Abstract
Moiré superlattices based on van der Waals bilayers1,2,3,4 created at small twist angles lead to a long wavelength pattern with approximate translational symmetry. At large twist angles (θt), moiré patterns are, in general, incommensurate except for a few discrete angles. Here we show that large-angle twisted bilayers offer distinctly different platforms. More specifically, by using twisted tungsten diselenide bilayers, we create the incommensurate dodecagon quasicrystals at θt = 30° and the commensurate moiré crystals at θt = 21.8° and 38.2°. Valley-resolved scanning tunnelling spectroscopy shows disparate behaviours between moiré crystals (with translational symmetry) and quasicrystals (with broken translational symmetry). In particular, the K valley shows rich electronic structures exemplified by the formation of mini-gaps near the valence band maximum. These discoveries demonstrate that bilayers with large twist angles offer a design platform to explore moiré physics beyond those formed with small twist angles.
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Data availability
Source data that reproduce the plots in the main text and extended data figures are provided with this paper. Source data that reproduce the plots in the Supplementary Information are available on request.
Code availability
The DFT calculations presented in the paper were carried out using publicly available electronic structure codes (referenced in Methods). All other codes in the Supplementary Information are available upon reasonable request.
References
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).
Zhang, C. et al. Interlayer couplings, Moiré patterns, and 2D electronic superlattices in MoS2/WSe2 hetero-bilayers. Sci. Adv. 3, e1601459 (2017).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Jin, C. et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature 567, 76–80 (2019).
Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567, 71–75 (2019).
Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. Nature 567, 66–70 (2019).
Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).
Zhang, Z. et al. Flat bands in twisted bilayer transition metal dichalcogenides. Nat. Phys. 16, 1093–1096 (2020).
Zhang, L. et al. Twist-angle dependence of moiré excitons in WS2/MoSe2 heterobilayers. Nat. Commun. 11, 5888 (2020).
Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).
Li, H. et al. Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices. Nat. Mater. 20, 945–950 (2021).
Bai, Y. et al. Excitons in strain-induced one-dimensional moiré potentials at transition metal dichalcogenide heterojunctions. Nat. Mater. 19, 1068–1073 (2020).
Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019).
Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).
Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021).
Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021).
Wang, X. et al. Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides. Nat. Nanotechnol. 17, 367–371 (2022).
Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).
Weston, A. et al. Interfacial ferroelectricity in marginally twisted 2D semiconductors. Nat. Nanotechnol. 17, 390–395 (2022).
Rozhkov, A. V., Sboychakov, A. O., Rakhmanov, A. L. & Nori, F. Electronic properties of graphene-based bilayer systems. Phys. Rep. 648, 1–104 (2016).
Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Continuum model of the twisted graphene bilayer. Phys. Rev. B 86, 155449 (2012).
Mele, E. J. Commensuration and interlayer coherence in twisted bilayer graphene. Phys. Rev. B 81, 161405 (2010).
Sboychakov, A. O., Rakhmanov, A. L., Rozhkov, A. V. & Nori, F. Electronic spectrum of twisted bilayer graphene. Phys. Rev. B 92, 075402 (2015).
Rozhkov, A. V., Sboychakov, A. O., Rakhmanov, A. L. & Nori, F. Single-electron gap in the spectrum of twisted bilayer graphene. Phys. Rev. B 95, 045119 (2017).
Ahn, S. J. et al. Dirac electrons in a dodecagonal graphene quasicrystal. Science 361, 782–786 (2018).
Yao, W. et al. Quasicrystalline 30° twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling. Proc. Natl Acad. Sci. USA 115, 6928–6933 (2018).
Pezzini, S. et al. 30°-twisted bilayer graphene quasicrystals from chemical vapor deposition. Nano Lett. 20, 3313–3319 (2020).
Nguyen, P. V. et al. Visualizing electrostatic gating effects in two-dimensional heterostructures. Nature 572, 220–223 (2019).
Bisri, S. Z., Shimizu, S., Nakano, M. & Iwasa, Y. Endeavor of iontronics: from fundamentals to applications of ion‐controlled electronics. Adv. Mater. 29, 1607054 (2017).
Zhang, C. et al. Probing critical point energies of transition metal dichalcogenides: surprising indirect gap of single layer WSe2. Nano Lett. 15, 6494–6500 (2015).
Koren, E. et al. Coherent commensurate electronic states at the interface between misoriented graphene layers. Nat. Nanotechnol. 11, 752–757 (2016).
Inbar, A. et al. The quantum twisting microscope. Nature 614, 682–687 (2023).
Chari, T., Ribeiro-Palau, R., Dean, C. R. & Shepard, K. Resistivity of rotated graphite–graphene contacts. Nano Lett. 16, 4477–4482 (2016).
Zhao, X. et al. Strong moiré excitons in high-angle twisted transition metal dichalcogenide homobilayers with robust commensuration. Nano Lett. 22, 203–210 (2022).
Weston, A. et al. Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol. 15, 592–597 (2020).
McCreary, K. M. et al. Stacking-dependent optical properties in bilayer WSe2. Nanoscale 14, 147–156 (2022).
Hsu, W.-T. et al. Quantitative determination of interlayer electronic coupling at various critical points in bilayer MoS2. Phys. Rev. B 106, 125302 (2022).
Hsu, W.-T. et al. Tailoring excitonic states of van der Waals bilayers through stacking configuration, band alignment, and valley spin. Sci. Adv. 5, eaax7407 (2019).
Zhang, Y., Devakul, T. & Fu, L. Spin-textured Chern bands in AB-stacked transition metal dichalcogenide bilayers. Proc. Natl Acad. Sci. USA 118, e2112673118 (2021).
Uri, A. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023).
Lin, Y.-C. et al. Realizing large-scale, electronic-grade two-dimensional semiconductors. ACS Nano 12, 965–975 (2018).
Giannozzi, P. et al. Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
Giannozzi, P. et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys. Condens. Matter 29, 465901 (2017).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Hamann, D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Phys. Rev. B 88, 085117 (2013).
van Setten, M. J. et al. The PseudoDojo: training and grading a 85 element optimized norm-conserving pseudopotential table. Comput. Phys. Commun. 226, 39–54 (2018).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H–Pu. J. Chem. Phys. 132, 154104 (2010).
Medeiros, P. V. C., Stafström, S. & Björk, J. Effects of extrinsic and intrinsic perturbations on the electronic structure of graphene: retaining an effective primitive cell band structure by band unfolding. Phys. Rev. B 89, 041407 (2014).
Medeiros, P. V. C., Tsirkin, S. S., Stafström, S. & Björk, J. Unfolding spinor wave functions and expectation values of general operators: introducing the unfolding-density operator. Phys. Rev. B 91, 041116 (2015).
Iraola, M. et al. IrRep: symmetry eigenvalues and irreducible representations of ab initio band structures. Comput. Phys. Commun. 272, 108226 (2022).
Acknowledgements
This work was primarily supported by the NSF through the Center for Dynamics and Control of Materials: an NSF Materials Research Science and Engineering Center under cooperative agreement nos. DMR-1720595, DMR-2308817 and the US Air Force grant no. FA2386-21-1-4061. Other supports were from NSF grant nos. DMR-1808751 and DMR-2219610 and the Welch Foundation F-2164. V.-A.H. and F.G. were supported by the Welch Foundation (grant no. F-2139−20230405) and the National Science Foundation (grant no. 2103991). V.-A.H. and F.G. used the resources of the National Energy Research Scientific Computing Center and the Argonne Leadership Computing Facility, which are DOE Office of Science User Facilities supported by the Office of Science of the US Department of Energy (DOE) (contract nos. DE-AC02-05CH11231 and DE-AC02-06CH11357, respectively). V.-A.H. and F.G. acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing access to Frontera, Lonestar6 and Texascale Days, which contributed to the research results reported in this paper (http://www.tacc.utexas.edu). Work at Penn State University was 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 (CEFMS) of NYCU and the Yushan Young Scholar Program from the Ministry of Education of Taiwan. E.R. and S.H.R. acknowledge the funding of the QSA, supported by the US DOE, Office of Science, National Quantum Information Science Research Centers. This research used the resources of the Advanced Light Source, which is a DOE, Office of Science User Facility (contract no. DE-AC02-05CH11231). Z.L., X. Liu and X. Li acknowledge support from the National Science Foundation (NSF grant no. ECCS-2130552), the DOE, Office of Basic Energy Sciences (grant no. DE-SC0019398) and the Welch Foundation (grant no. F-1662). K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant nos. 20H00354, 21H05233 and 23H02052) and the World Premier International Research Center Initiative (WPI), NEXT, Japan.
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Contributions
C.-K.S. conceived the experiment. Y.L. and F.Z. carried out the STM and STS measurements. V.-A.H. and F.G. performed the DFT 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. helped anneal the sample and perform the low-energy electron diffraction measurements. Q.G., B.K. and E.K. performed the theoretical model calculations. Z.L. and X. Liu prepared the exfoliated the sample. X. Li was involved in the discussion. K.W. and T.T. synthesized the hBN bulk crystals. S.H.R. and E.R. performed the nano-ARPES measurement and analysed the ARPES data. C.J. and A.B. helped with the nano-ARPES set-up. Y.L., F.Z. and C.-K.S. analysed the STM data. Y.L., F.Z. and C.-K.S. wrote the paper with contributions from all the authors.
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Extended data figures and tables
Extended Data Fig. 1 Comparison of valley assignments between CCSTS and nano-ARPES.
a, The zoom-in CCSTS taken on quasicrystal. (same as Fig. 3d) b, The zoom in K-valley spectrum from the nano-ARPES measurements. c, The zoom in Γ-valley spectrum from the nano-ARPES measurements.
Extended Data Fig. 2 nano-ARPES results for quasicrystal.
a, The constant energy surface at E = −1.9 eV. The trigonal K-valleys are labeled by the blue and red solid dots. The blue and red hole circles label the expected locations of the 1st-order Umklapp scatterings. However, no clear features of Umklapp replicas are observed. b, Energy distribution curve (EDC) at the Γ. Two peaks represent Γ1 (−1.97 eV) and Γ2 (−2.44 eV) respectively. c, The measured band structure across the Γ (ky = 0 Å−1). d, The measured band structure across the K (ky = −0.4 Å−1), the KVBM is estimated to be at −1.7 eV. e, The measured band structure along the Kt-Kb direction (kx = −1.4 Å−1) shows the bands crossing. The energy level of the crossing point is determined by the saddle point, at E = −2.23 eV.
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Supplementary Figs 1–29 and Supplementary Notes 1–7.
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Li, Y., Zhang, F., Ha, VA. et al. Tuning commensurability in twisted van der Waals bilayers. Nature 625, 494–499 (2024). https://doi.org/10.1038/s41586-023-06904-w
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DOI: https://doi.org/10.1038/s41586-023-06904-w
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