repulsion U = 23 meV. Dashed black lines highlight the cascade features, which are fewer in number than in the experimental data because the theoretical model has only 2 flavours (compared to fourfold degeneracy in MATBG). b, The calculated chemical potential μ as a function of ν shows jumps at integer fillings. The inset schematically depicts the 4 AA sites, with on-site repulsion U and hopping t. c, Calculated energy difference E between cascade features and the chemical potential jump at ν = 1 as a function of U.

Extended Data Fig. 4 Undertwisted and overtwisted bilayer graphene.

a, dI/dV(Vb, Vg) measured at the centre of an AA site for 0.85° (0.3% strain) twisted bilayer graphene (device B; initial tunnelling parameters: Vb = −40 mV, I = 30 pA, 381.7 Hz sinusoidal modulation of Vrms = 1.2 mV). These data do not show the cascade features. b, Same as a but divided by I/Vb. c, dI/dV(Vb, Vg) measured at the centre of an AA site for 1.21° (0.2% strain) twisted bilayer graphene (device A; initial tunnelling parameters: Vb = −100 mV, I = 1.5 nA, 4.121 kHz sinusoidal modulation of Vrms = 0.5 mV). These data do not show the cascade features (except possibly very faintly for negative ν). A dip of unknown origin at EF is visible at all carrier densities. The persistence of this dip disqualifies it from being a superconducting or correlated-insulating gap. This dip is likely to be a Coulomb gap or a tip artefact7. d, Same as c but divided by I/Vb.

Extended Data Fig. 5 dI/dV on an AB site and on an AA site in a magnetic field.

a, dI/dV spectra acquired at the centre of an AB region in 1.05° MATBG in device B (Vg = 38 V to −40 V). These data are the same as those shown in main text Fig. 3a and clearly show gate-dependent oscillations in the remote bands. Sequential curves are vertically offset by 1 nS for clarity. b, dI/dV curves acquired at the centre of a nearby AA site at zero magnetic field (initial tunnelling parameters: Vb = −150 mV, I = 0.5 nA, 4.121 kHz sinusoidal modulation of Vrms = 1 mV). Sequential curves are vertically offset by 10 nS for clarity. c, dI/dV(Vb, Vg) measured at the centre of the same AA site as in b at zero magnetic field. d, dI/dV(Vb, Vg) measured at the centre of an AA site at B = 9 T (initial tunnelling parameters for c and d: Vb = −100 mV, I = 0.5 nA, 4.121 kHz sinusoidal modulation of Vrms = 0.6 mV). The out-of-plane magnetic field suppresses the cusps at integer ν and strongly modifies the cascade features near EF. This suggests that a perpendicular field may change the localized nature of electrons in the flat bands; Landau quantization of the remote bands may also change their interactions with the flat bands.

Extended Data Fig. 6 Tip-induced collapse of a flat band.

Measurements on MATBG can strongly depend on the STM tip, so we calibrate the tip against the Shockley surface state of Cu(111). Damage to or contamination of the tip causes distortions to observed features. Copper and tungsten, incidentally, have work functions similar to that of graphene. a, dI/dV(Vb, Vg) measured at the centre of an AA site in device A, with a twist angle of 1.01° and 0.2% strain (same data as Extended Data Fig. 2a). The lack of symmetry in dI/dV(Vb, Vg) indicates that the tip condition is already non-optimal. b, Same as a after lightly crashing the tip into the sample (initial tunnelling parameters: Vb = −200 mV, I = 0.5 nA, 551.7 Hz sinusoidal modulation of Vrms = 1 mV). The crash further degraded the tip, causing the valence flat band to appear to cross EF without pinning. This can be caused by changes in tip-induced band bending7 and tip-induced doping37.

Extended Data Fig. 7 Device fabrication.

a, Schematic showing step-by-step device fabrication procedure for device A (device B is similar, but without PMMA). See Methods for details. b, In order to do STM measurements, MATBG must be ‘face up’. To achieve this, we transfer a MATBG/h-BN stack from its initial PVA handle to a secondary handle. Water injected between the two glass handles (each a standard 75 mm × 25 mm microscope glass slide) via a syringe dissolves the PVA. The glass slides are fixed on linear translation stages and are underneath an optical microscope. c, 500 nm × 500 nm STM topograph of device B showing a moiré pattern and physical corrugation in the underlying substrate (Vb = −400 mV, I = 100 pA). d, Zoomed-in image of the centre of the area in c. Each device has regions that are magic angle (which always show cascade features) and regions that are not magic angle (which do not show cascade features). Regions of a device that are magic angle are typically so over at least a square micrometre.

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Wong, D., Nuckolls, K.P., Oh, M. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene. Nature 582, 198–202 (2020). https://doi.org/10.1038/s41586-020-2339-0

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