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Robust supermoiré pattern in large-angle single-twist bilayers

Nature Physics volume 21, pages 1085–1092 (2025)Cite this article

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Abstract

Forming long-wavelength moiré superlattices in van der Waals bilayers that have a small-angle twist between the two layers has been a key approach for creating moiré flat bands. However, for small twist angles, strong lattice reconstruction creates domain walls and other forms of disorder in the moiré pattern, posing considerable challenges for engineering such platforms. At large twist angles, the lattices are more rigid, but it is difficult to produce flat bands in shorter-wavelength moiré superlattices. Here we introduce an approach for tailoring robust supermoiré structures in bilayers of transition-metal dichalcogenides using only a single twist near a commensurate angle. Structurally, we show the spontaneous formation of a periodic arrangement of three inequivalent commensurate moiré stackings, where the angle deviation from the commensurate angle determines the periodicity. Electronically, we reveal a large set of van Hove singularities that indicate strong band hybridization, leading to flat bands near the valence band maximum. Our study extends the study of the interplay among band topology, quantum geometry and moiré superconductivity to the large twist angle regime.

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Fig. 1: CM and SM patterns in near 32.2° tWSe2.
Fig. 2: Gliding translation relation and SM symmetry.
Fig. 3: Dense VHSs in the SM lattice.
Fig. 4: Continuum model for the SM lattice.
Fig. 5: Quasiparticle interference in a θt ≈ 31.1° SM.

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Data availability

Source data for the plots in Supplementary Information are available upon request. Other data are available via figshare at https://doi.org/10.6084/m9.figshare.28558070 (ref. 59). Source data are provided with this paper.

Code availability

The codes described in Supplementary Information are available upon reasonable request.

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Acknowledgements

We acknowledge Y.-W. Son for a useful discussion. Y.L., F.Z., H.K. and C.-K.S. are mainly supported by the NSF through the Center for Dynamics and Control of Materials: an NSF Materials Research Science and Engineering Center (Cooperative Agreement No. DMR-2308817) and partially by the US Air Force (Grant No. FA2386-21-1-4061), NSF (Grant Nos. DMR-1808751 and DMR-2219610) and the Welch Foundation (Grant No. F-2164). C.S. and Y.H. acknowledge support from the NSF (Grant No. CMMI-2239545) and the Welch Foundation (Grant No. C-2065). V.-A.H. and F.G. were supported by the Welch Foundation (Grant No. F-2139-20230405). V.-A.H. and F.G. acknowledge the National Energy Research Scientific Computing Center (a DOE Office of Science User Facility supported under Contract No. DE-AC02-05CH11231) and the Texas Advanced Computing Center at the University of Texas at Austin for providing computational resources. X. Liu and X. Li were partially supported by the Department of Energy, Office of Basic Energy Sciences (Grant No. DE-SC0019398 for device fabrication) and the Welch Foundation chair (Grant No. F-0014 for sample preparation). C.D. and J.A.R. were supported by the Penn State Center for Nanoscale Science (NSF Grant No. DMR-2011839) and the Penn State 2DCC-MIP (NSF Grant No. DMR-1539916). Y.-C.L. acknowledges the support from the Center for Emergent Functional Matter Science of NYCU and the Yushan Young Scholar Program from the Ministry of Education of Taiwan. L.N.H., M.H., J.H. and K.B are supported by the NSF MRSEC programme at Columbia through the Center for Precision-Assembled Quantum Materials (NSF Grant No. DMR-2011738). Y.X., Q.G. and E.K. are supported by the NSF through the Center for Dynamics and Control of Materials: an NSF Materials Research Science and Engineering Center (Cooperative Agreement No. DMR-2308817).

Author information

Author notes

  1. These authors contributed equally: Yanxing Li, Chuqiao Shi, Fan Zhang, Xiaohui Liu.

Authors and Affiliations

  1. Department of Physics, The University of Texas at Austin, Austin, TX, USA

    Yanxing Li, Fan Zhang, Xiaohui Liu, Yuan Xue, Viet-Anh Ha, Nicolás Morales-Durán, Hyunsue Kim, Xiaoqin Li, Feliciano Giustino & Chih-Kang Shih

  2. Department of Materials Science and NanoEngineering, Rice University, Houston, TX, USA

    Chuqiao Shi & Yimo Han

  3. Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

    Viet-Anh Ha & Feliciano Giustino

  4. Department of Physics, Harvard University, Cambridge, MA, USA

    Qiang Gao & Eslam Khalaf

  5. Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA, USA

    Chengye Dong & Yu-Chuan Lin

  6. Two-Dimensional Crystal Consortium, The Pennsylvania State University, University Park, PA, USA

    Chengye Dong & Joshua A. Robinson

  7. Department of Materials Science and Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan

    Yu-Chuan Lin

  8. Department of Applied Physics and Applied Mathematics, Columbia University, New York City, NY, USA

    Luke N. Holtzman & Katayun Barmak

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

    Nicolás Morales-Durán

  10. Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA

    Yi Jiang

  11. Department of Mechanical Engineering, Columbia University, New York City, NY, USA

    Madisen Holbrook & James Hone

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Contributions

C.-K.S. and Y.L. conceived the experiment. Y.L. and F.Z. carried out the STM and STS measurements. C.S. and Y.H. carried out the STEM measurements. C.S. and Y.J. performed the electron ptychography reconstruction. Y.X., Q.G. and E.K. performed the theoretical model calculations. N.M.-D. was involved in the discussion of the continuum model of TMDs. V.-A.H. and F.G. performed the density functional theory calculations. Y.-C.L. synthesized the twisted WSe2 bilayers. C.D. prepared the graphitic buffer layer/SiC. J.A.R. supervised the sample preparation effort. H.K. annealed and pretreated the samples. X. Liu and Y.L. prepared the exfoliated samples for STEM and STM under the supervision of X. Li. L.N.H., M.H., K.B. and J.H. synthesized the WSe2 bulk crystals. Y.L., F.Z., Y.H., C.S. and C.-K.S. analysed the STM and STEM data. Y.L., C.S., Y.H., E.K. and C.-K.S. wrote the paper with contributions from all authors.

Corresponding authors

Correspondence to Yimo Han or Chih-Kang Shih.

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Nature Physics thanks Jiamin Xue 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 STM/S measurement of a fabricated θt ≈ 31.5° twisted bilayer WSe2 SM.

a, An STM image of the 31.5° twisted bilayer WSe2, showing a <1011 cm−2 low density of defects thanks to the high-quality bulk crystal WSe2 (VBias = −0.98 V, I = 100 pA). b, An in-gap taken STM topography image of θt ≈ 31.5° tWSe2 (VBias = −0.6 V, I = 100 pA), revealing unambiguous SM patterns. c, The FFT image of b. The black squares, blue circles, and red diamonds represent top layer WSe2 Bragg peaks, second order and first order tWSe2 moiré, with SM splitting, respectively. d, Filtered iFFT image using the FFT peaks in red diamonds. e, The logarithmic scale constant height STS taken in the pristine region (Vset = −2.0 V, I = 100 pA, Vamp = 29 mV). Blue and green arrows represent the \({\Gamma }_{1}\) and KVBM respectively. f, The constant current STS taken at the same site (Vint = −1.4 V, I = 50 pA, Vamp = 19 mV). Blue and green arrows represent the \({\Gamma }_{1}\) and KVBM respectively. Scale bars: 3 nm.

Extended Data Fig. 2 Quasiparticle interferences measurement and analysis on the fabricated θt ≈ 31.5° twisted bilayer WSe2 SM.

a–c, The constant current dI/dV mappings taken at −1.0 V (a), −0.98 V (b), and −0.96 V (c) (I = 100 pA, Vamp = 20 mV). d, Schematic of the intra-K-valley scattering in θt ≈ 31.5° SM. The purple and grey BZs depict the CM and SM BZs, respectively. ∆SM is the SM minigap while ∆CM is the CM minigap. The blue arrow exhibits the q vector from intra-K-valley scatterings. The SM γ overlaps with CM Γm and Km valleys. e–g, The FFT images of a-c, Red boxes label the regions of quasiparticle interference q vector rings near the center. Insets: magnified near-zero FFT frequency images in the red boxes, showing the rings of quasiparticle interference. h, E v.s. q/2 plot extracted from the quasiparticle interferences of θt ≈ 31.5° tWSe2 sample. The quasiparticle interferences measured under 77K can infer the intra-K-valley scatterings information. The blue curve shows the parabolic fitting of K-valley band dispersion from the three quasiparticle interference measurement dots. The fitted effective mass and KVBM energy levels are meff = 0.55 ± 0.1, EKVBM = −0.925 ± 0.1 eV. The vertical error bars, shown as grey lines, are set as 20 meV in E due to local field fluctuation. The horizontal error bars, also shown as grey lines, are from the q estimations in each Fourier transferred images, which is limited by the image resolution, field of view and quasiparticle interference ring width. Scale bars: a, b, c, 3 nm.

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Li, Y., Shi, C., Zhang, F. et al. Robust supermoiré pattern in large-angle single-twist bilayers. Nat. Phys. 21, 1085–1092 (2025). https://doi.org/10.1038/s41567-025-02914-9

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