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# Mapping the twist-angle disorder and Landau levels in magic-angle graphene

## Abstract

The recently discovered flat electronic bands and strongly correlated and superconducting phases in magic-angle twisted bilayer graphene (MATBG)1,2 crucially depend on the interlayer twist angle, θ. Although control of the global θ with a precision of about 0.1 degrees has been demonstrated1,2,3,4,5,6,7, little information is available on the distribution of the local twist angles. Here we use a nanoscale on-tip scanning superconducting quantum interference device (SQUID-on-tip)8 to obtain tomographic images of the Landau levels in the quantum Hall state9 and to map the local θ variations in hexagonal boron nitride (hBN)-encapsulated MATBG devices with relative precision better than 0.002 degrees and a spatial resolution of a few moiré periods. We find a correlation between the degree of θ disorder and the quality of the MATBG transport characteristics and show that even state-of-the-art devices—which exhibit correlated states, Landau fans and superconductivity—display considerable local variation in θ of up to 0.1 degrees, exhibiting substantial gradients and networks of jumps, and may contain areas with no local MATBG behaviour. We observe that the correlated states in MATBG are particularly fragile with respect to the twist-angle disorder. We also show that the gradients of θ generate large gate-tunable in-plane electric fields, unscreened even in the metallic regions, which profoundly alter the quantum Hall state by forming edge channels in the bulk of the sample and may affect the phase diagram of the correlated and superconducting states. We thus establish the importance of θ disorder as an unconventional type of disorder enabling the use of twist-angle gradients for bandstructure engineering, for realization of correlated phenomena and for gate-tunable built-in planar electric fields for device applications.

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## Data availability

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

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## Acknowledgements

We thank A. Stern and E. Berg for valuable discussions and M. F. da Silva for constructing the COMSOL simulations. This work was supported by the Sagol WIS–MIT Bridge Program, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 785971), by the Israel Science Foundation (ISF, grant no. 994/19), by the Minerva Foundation with funding from the Federal German Ministry of Education and Research, and by the Leona M. and Harry B. Helmsley Charitable Trust grant no. 2018PG-ISL006. Y.C., P.J.-H. and E.Z. acknowledge the support of the MISTI (MIT International Science and Technology Initiatives) MIT–Israel Seed Fund. Work at MIT was supported by the National Science Foundation (NSF, grant no. DMR-1809802), the Center for Integrated Quantum Materials under NSF grant no. DMR-1231319 and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF4541 to P.J.-H. for device fabrication, transport measurements and data analysis. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF ECCS award no. 1541959. D.R.-L acknowledges partial support from Fundaciò Bancaria ‘la Caixa’ (LCF/BQ/AN15/10380011) and from the US Army Research Office grant no. W911NF-17-S-0001. M.K. acknowledges the financial support of JSPS KAKENHI grant no. JP17K05496. J.A.C. and P.M. were supported by the Science and Technology Commission of Shanghai Municipality grant no. 19ZR1436400, the NYU–ECNU Institute of Physics at NYU Shanghai and New York University Global Seed Grants for Collaborative Research. J.A.C. acknowledges support from the National Science Foundation of China grant no. 11750110420. This research was carried out on the High Performance Computing resources at NYU Shanghai. 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.

## Author information

Authors

### Contributions

A.U., S.G. and E.Z. designed the experiment. A.U., S.G. and Y.C. performed the measurements. A.U. and S.G. performed the analysis. Y.C., D.R.-L. and P.J.-H. designed and provided the samples and contributed to the analyses of the results. K.B. fabricated the SOTs. Y.M. fabricated the tuning forks. J.A.C. performed the tight-binding calculations with P.M. and M.K., and K.W. and T.T. fabricated the hBN. A.U., S.G. and E.Z. wrote the manuscript. All authors participated in discussions and in writing of the manuscript.

### Corresponding authors

Correspondence to P. Jarillo-Herrero or E. Zeldov.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Emanuel Tutuc, Fan Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Optical image of MATBG devices.

a, Optical image of device A showing hBN/MATBG/hBN (green), the underlying PdAu backgate (light brown) and the marked electrodes used for four-probe Rxx measurements. b, Optical image of device B (cyan) on the PdAu backgate (light blue) with marked electrodes.

### Extended Data Fig. 2 Transport measurements at T = 300 mK.

a, Four-probe measurement of Rxx(Vbg) versus Ba in device A using an excitation current of 10 nA with the corresponding traces of the Landau fan diagram at the bottom. The green solid lines show the segments that can be traced in the data and the dotted lines indicate their extrapolation to the origin. b, As in a for device B. The purple colour marks the regions where the Rxx signal was slightly negative.

### Extended Data Fig. 3 Transport measurements in the superconducting state at T = 300 mK.

ab,Colour rendering of Rxx measured in the vicinity of –ns/2 versus Ba and ne at low fields using an r.m.s. excitation current of 5 nA in device A (a) and 4 nA in device B (b). A zero-resistance superconducting state (black) is observed in device B. c, dV/dI versus Idc characteristics at various carrier concentrations ne in the superconducting state in device B at Ba = 0 T using an r.m.s. a.c. excitation Iac = 10 nA.

### Extended Data Fig. 4 Numerical simulation demonstrating current imaging by measuring $${{B}}_{{z}}^{{\rm{ac}}}$$.

a, Current distribution Jy(x – x0) of a Δx = 50-nm-wide channel carrying Iy = 1 μA in the $$\hat{{\bf{y}}}$$ direction. b, Calculated −Bz(x – x0) at a height of 70 nm above the sample, convoluted with a 220-nm-diameter SOT sensing area. c, Calculated $${B}_{z}^{{\rm{ac}}}(x-{x}_{0})$$ for an r.m.s. $${x}_{0}^{{\rm{ac}}}$$ = 54 nm spatial modulation of the channel position. The dashed profile corresponds to a current strip of width Δx = 150 nm carrying the same current, showing that the spatial resolution is limited by the SOT diameter. df, As in ac but for three counter-propagating currents spaced 150 nm apart. g, Analysis of the $${B}_{z}^{{\rm{ac}}}$$ peak of an incompressible strip. $${B}_{z}^{{\rm{ac}}}(x)$$ signal (blue) acquired along the line indicated in Fig. 3a for Vbg = −10.54 V (a single vertical line from Fig. 2a) showing the ν = −12 incompressible peak, along with a numerical fit (red). The fit uses the experimental values of $${V}_{{\rm{bg}}}^{{\rm{ac}}}$$, hSOT and the SOT diameter with a single fitting parameter of the total current in the incompressible strip resulting in IT = 1.3 μA. An incompressible strip of width Δx = 50 nm was used for the fit. The mean value of $${B}_{z}^{{\rm{ac}}}(x)$$ was subtracted from the data. The asymmetry in $${B}_{z}^{{\rm{ac}}}(x)$$ away from the peak is caused by the presence of counterflowing nontopological currents INT of lower density in the adjacent compressible strips.

### Extended Data Fig. 5 The origin of equilibrium currents in the compressible and incompressible quantum Hall states.

a, Semiclassical picture of cyclotron orbits of holes with mutually canceling neighbouring currents, resulting in zero bulk current. b, In the presence of an in-plane electric field Ex (+ and − signs represent external charges) the cyclotron orbits acquire a drift velocity resulting in a non-zero $${J}_{y}^{{\rm{T}}}$$ in the incompressible state. c, In the compressible regime the external in-plane electric field is screened by establishing a charge-density gradient, giving rise to $${J}_{y}^{{\rm{NT}}}$$ flowing in the opposite direction (cyan arrows).

### Extended Data Fig. 6 Determination of the accuracy of the twist-angle measurement.

a, Traces of $$-{B}_{z}^{{\rm{ac}}}$$ versus Vbg in device A (from Fig. 1f) acquired with a step size ΔVbg = 4.7 mV and an r.m.s. $${V}_{{\rm{bg}}}^{{\rm{ac}}}$$ = 15 mV. The positions of the $${V}_{{\rm{bg}}}^{N=-3}$$ and $${V}_{{\rm{bg}}}^{N=-4}$$ peaks can be determined to an accuracy better than ±ΔVbg (one step size), corresponding to a relative θ accuracy of δθ = ±0.0002°. b, As in a, taken from Supplementary Video 1 at a pixel position (xy) = (2.53 μm, 5.9 μm) with step size ΔVbg = 40 mV and an r.m.s. $${V}_{{\rm{bg}}}^{{\rm{ac}}}$$ = 35 mV, resulting in a relative θ accuracy of δθ = ±0.002° in the imaging mode. The larger $${B}_{z}^{{\rm{ac}}}$$ signal and the broader IT peaks in b compared to a are due to larger $${V}_{{\rm{bg}}}^{{\rm{ac}}}$$ excitation (see Methods section ‘Measurement parameters’).

### Extended Data Fig. 7 Resolving the local quantum Hall states in flat and dispersive bands in device A.

Global Rxx (purple, right axis) and local $${B}_{z}^{{\rm{ac}}}$$ (blue, left axis) measured at a point in the bulk of device A versus the electron density ne at Ba = 1.19 T. The sharp $${B}_{z}^{{\rm{ac}}}$$ peaks reflect the IT current in incompressible strips with sign determined by the sign of σyx, magnitude by the Landau level energy gap and separation by the Landau level degeneracy (red bars). The dispersive bands are shaded in yellow and the signal in the flat bands is amplified six times for clarity.

### Extended Data Fig. 8 Landau level tomography.

a, Slices of the 3D dataset $${B}_{z}^{{\rm{ac}}}(x,y,{V}_{{\rm{bg}}})$$ along various planes for device A. The bright signals denote the 2D manifolds tracing the incompressible states. The black lines trace the N = −4 incompressible manifold used to determine ns(xy) and θ(xy). It separates fourfold degenerate Landau levels below it from an eightfold degenerate Landau level above it (wide dark blue band). The region in the centre of the sample that shows no Landau levels corresponds to the grey-blue area in Fig. 3b where no MATBG physics is resolved. b, Representative horizontal slices of the data from Supplementary Video 1 showing the evolution of the Landau levels with Vbg. c, As in a, for device B. For the range of gate voltages shown, εF lies in the p dispersive band for the entire sample. The black lines show an example of a trace of the incompressible manifold lying above an eightfold degenerate Landau level. d, Representative horizontal slices of the data from Supplementary Video 3. An interactive interface for tomographic visualization of the data is available at ref. 30.

### Extended Data Fig. 9 Histogram of the charge disorder in device B.

Histogram of δnd(r) data from Fig. 3h along with a Gaussian fit (black) with a standard deviation Δnd = 2.59 × 1010 cm−2.

### Extended Data Fig. 10 Landau level crossings of the dispersive bands.

a, Numerically calculated Landau level energies as a function of magnetic field for a fixed θ = 1.05°. An example level crossing is highlighted in red. b, Numerically calculated Landau level energies as a function of θ for a fixed Ba = 1.22 T. An example level crossing is highlighted in red. ce, The Ba = 0 bandstructure of bilayer graphene for θ = 1.05° (c), θ = 1.16° (d) and θ = 1.27° (e). The blue and red lines indicate the bands that arise from the positive and negative valleys, respectively.

## Supplementary information

### Supplementary Video 1 | Evolution of the Landau levels upon sweeping the back gate through the bottom of the p dispersive band in device A

Zoom on the central part of the Hall bar in device A (purple rectangle in Fig. 3a) upon sweeping Vbg=−8.58 to −11.50 V in fine steps of ΔVbg= 40 mV through the bottom of the p dispersive band. No LLs are visible in the first few frames when |ne|<nS(r) everywhere. On increasing |Vbg|, the hole concentration reaches nS(r) first where θ(r) is the lowest, which occurs in the lower-right corner (the darkest region in Fig. 3b and the 3D triangular structure in Fig. 3d). There the incompressible strips (light blue to red-yellow) are formed with arc-like shape and “climb” the amphitheater-like θ(r) landscape following the equi-θ(r) contours. Note that the incompressible arcs do not form closed contours as expected in conventional QH state and the equilibrium IT currents flowing along the arcs apparently “return” through the INT counterflowing in the adjacent compressible regions (dark blue). Similar dynamics occurs in the top-right corner (dark brown in Fig. 3b) where parallel stripes appear and climb towards bottom-left until they reach a saddle point. There stripes approaching from opposite slopes merge at the saddle point and then split and climb the other two slopes with perpendicular orientation (bright yellow in Fig. 3b), as seen in frames Vbg=−10.46 V to −10.7 V and then repeated for the next LL at Vbg=−11.22 V to −11.50 V.

### Supplementary Video 2 | Evolution of the Landau levels upon sweeping the back gate in the p dispersive band in a larger area of device A

The video presents a sequence of large area $${B}_{z}^{{ac}}$$ images in the p dispersive band in device A upon sweeping Vbg=−16.4 to −17.0 V in coarse steps of ΔVbg= 0.1 V (see Measurement parameters in Methods). In this range, εF lies relatively high in the p dispersive band corresponding to |ne|≈ 1.7nS (see Fig. 2a). The edges of the device are indicated by the black lines and the first frame of the video is described in Fig. 3a. The video shows the evolution of the compressible (light blue to red-yellow) and incompressible (dark blue) strips upon increasing the hole concentration. Note that large parts of the sample, including the entire top part, bottom left area, and smaller regions in the center containing bubbles, do not show LLs and MATBG physics at all. These are the areas that are highly disordered or have a very different twist angle.

### Supplementary Video 3 and Supplementary Video 4 | Evolution of the Landau levels upon sweeping the back gate in the p and n dispersive bands in device B respectively

The two videos show the evolution of the LLs in the p and n dispersive bands in device B (see Measurement parameters in Methods). The data in Supplementary Video 3 is negated so that the incompressible strips appear bright in all the videos. Similarly to device A the LLs climb the θ(r) slopes which are particularly large in the top and bottom regions of device B (dark brown in Fig. 3f). The steep slopes of LLs in these regions are clearly visible in the tomography (Extended Data Fig. 8c).

### Supplementary Video 3 and Supplementary Video 4 | Evolution of the Landau levels upon sweeping the back gate in the p and n dispersive bands in device B respectively

The two videos show the evolution of the LLs in the p and n dispersive bands in device B (see Measurement parameters in Methods). The data in Supplementary Video 3 is negated so that the incompressible strips appear bright in all the videos. Similarly to device A the LLs climb the θ(r) slopes which are particularly large in the top and bottom regions of device B (dark brown in Fig. 3f). The steep slopes of LLs in these regions are clearly visible in the tomography (Extended Data Fig. 8c).

### Supplementary Video 5 | Landau level tomography in device A

The video shows an alternate method for visualizing the data presented in Supplementary Video 1, as described in Tomography section in Methods. The $${B}_{z}^{{ac}}$$ data is shown in the y-Vbg plane at different x, then in x-Vbg plane at different y and finally in the x-y plane at different Vbg. The angle inhomogeneity is seen as a change in Vbg of each incompressible stripe (red) in the x-Vbg or y-Vbg planes. Regions which do not have incompressible regions for any value of Vbg correspond to bubbles seen in the AFM image (Figure 3a inset).

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Uri, A., Grover, S., Cao, Y. et al. Mapping the twist-angle disorder and Landau levels in magic-angle graphene. Nature 581, 47–52 (2020). https://doi.org/10.1038/s41586-020-2255-3

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