Abstract
Magic-angle twisted bilayer graphene (MATBG) exhibits a range of correlated phenomena that originate from strong electron–electron interactions. These interactions make the Fermi surface highly susceptible to reconstruction when ±1, ±2 and ±3 electrons occupy each moiré unit cell, and lead to the formation of various correlated phases1,2,3,4. Although some phases have been shown to have a non-zero Chern number5,6, the local microscopic properties and topological character of many other phases have not yet been determined. Here we introduce a set of techniques that use scanning tunnelling microscopy to map the topological phases that emerge in MATBG in a finite magnetic field. By following the evolution of the local density of states at the Fermi level with electrostatic doping and magnetic field, we create a local Landau fan diagram that enables us to assign Chern numbers directly to all observed phases. We uncover the existence of six topological phases that arise from integer fillings in finite fields and that originate from a cascade of symmetry-breaking transitions driven by correlations7,8. These topological phases can form only for a small range of twist angles around the magic angle, which further differentiates them from the Landau levels observed near charge neutrality. Moreover, we observe that even the charge-neutrality Landau spectrum taken at low fields is considerably modified by interactions, exhibits prominent electron–hole asymmetry, and features an unexpectedly large splitting between zero Landau levels (about 3 to 5 millielectronvolts). Our results show how strong electronic interactions affect the MATBG band structure and lead to correlation-enabled topological phases.
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The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
We acknowledge discussions with A. Young, G. Refael and S. Mashhadi. The device nanofabrication was performed at the Kavli Nanoscience Institute (KNI) at Caltech. This work was supported by NSF through grants DMR-2005129 and DMR-1723367 and by the Army Research Office under grant award W911NF-17-1-0323. Part of the initial STM characterization was supported by CAREER DMR-1753306. Nanofabrication performed by Y.Z. was supported by the DOE-QIS programme (DE-SC0019166). J.A. and S.N.-P. also acknowledge the support of IQIM (an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through grant GBMF1250). Y.P. acknowledges support from a startup fund from California State University, Northridge. A.T., C.L. and J.A. are grateful for support from the Walter Burke Institute for Theoretical Physics at Caltech and the Gordon and Betty Moore Foundation’s EPiQS Initiative, grant GBMF8682. Y.C. and H.K. acknowledge support from the Kwanjeong fellowship.
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Y.C. and H.K. fabricated samples with the help of R.P., Y.Z. and H.A., and performed STM measurements. Y.C., H.K. and S.N.-P. analysed the data. Y.P. and A.T. implemented the models. Y.P., A.T. and C.L. provided theoretical analysis, supervised by J.A.; K.W. and T.T. provided materials (hBN). S.N-P. supervised the project. Y.C., H.K, Y.P., A.T., C.L., J.A. and S.N-P. wrote the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Optical image of the device and STM topography of WSe2/hBN boundary.
a, Optical image of the device. The blue dashed line indicates the region with twisted bilayer graphene (TBG). b, STM topography of a WSe2 boundary underneath twisted bilayer graphene (Vbias = −400 mV, I = 20 pA). c, Linecut along the yellow dashed line in b. The height difference is around 0.8 nm, which matches the monolayer WSe2 thickness. Small ripples originate from the moiré pattern.
Extended Data Fig. 2 AA site point spectra versus gate voltage with magnetic field.
a, B = 0 T. b, B = 4 T. c, B = 8 T. The area (with θ = 1.03°) is the same as that shown in Fig. 1c–e. The VHSs are more pronounced compared to the AB site, making LL features harder to resolve.
Extended Data Fig. 3 Evolution of cascade with magnetic field.
a–e, Point spectra versus gate voltage on an AB site (θ = 1.03°) at B = 4 T (a), B = 5 T (b), B = 6 T (c), B = 7 T (d) and B = 8 T (e). Coloured dashed lines indicate positions calculated with the corresponding Chern numbers. f, Tracking of cascades on the conduction band at different magnetic fields. As the field is increased, the shifting of cascade onsets follows the evolution of Chern insulators. a and e show the same data as Fig. 1d, e; b–d are at intermediate fields on the same area.
Extended Data Fig. 4 Quantum Hall ferromagnetic states at the zLL.
a, Point spectra on an AB site at θ = 1.01° as a function of Vgate for magnetic field B = 7 T. b, Zoom-in around the CNP, where zLLs (0+ and 0−) cross the Fermi energy. The fourfold degeneracy of each zLL is broken by exchange interactions at the Fermi energy, creating gaps at νLL = −4, −3,…, +4. c, Linecuts obtained when the 0− and 0+ LLs cross the Fermi energy, with vertical offset for clarity. Shifting of the spectral weight of the fourfold-degenerate 0− and 0+ LLs cross the Fermi level is clearly visible. Numbers in blue (red) correspond to the number of spin–valley flavours of 0− (0+) LLs below and above the Fermi level (Vbias = 0) as they cross EF near νLL = 4, 3, 2, 1, 0, −1, −2, −3, −4.
Extended Data Fig. 5 Chern insulating phases in the valence band.
a, Full gate voltage range of Fig. 2d at B = 7 T, from ν = −4 to ν = 4. b–g, Zoom-in of Extended Data Fig. 5a around C = −3 (b), C = −2 (d) and C = −1 (f), and the corresponding linecuts (c, e, g) at various Vgate values with vertical offset. Shifting of the spectral weight across the Fermi level is clearly visible in c and e. The C = −1 state is not visible at this angle (θ = 1.02°) and this magnetic field (B = 7 T).
Extended Data Fig. 6 C = −1 phase in a high-resolution angle-dependent LDOS map at B = 7 T.
Measurements were taken over a 20 nm × 200 nm area in which the twist angle varies continuously from θ = 1.01° to 1.18° by using Vbias = 0 mV and changing the spatial position and carrier density (Vgate). The signature of the C = −1 phase and νLL = ±1, ±3, which are not visible in Fig. 3b, are now observed owing to the enhanced Vgate resolution, and the electron–hole asymmetry between Chern phases is even more pronaunced. We note that the visibility of νLL = ±1, ±3 shows strong spatial dependence, whereas the visibility of νLL = ±2, ±4 is high throughout the range.
Extended Data Fig. 7 Spectrum of a continuum model taking into account the effects of a displacement field.
a–c, The spectra correspond to twist angles of θ = 1.03° (a), θ = 1.06° (b) and θ = 1.09° (c). The displacement field, expressed as an energy difference between the two graphene monolayers, is 10 meV, and the ratio between AA and AB tunnelling amplitudes, w0/w1 = 0.4, is the same for all results. A similar spectrum is obtained for larger w0/w1 ratios. The simple continuum model, known to predict correct trends, can capture energetically offset Dirac cones by including a displacement field. However, it underestimates the bandwidth of the flat bands (or equivalently the separation between VHSs) and exhibits an unrealistically strong dependence on the twist angle; even at angles slightly far from the magic angle (θ = 1.06−1.09°), the model predicts a 2−4 times smaller bandwidth than that observed in the data. All spectra feature a well separated LL starting from each band edge that moves inwards with the field. This LL originates from the pocket near the centre of the moiré Brillouin zone and was not observed in the experiment. The bandwidth We (Wh) corresponding to the conductance (valence) flat VHS sub-band is used for estimating the U/W values in Extended Data Fig. 8.
Extended Data Fig. 8 Estimated ratio between correlation and kinetic-energy scales, U/W, as a function of twist angle and magnetic field.
a, b, Estimates from the continuum model for the conduction (a) and valence (b) bands for w0/w1 = 0.4. c, d, The same calculation for w0/w1 = 0.7. e, Analytical estimate from equation (2). The bandwidth in a, b,c,d is evaluated as the total width of all levels that comprise the sub-band that includes the VHS (and carry C = −1 Chern number). Points very close to the magic angle are not shown owing to a lack of clear energy separation from adjacent levels. The plotted constant-U/W curves outline regions where U/W exceeds a threshold value, and qualitatively reproduce the structure of the experimental phase diagram in which Chern insulating phases are observed (Fig. 3f). In e, the large η values are necessary to compensate for the crude approximation of the bandwidth from equation (3), which underestimates the bandwidth compared to the numerical simulation in a, b,c,d.
Extended Data Fig. 9 Possible MATBG band-structure scenarios obtained using the ten-band model and corresponding LL spectra.
a, b, Scenario (i); c, d, scenario (ii); e, f, scenario (iii); g, h, scenario (iv) (see Methods section ‘LL spectrum for low fields and MATBG band structure’). The inset in a shows the relevant high-symmetry momenta in the moiré Brillouin zone. Colours in b, d, f and h represent the normalized density of states (DOS) obtained via Lorentzian broadening of the Landau spectrum by 0.3 meV. i, A possible fit to the data in Fig. 4j corresponding to scenario (ii), yielding vD1 = 2.3 × 105 m s−1 and vD2 = 1.9 × 105 m s−1 for the LLs originating from two two Dirac cones (here marked with red and blue solid lines). The dashed lines correspond to expected LLs that were not observed, possibly owing to the relative positions of the Dirac cones with respect to the VHS. We note that this fitting scenario is not unique. However, all analysed scenarios give vD that exceeds 105 m s−1.
Extended Data Fig. 10 Landau fan diagram overlaid on experimental data for different choices of gate voltages corresponding to ν = ±4.
a, LDOS Landau fan with the same gate voltages Vν=−4 and Vν=4, corresponding to ν = ±4, as in Fig. 2c. The same gate voltages (Vν=−4 = −5.6 V and Vν=4 = 6.2 V) are used in b and d. b, c, LDOS Landau fan zoomed-in around the hole-side dispersive band with different choice of gate voltages. The choice in c (Vν=−4 = −5.7 V and Vν=4 = 6.2 V) does not reproduce the data well. d, e, Similar comparison around the CNP, where e corresponds to Vν=−4 = –5.7 V and Vν=4 = 6.3 V. From this comparison, we extract the error in determining ν = ±4 to be 0.1 V. All LDOS data are normalized by an average LDOS value for each magnetic field. f, Linecut of the LDOS fan in a along the LL filling of −2 emanating from ν = −2, averaged over a Vgate window of 0.04 V. The state above B = 6 T that we identify as a Chern insulating phase is distinguished from half-filling LLs by the abrupt drop in LDOS.
Supplementary information
Supplementary Information
The file includes the continuum and ten-band model details that are used to calculate the spectrum in Fig. 1g, Extended Data Fig. 7, Extended Data Fig. 8 and Extended Data Fig. 9.
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Choi, Y., Kim, H., Peng, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021). https://doi.org/10.1038/s41586-020-03159-7
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DOI: https://doi.org/10.1038/s41586-020-03159-7
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