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Imaging two-dimensional generalized Wigner crystals


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, and are also available from the corresponding authors upon reasonable request.


  1. Wigner, E. On the interaction of electrons in metals. Phys. Rev. 46, 1002–1011 (1934).

    ADS  CAS  Article  Google Scholar 

  2. Goldman, V., Santos, M., Shayegan, M. & Cunningham, J. Evidence for two-dimentional quantum Wigner crystal. Phys. Rev. Lett. 65, 2189–2192 (1990).

    ADS  CAS  Article  Google Scholar 

  3. Jang, J., Hunt, B. M., Pfeiffer, L. N., West, K. W. & Ashoori, R. C. Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal. Nat. Phys. 13, 340–344 (2017).

    CAS  Article  Google Scholar 

  4. Zhou, H., Polshyn, H., Taniguchi, T., Watanabe, K. & Young, A. Solids of quantum Hall skyrmions in graphene. Nat. Phys. 16, 154–158 (2020).

    CAS  Article  Google Scholar 

  5. Shapir, I. et al. Imaging the electronic Wigner crystal in one dimension. Science 364, 870–875 (2019).

    ADS  CAS  Article  Google Scholar 

  6. Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).

    ADS  CAS  Article  Google Scholar 

  7. Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Nat. Mater. 20, 940–944 (2021).

    ADS  CAS  Article  Google Scholar 

  8. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).

    ADS  CAS  Article  Google Scholar 

  9. Huang, X. et al. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 17, 715–719 (2021).

    CAS  Article  Google Scholar 

  10. Deshpande, V. V. & Bockrath, M. The one-dimensional Wigner crystal in carbon nanotubes. Nat. Phys. 4, 314–318 (2008).

    CAS  Article  Google Scholar 

  11. Crandall, R. & Williams, R. Crystallization of electrons on the surface of liquid helium. Phys. Lett. A 34, 404–405 (1971).

    ADS  CAS  Article  Google Scholar 

  12. Williams, R., Crandall, R. & Willis, A. Surface states of electrons on liquid helium. Phys. Rev. Lett. 26, 7–9 (1971).

    Google Scholar 

  13. Grimes, C. & Adams, G. Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons. Phy. Rev. Lett. 42, 795–798 (1979).

    Article  Google Scholar 

  14. Williams, F. Collective aspects of charged-particle systems at helium interfaces. Surface Sci. 113, 371–388 (1982).

    ADS  CAS  Article  Google Scholar 

  15. Lam, P. K. & Girvin, S. Liquid-solid transition and the fractional quantum-Hall effect. Phy. Rev. B 30, 473–475 (1984).

    Article  Google Scholar 

  16. Levesque, D., Weis, J. & MacDonald, A. Crystallization of the incompressible quantum-fluid state of a two-dimensional electron gas in a strong magnetic field. Phys. Rev. B 30, 1056–1058 (1984).

    Article  Google Scholar 

  17. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett.s 48, 1559–1562 (1982).

    Google Scholar 

  18. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    Article  Google Scholar 

  19. Pan, H., Wu, F. & Sarma, S. D. Quantum phase diagram of a moiré-Hubbard model. Phys. Rev. B 102, 201104 (2020).

    ADS  CAS  Article  Google Scholar 

  20. Hubbard, J. Generalized Wigner lattices in one dimension and some applications to tetracyanoquinodimethane (TCNQ) salts. Phys.l Rev. B 17, 494–505 (1978).

    Google Scholar 

  21. Li, H. et al. Imaging local discharge cascades for correlated electrons in WS2/WSe2 moiré superlattices. Nat. Phys. (2021).

  22. Li, H. et al. Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices. Nat. Mater. 20, 945–950 (2021).

    ADS  CAS  Article  Google Scholar 

  23. Zhang, Y. et al. Giant phonon-induced conductance in scanning tunnelling spectroscopy of gate-tunable graphene. Nat. Phys. 4, 627–630 (2008).

    CAS  Article  Google Scholar 

  24. Jung, S. et al. Evolution of microscopic localization in graphene in a magnetic field from scattering resonances to quantum dots. Nat. Phys. 7, 245–251 (2011).

    CAS  Article  Google Scholar 

  25. Decker, R. et al. Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy. Nano Lett. 11, 2291–2295 (2011).

    ADS  CAS  Article  Google Scholar 

  26. Wong, D. et al. Spatially resolving density-dependent screening around a single charged atom in graphene. Phys. Rev. B 95, 205419 (2017).

    ADS  Article  Google Scholar 

  27. Yang, F. et al. Experimental determination of the energy per particle in partially filled Landau levels. Phys. Rev. Lett. 126, 156802 (2021).

    ADS  CAS  Article  Google Scholar 

  28. Li, T. et al. Charge-order-enhanced capacitance in semiconductor moiré superlattices. Nat. Nanotechnol. (2021).

  29. Tomarken, S. L. et al. Electronic compressibility of magic-angle graphene superlattices. Phys. Rev. Lett. 123, 046601 (2019).

    ADS  CAS  Article  Google Scholar 

  30. Zondiner, U. et al. Cascade of phase transitions and Dirac revivals in magic-angle graphene. Nature 582, 203–208 (2020).

    ADS  CAS  Article  Google Scholar 

  31. Pierce, A. T. et al. Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene. Preprint at (2021).

  32. Pradhan, N. A., Liu, N., Silien, C. & Ho, W. Atomic scale conductance induced by single impurity charging. Phys. Rev. Lett. 94, 076801 (2005).

    ADS  CAS  Article  Google Scholar 

  33. Brar, V. W. et al. Gate-controlled ionization and screening of cobalt adatoms on a graphene surface. Nat. Phys. 7, 43–47 (2011).

    CAS  Article  Google Scholar 

  34. Wong, D. et al. Characterization and manipulation of individual defects in insulating hexagonal boron nitride using scanning tunnelling microscopy. Nat. Nanotechnol. 10, 949–953 (2015).

    ADS  CAS  Article  Google Scholar 

  35. Teichmann, K. et al. Controlled charge switching on a single donor with a scanning tunneling microscope. Phys. Rev. Lett. 101, 076103 (2008).

    ADS  CAS  Article  Google Scholar 

  36. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

    ADS  CAS  Article  Google Scholar 

  37. Schutte, W., De Boer, J. & Jellinek, F. Crystal structures of tungsten disulfide and diselenide. J. Solid State Chem. 70, 207–209 (1987).

    ADS  CAS  Article  Google Scholar 

  38. Kerelsky, A. et al. Maximized electron interactions at the magic angle in twisted bilayer graphene. Nature 572, 95–100 (2019).

    ADS  CAS  Article  Google Scholar 

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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 and Affiliations



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).

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