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Moiré periodic and quasiperiodic crystals in heterostructures of twisted bilayer graphene on hexagonal boron nitride

Nature Materials volume 24, pages 1019–1026 (2025)Cite this article

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Abstract

Stacking two atomic crystals with a twist between their crystal axes produces moiré potentials that modify the electronic properties. Here we show that double-moiré potentials generated by superposing three atomic crystals create a unique class of tunable quasiperiodic structures that alter the symmetry and spatial distribution of the electronic wavefunctions. By using scanning tunnelling microscopy and scanning tunnelling spectroscopy to study twisted bilayer graphene on hexagonal boron nitride, we unveil a moiré phase diagram defined by the lattice constants of the two moiré lattices (graphene-on-graphene and graphene-on-hexagonal boron nitride), comprising both commensurate periodic and incommensurate quasiperiodic crystals. Remarkably, the 1:1 commensurate crystals, which should theoretically exist at only one point on this phase diagram, are observed over a wide range, demonstrating an unexpected self-alignment mechanism. The incommensurate crystals include quasicrystals, which are quasiperiodic and feature a Bravais-forbidden dodecagonal symmetry, and intercrystals, which are also quasiperiodic but lack forbidden symmetries. This rich variety of tunable double-moiré structures offers a synthetic platform for exploring the unique electronic properties of quasiperiodic crystals, which are rarely found in nature.

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Fig. 1: Imaging GG and GBN double-moiré patterns and the preferred local stacking order.
Fig. 2: Double-moiré patterns.
Fig. 3: Phase diagram of LGBN versus LGG.
Fig. 4: Mechanism of moiré self-alignment.
Fig. 5: Mapping the electronic states of moiré crystals, MICs and MQCs.

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All data are available in the article and its Supplementary Information.

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Acknowledgements

We thank P. Steinhardt, E. Kaxiris and Z. Zhang for insightful discussions, E. Kim and K. Mallayya for help with machine learning tools, N. Tilak for help with the sample fabrication, D. Guerci and J. Wilson for collaborations on related topics and S. Fang for discussions at the early stages of this work. Funding for this project was provided by the Department of Energy grant no. DOE-FG02-99ER45742 (X.L., A.M.C. and E.Y.A.); Gordon and Betty Moore Foundation EPiQS initiative grant GBMF9453 (X.L., A.M.C. and E.Y.A.); Rutgers University, SAS (G.L.); NSF CAREER grant no. DMR-1941569 (J.H.P.) and Sloan Research Fellowship through the Alfred P. Sloan Foundation (J.H.P.); Aspen Center for Physics at which part of this work was performed, which is supported by the National Science Foundation grant no. PHY-1607611 (J.H.P.); and Kavli Institute of Theoretical Physics supported in part by NSF under Grants NSF PHY-1748958 and PHY-2309135 (J.H.P. and E.Y.A.).

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Authors and Affiliations

  1. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, USA

    Xinyuan Lai, Guohong Li, Angela M. Coe, Jedediah H. Pixley & Eva Y. Andrei

  2. Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA

    Jedediah H. Pixley

  3. Center for Materials Theory, Rutgers University, Piscataway, NJ, USA

    Jedediah H. Pixley

  4. Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe & Takashi Taniguchi

Authors
  1. Xinyuan Lai
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Contributions

E.Y.A. conceived and supervised the project. X.L. fabricated, characterized the samples and performed the STM measurements with input from G.L., A.M.C. and E.Y.A. X.L., G.L., J.H.P. and E.Y.A. analysed and interpreted the results. T.T. and K.W. synthesized the hBN crystals. X.L., G.L. and E.Y.A. wrote the paper with input from all authors. All authors discussed the results.

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Correspondence to Eva Y. Andrei.

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Nature Materials thanks Xiaohui Qiu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Classification of 2D moiré periodic and quasiperiodic crystals.

A representative FFT and the corresponding topography (insets) of moiré crystal, MIC and MQC is shown on the right of the table. The right panel schematically illustrates the different moiré structures in GG/GBN. The scale bars are 0.16 nm-1 for the FFTs and 16 nm for the topographies.

Supplementary information

Supplementary Information

Supplementary Figs. 1–21, Tables 1–3 and Discussion.

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Lai, X., Li, G., Coe, A.M. et al. Moiré periodic and quasiperiodic crystals in heterostructures of twisted bilayer graphene on hexagonal boron nitride. Nat. Mater. 24, 1019–1026 (2025). https://doi.org/10.1038/s41563-025-02222-w

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