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Imaging two-dimensional generalized Wigner crystals

Abstract

The Wigner crystal1 has fascinated condensed matter physicists for nearly 90 years2,3,4,5,6,7,8,9,10,11,12,13,14. Signatures of two-dimensional (2D) Wigner crystals were first observed in 2D electron gases under high magnetic field2,3,4, and recently reported in transition metal dichalcogenide moiré superlattices6,7,8,9. Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Conventional scanning tunnelling microscopy (STM) has sufficient spatial resolution but induces perturbations that can potentially alter this fragile state. Here we demonstrate real-space imaging of 2D Wigner crystals in WSe2/WS2 moiré heterostructures using a specially designed non-invasive STM spectroscopy technique. This employs a graphene sensing layer held close to the WSe2/WS2 moiré superlattice. Local STM tunnel current into the graphene layer is modulated by the underlying Wigner crystal electron lattice in the WSe2/WS2 heterostructure. Different Wigner crystal lattice configurations at fractional electron fillings of n = 1/3, 1/2 and 2/3, where n is the electron number per site, are directly visualized. The n = 1/3 and n = 2/3 Wigner crystals exhibit triangular and honeycomb lattices, respectively, to minimize nearest-neighbour occupations. The n = 1/2 state spontaneously breaks the original C3 symmetry and forms a stripe phase. Our study lays a solid foundation for understanding Wigner crystal states in WSe2/WS2 moiré heterostructures and provides an approach that is generally applicable for imaging novel correlated electron lattices in other systems.

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Fig. 1: STM measurement of Wigner crystal states in a dual-gated WSe2/WS2 moiré superlattice.
Fig. 2: Imaging Mott and generalized Wigner crystal states.
Fig. 3: Evolution of dI/dV maps for the n = 2/3 state with increased Vbias.

Data availability

The data supporting the findings of this study can be found at https://github.com/HongyuanLiCMP/Imaging_Generalized_Wigner_Crystals_data, and are also available from the corresponding authors upon reasonable request.

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Acknowledgements

This work was primarily funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 (van der Waals heterostructure program KCFW16) (device electrode preparation and STM spectroscopy). Support was also provided by the US Army Research Office under MURI award W911NF-17-1-0312 (device layer transfer), and by the National Science Foundation Award DMR-1807233 (surface preparation). S.T. acknowledges support from DOE-SC0020653, NSF DMR 2111812, NSF DMR 1552220, NSF 2052527, DMR 1904716 and NSF CMMI 1933214 for WSe2 and WS2 bulk crystal growth and analysis. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant number JPMXP0112101001, JSPS KAKENHI grant number JP20H00354 and the CREST(JPMJCR15F3), JST for bulk hBN crystal growth and analysis. E.C.R. acknowledges support from the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. S.L. acknowledges support from Kavli ENSI Heising Simons Junior Fellowship. We also thank M. H. Naik for sharing unpublished theoretical simulation data on the WSe2/WS2 moiré superlattice.

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Authors

Contributions

M.F.C. and F.W. conceived the project. H.L. and S.L. performed the STM measurement, H.L., E.C.R., D.W., W.Z. and S.K. fabricated the heterostructure device and performed the SHG measurement. K.Y., M.B. and S.T. grew WSe2 and WS2 crystals. K.W. and T.T. grew the hBN single crystal. All authors discussed the results and wrote the manuscript.

Corresponding authors

Correspondence to Shaowei Li, Michael F. Crommie or Feng Wang.

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The authors declare no competing interests.

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Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Comparison of single dI/dV spectra obtained at VTG = 0 and VTG = 0.53 V.

a, VTG = 0. b, VTG = 0.53 V. In a we display dI/dV spectra obtained when the graphene doping is near the CNP. A strong spectral change is observed when the graphene transitions from hole-doped to electron-doped. The Dirac point positions are denoted by vertical arrows. In b we display typical dI/dV spectra at n = 1/3, 1/2, 2/3 and 1 for correlated states (red) as well as for three other filling factors that lack correlated states (black). In each panel the dI/dV spectra are shifted vertically for clarity. The spectra indicate that the graphene sensing layer is more electron doped when the moiré heterostructure is in a correlated insulator state.

Extended Data Fig. 2 Moiré site dependence of the dI/dV spectra.

a, A typical STM topographic image of the moiré superlattice seen through the graphene sensing layer. bd, Position dependent dI/dV spectra measured along the red linecut shown in a with VTG = 0.7 V and VBG = 19 V (b), 26.5 V and 35 V (d).

Extended Data Fig. 3 Raw images and FFT filtering of the dI/dV maps for the generalized Wigner crystal states.

a, Raw dI/dV map of the n = 2/3 state. b, FFT image of a. c, Real space dI/dV map after FFT filtering of a. In the filtering process, we removed the Fourier components within the six red circles indicated in b. This FFT filtering suppresses the periodic feature associated with the moiré superlattice. d, FFT image of c. e, Raw dI/dV map of the n = 1/3 state. f, FFT image of e. g, Real space dI/dV map after FFT filtering of e. The Fourier components within the red circles shown in f have been filtered out. h, FFT image of g. i, Raw dI/dV map of the n = 1/2 state. j, FFT image of i. k, Real-space dI/dV map after FFT filtering of i. The Fourier components within the red circles shown in j have been filtered out. l, FFT image of k.

Extended Data Fig. 4 Uniaxial strain of the moiré superlattice.

a, Topography image shown in Fig. 1b. b, the corresponding FFT image. c, d, 1D height modulation along three directions (c; denoted by the red arrows in a), and the corresponding FFT results (d). The different moiré periods along the three different directions yields a uniaxial strain of 0.39% along the pink double-arrowed line (a, b). As a comparison, the stripe direction of the n = 1/2 Wigner crystal state is denoted by the yellow double-arrowed line.

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Li, H., Li, S., Regan, E.C. et al. Imaging two-dimensional generalized Wigner crystals. Nature 597, 650–654 (2021). https://doi.org/10.1038/s41586-021-03874-9

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