Abstract
Magic-angle twisted bilayer graphene exhibits correlated phenomena such as superconductivity and Mott insulating states related to the weakly dispersing flat band near the Fermi energy. Such a flat band is expected to be sensitive to both the moiré period and lattice relaxations. Thus, clarifying the evolution of the electronic structure with the twist angle is critical for understanding the physics of magic-angle twisted bilayer graphene. Here we combine nano-spot angle-resolved photoemission spectroscopy and atomic force microscopy to resolve the fine electronic structure of the flat band and remote bands, as well as their evolution with twist angle from 1.07° to 2.60°. Near the magic angle, the dispersion is characterized by a flat band near the Fermi energy with a strongly reduced band width. Moreover, we observe a spectral weight transfer between remote bands at higher binding energy, which allows to extract the modulated interlayer spacing near the magic angle. Our work provides direct spectroscopic information on flat band physics and highlights the important role of lattice relaxations.
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All relevant data of this study are available within the paper and its Supplementary Information files. Source data are provided with this paper.
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Acknowledgements
This work is supported by the National Key R&D Program of China (nos 2021YFA1400100 and 2020YFA0308800), the National Natural Science Foundation of China (nos 12234011, 52025024, 92250305, 52388201, 12327805 and 12304226) and the Basic Science Center Program of NSFC (no. 52388201). H.Z. acknowledges support from the Shuimu Tsinghua Scholar project and the project funded by China Postdoctoral Science Foundation (grant no. 2022M721887). Z.S. and Y.W. were supported by the National Natural Science Foundation of China (General Program no. 12274005), National Key Research and Development Program of China (no. 2021YFA1401900) and Innovation Program for Quantum Science and Technology (no. 2021ZD0302403). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT (grant no. JPMXP0112101001), JSPS KAKENHI (grant no. JP20H00354) and the CREST (JPMJCR15F3), JST. We acknowledge SOLEIL for the provision of synchrotron radiation facilities of beamline ANTARES.
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Contributions
S. Zhou designed the research project. Qian Li and Hongyun Zhang prepared the samples. Hongyun Zhang, Qian Li, W.C., C.B., Q. Liu, T.L., Haoxiong Zhang, J.A., P.D. and S. Zhou performed the NanoARPES measurements and analysed the ARPES data. Qian Li performed the AFM measurements, with assistance from S. Zhang, Qunyang Li and P.Y. K.W. and T.T. grew the BN crystals. Y.W., W.D. and Z.S. performed the numerical calculations. Qian Li, Hongyun Zhang, P.Y. and S. Zhou wrote the manuscript, and all authors commented on the manuscript.
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Extended data
Extended Data Fig. 1 Sample preparation.
a, Monolayer graphene exfoliated on SiO2/Si. b, The bottom graphene is picked up by BN attached to PDMS stamp. c, The twist angle between two layers of graphene is achieved by rotating the other half of graphene on the SiO2/Si. Then the top graphene is picked up by the PDMS/BN/G heterostructure. d, The heterostructure is flipped over, and the BN/tBLG sample is picked up by another PDMS stamp (see the live video in the Supplementary information). e, The heterostructure is transferred onto a gold-coated substrate. f, Two narrow pieces of graphite are used to connect graphene and the gold-coated substrate. The scale bars in a-f are 100 μm.
Extended Data Fig. 2 Determination of twist angle from ARPES and AFM measurements.
a, Optical image of 1.93∘ tBLG sample. The red dot indicates the measurement position. Red and orange curves indicate the top and bottom graphene layers, respectively. b, AFM image to show the measurement position as indicated by the red oval. c, d, L-AFM image and the fast Fournier transform (FFT) filtered image measured before the NanoARPES experiment. The scale bar is 20 nm. e, f, C-AFM image and the FFT-filtered image measured after NanoARPES experiment. g, Extracted line profiles from d and f along the blue and red lines, which suggests that the moiré period of 7.3 nm not change during the NanoARPES measurement. h, Fermi surface measured at the red dot in a, from which the separation between Kt and Kb points agrees well with the extracted moiré period from AFM measurement.
Extended Data Fig. 3 AFM characterization of the moiré superlattices for twist angles θ ranging from 1.07∘ to 2.60∘.
a-i, AFM characterization for P1 (θ = 1.07∘) and P2 (θ = 1.31∘) on sample S1. The white scale bar in a is 20 nm. j-r, AFM characterization for P3 (θ = 1.93∘) and P4 (θ = 2.22∘) on sample S2. s-w, AFM characterization for P5 (θ = 2.60∘) on sample S3. The first four columns are the AFM images, and fast Fourier transform (FFT) of AFM images, and FFT-filtered images, and ARPES spatial images with P1-P5 marked by red spots. The fifth column shows the optical images of the three samples S1-S3. The scale bars in a, e, j, n and s are 20 μm.
Extended Data Fig. 4 Extracting the bandwidth of the flat band from experimental dispersion images at different twist angles.
a-e, Measured dispersion images from 1.07∘ to 2.60∘ tBLG along the cut H direction as indicated by the black line in the inset. f-j, Calculated dispersion images from 1.07∘ to 2.60∘. The gray broken lines indicate the momentum range of the flat band edge, where gaps open between the flat band and remote bands. k-o, EDCs from a-e. The red tick marks indicate the peak positions, which are used for extracting the flat band dispersion.
Extended Data Fig. 5 Dependence of the momentum separation between the remote bands and momentum position of the vHS on the interlayer tunneling parameters ωAB and α.
a, Experimental energy contour at -0.5 eV for the 2.22∘ tBLG. The red arrow indicates the momentum separation between remote bands p1 and p2. b-d, Calculated energy contours using ωAB of 80, 120 and 160 meV, respectively. The energy separation gradually increases with increasing of ωAB, from which the calculated result at ωAB = 120 meV agrees well with the experiment result. e, Experimental energy contour at -0.1 eV for the 2.22∘ tBLG. f-h, Calculated energy contours using α of 0.5, 0.8 and 1.0, respectively. The position of the vHS shifts away from the moiré Brillouin zone border (gray dashed hexagons), from which the calculated results at α = 0.8 agrees well with the experimental results.
Extended Data Fig. 6 Evolution of the energy separation between remote bands p1 and p2 with twist angle.
a, Dispersion image at twist angle of 1.07∘ measured along the cut V direction as indicated by the black line in the inset. b, c, The comparison of the extracted ΔE1 and ΔE2 from experimental results with calculations using different parameters of ωAB and α. d, The comparison of energy separation between experimental and calculated results at different ky under the twist angle of 1.07∘. The gray broken line indicates that overall linear scaling of the energy separation with ky. The error bar of experimental results in b-d is 3-4 meV, which is too small to see. Data in b, c are presented as difference between fitting values from 5 samples with different twist angles, with error bars representing the standard error.
Extended Data Fig. 7 Reproducibility of the experimental spectral weight ratio between p1 and p2 at different sample positions and after different beam exposure time.
a, ARPES spatial intensity image to show the two different positions A and B on sample 1, indicated by red and blue marks. b, c, ARPES energy contours at -0.5 eV for positions A and B. d, ARPES spatial intensity image to show the position C on sample 2, indicated by the green mark. e, ARPES measured energy contours at -0.5 eV for position C. f, Extracted momentum distribution curves (MDCs) from b, c, e along the red, blue and green broken lines. The extracted ratio between p1 and p2 from these three positions gives the same value, showing that the ratio is reproducible on different sample positions and different samples with the same twist angle. g, h, The dispersion images measured along the direction shown in inset of h at the beginning and after 6 hours measurement. i, j, ARPES intensity maps measured at the beginning of NanoARPES experiment and after 6 hours beam exposure. k, Extracted MDCs from i and j along the blue and red broken lines. The spectral weight ratio between p1 and p2 does not change with the beam exposure time.
Extended Data Fig. 8 Extracting the interlayer spacing of 1.07∘ tBLG from the spectral weight ratio.
a, Energy contour measured at -0.6 eV in 1.07∘ tBLG. b-e Calculated energy contours by using interlayer spacing c = 3.35 Å, 3.40 Å, 3.42 Å and 3.50 Å, respectively. f-j, Extracted MDCs from the calculated dispersion images in a-e along the black dashed lines at -0.6 eV.
Extended Data Fig. 9 Extracting the interlayer spacing for tBLG at twist angles ranging from 1.07∘ to 2.60∘.
a-e, ARPES intensity maps measured at -0.60 eV at twist angles from 1.07∘ to 2.60∘. f-j, Calculated energy contours after adjusting the interlayer spacing to fit the spectral ratio in a-e. k, Extracted MDCs at -0.6 eV, as indicated by the black dashed line in a, from which the interlayer spacing is obtained.
Extended Data Fig. 10 Lattice relaxation revealed by local vertical conductivity from C-AFM measurements.
a-d, C-AFM images measured at twist angles of 2.18∘, 1.27∘, 1.07∘ and 0.51∘. e-h, Current profiles measured along the colored lines in a-d, which are normalized by the local current at AA-stacking (maximum current). The dashed line marks the minimum current.
Supplementary information
Supplementary Information
Supplementary Figs. 1–4 and discussion.
Supplementary Video 1
A video showing the flipping process whereby the PDMS/BN/tBLG structure was flipped over and picked up by another PDMS stamp.
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Source Data Fig. 1
Statistical source data.
Source Data Fig. 2
Statistical source data.
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Statistical source data.
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Source Data Fig. 5
Statistical source data.
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Li, Q., Zhang, H., Wang, Y. et al. Evolution of the flat band and the role of lattice relaxations in twisted bilayer graphene. Nat. Mater. (2024). https://doi.org/10.1038/s41563-024-01858-4
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DOI: https://doi.org/10.1038/s41563-024-01858-4
