Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Exciton density waves in Coulomb-coupled dual moiré lattices

Nature Materials volume 22, pages 175–179 (2023)Cite this article

Subjects

Abstract

Strongly correlated bosons in a lattice are a platform that can realize rich bosonic states of matter and quantum phase transitions1. While strongly correlated bosons in a lattice have been studied in cold-atom experiments2,3,4, their realization in a solid-state system has remained challenging5. Here we trap interlayer excitons–bosons composed of bound electron–hole pairs, in a lattice provided by an angle-aligned WS2/bilayer WSe2/WS2 multilayer. The heterostructure supports Coulomb-coupled triangular moiré lattices of nearly identical period at the top and bottom interfaces. We observe correlated insulating states when the combined electron filling factor of the two lattices, with arbitrary partitions, equals \(\frac{1}{3},\frac{2}{3},\frac{4}{3}\) and \(\frac{5}{3}\). These states can be interpreted as exciton density waves in a Bose–Fermi mixture of excitons and holes6,7. Because of the strong repulsive interactions between the constituents, the holes form robust generalized Wigner crystals8,9,10,11, which restrict the exciton fluid to channels that spontaneously break the translational symmetry of the lattice. Our results demonstrate that Coulomb-coupled moiré lattices are fertile ground for correlated many-boson phenomena12,13.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Coulomb-coupled moiré lattices.
Fig. 2: Electrostatic phase diagram of coupled moiré lattices.
Fig. 3: Correlated insulating states in coupled moiré lattices.
Fig. 4: Exciton density waves.

Similar content being viewed by others

Data availability

Source data are provided with this paper. Additional data that support the findings of this study are available from the corresponding authors upon reasonable request.

References

  1. Dutta, O. et al. Non-standard Hubbard models in optical lattices: a review. Rep. Prog. Phys. 78, 066001 (2015).

    Article  Google Scholar 

  2. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).

    Article  CAS  Google Scholar 

  3. Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    Article  CAS  Google Scholar 

  4. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313–2363 (2010).

    Article  CAS  Google Scholar 

  5. Lagoin, C., Suffit, S., Baldwin, K., Pfeiffer, L. & Dubin, F. Mott insulator of strongly interacting two-dimensional semiconductor excitons. Nat. Phys. 18, 149–153 (2022).

    Article  CAS  Google Scholar 

  6. Büchler, H. P. & Blatter, G. Supersolid versus phase separation in atomic Bose-Fermi mixtures. Phys. Rev. Lett. 91, 130404 (2003).

    Article  Google Scholar 

  7. Titvinidze, I., Snoek, M. & Hofstetter, W. Supersolid Bose-Fermi mixtures in optical lattices. Phys. Rev. Lett. 100, 100401 (2008).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  9. Li, H. et al. Imaging two-dimensional generalized Wigner crystals. Nature 597, 650–654 (2021).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  12. Zhang, Y.-H. Doping a Mott insulator with excitons in moiré bilayer: fractional superfluid, neutral Fermi surface and Mott transition. Phys. Rev. B 106, 195120 (2022).

    Article  CAS  Google Scholar 

  13. Zhang, Y.-H., Sheng, D. N. & Vishwanath, A. SU(4) chiral spin liquid, exciton supersolid, and electric detection in moiré bilayers. Phys. Rev. Lett. 127, 247701 (2021).

    Article  CAS  Google Scholar 

  14. Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).

    Article  CAS  Google Scholar 

  15. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    Article  CAS  Google Scholar 

  16. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

    Article  CAS  Google Scholar 

  17. Andrei, E. Y. et al. The marvels of moiré materials. Nat. Rev. Mater. 6, 201–206 (2021).

    Article  CAS  Google Scholar 

  18. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  CAS  Google Scholar 

  19. Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    Article  CAS  Google Scholar 

  20. Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).

    Article  CAS  Google Scholar 

  21. Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).

    Article  CAS  Google Scholar 

  22. Huang, D., Choi, J., Shih, C.-K. & Li, X. Excitons in semiconductor moiré superlattices. Nat. Nanotechnol. 17, 227–238 (2022).

    Article  CAS  Google Scholar 

  23. Wilson, N. P., Yao, W., Shan, J. & Xu, X. Excitons and emergent quantum phenomena in stacked 2D semiconductors. Nature 599, 383–392 (2021).

    Article  CAS  Google Scholar 

  24. Burg, G. W. et al. Strongly enhanced tunneling at total charge neutrality in double-bilayer graphene-WSe2 heterostructures. Phys. Rev. Lett. 120, 177702 (2018).

    Article  CAS  Google Scholar 

  25. Butov, L. V. Condensation and pattern formation in cold exciton gases in coupled quantum wells. J. Phys. Condens. Matter 16, R1577–R1613 (2004).

    Article  CAS  Google Scholar 

  26. Eisenstein, J. P. & MacDonald, A. H. Bose–Einstein condensation of excitons in bilayer electron systems. Nature 432, 691–694 (2004).

    Article  CAS  Google Scholar 

  27. Fogler, M. M., Butov, L. V. & Novoselov, K. S. High-temperature superfluidity with indirect excitons in van der Waals heterostructures. Nat. Commun. 5, 4555 (2014).

    Article  CAS  Google Scholar 

  28. Liu, X. et al. Crossover between strongly coupled and weakly coupled exciton superfluids. Science 375, 205–209 (2022).

    Article  CAS  Google Scholar 

  29. Ma, L. et al. Strongly correlated excitonic insulator in atomic double layers. Nature 598, 585–589 (2021).

    Article  CAS  Google Scholar 

  30. Perali, A., Neilson, D. & Hamilton, A. R. High-temperature superfluidity in double-bilayer graphene. Phys. Rev. Lett. 110, 146803 (2013).

    Article  CAS  Google Scholar 

  31. Wu, F.-C., Xue, F. & MacDonald, A. H. Theory of two-dimensional spatially indirect equilibrium exciton condensates. Phys. Rev. B 92, 165121 (2015).

    Article  Google Scholar 

  32. Yu, E., Lozovik & Yudson, V. I. A new mechanism for superconductivity: pairing between spatially separated electrons and holes. Zh. Eksp. Teor. Fiz. 71, 738–753 (1976).

    Google Scholar 

  33. Gu, J. et al. Dipolar excitonic insulator in a moiré lattice. Nat. Phys. 18, 395–400 (2022).

    Article  CAS  Google Scholar 

  34. Zeng, Z. et al. Correlated interlayer exciton insulator in heterostructures of monolayer WSe2 and moiré WS2/WSe2. Nat. Phys. 18, 1214–1220 (2021).

    Article  Google Scholar 

  35. Chen, X. M. & Quinn, J. J. Excitonic charge-density-wave instability of spatially separated electron-hole layers in strong magnetic fields. Phys. Rev. Lett. 67, 895–898 (1991).

    Article  CAS  Google Scholar 

  36. Joglekar, Y. N., Balatsky, A. V. & Das Sarma, S. Wigner supersolid of excitons in electron-hole bilayers. Phys. Rev. B 74, 233302 (2006).

    Article  Google Scholar 

  37. Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).

    Article  Google Scholar 

  38. Tang, Y. et al. Dielectric catastrophe at the Mott and Wigner transitions in a moiré superlattice. Nat. Commun. 13, 4271 (2022).

    Article  CAS  Google Scholar 

  39. Xu, Y. et al. A tunable bilayer Hubbard model model in twisted WSe2. Nat. Nanotechnol. 17, 934–939 (2022).

    Article  CAS  Google Scholar 

  40. Kumar, A., Hu, N. C., MacDonald, A. H. & Potter, A. C. Gate-tunable heavy fermion quantum criticality in a moiré Kondo lattice. Phys. Rev. B 106, L041116 (2022).

    Article  CAS  Google Scholar 

  41. Zeng, Y., Wei, N. & MacDonald, A. H. Layer pseudospin magnetism in a transition-metal-dichalcogenide double-moiré system. Phys. Rev. B 106, 165105 (2022).

    Article  CAS  Google Scholar 

  42. Su, J.-J. & MacDonald, A. H. How to make a bilayer exciton condensate flow. Nat. Phys. 4, 799–802 (2008).

    Article  CAS  Google Scholar 

  43. Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe2 bilayers. Nano Lett. 18, 137–143 (2018).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We thank A. H. MacDonald, E. J. Mueller, Y.-H. Zhang and A. Vishwanath for fruitful discussions. This study was supported by the National Science Foundation (NSF) under DMR-2114535 (optical measurements), the US Office of Naval Research under award no. N00014-21-1-2471 (device fabrication) and the US Department of Energy, Office of Science, Basic Energy Sciences, under award number DE-SC0019481 (analysis). The study was also funded in part by the Gordon and Betty Moore Foundation. It made use of the Cornell NanoScale Facility, a National Nanotechnology Coordinated Infrastructure (NNCI) member supported by NSF grant NNCI-2025233. The growth of the hBN crystals was supported by the Elemental Strategy Initiative of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and the CREST programme (JPMJCR15F3) of the Japan Science and Technology Agency (JST).

Author information

Author notes

  1. These authors contributed equally: Yihang Zeng, Zhengchao Xia.

Authors and Affiliations

  1. Department of Physics, Cornell University, Ithaca, NY, USA

    Yihang Zeng, Roei Dery, Jie Shan & Kin Fai Mak

  2. School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA

    Zhengchao Xia, Jie Shan & Kin Fai Mak

  3. National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe & Takashi Taniguchi

  4. Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA

    Jie Shan & Kin Fai Mak

Authors
  1. Yihang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhengchao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roei Dery
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kenji Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Takashi Taniguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jie Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kin Fai Mak
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y.Z., Z.X. and R.D. fabricated the devices. Y.Z. and Z.X. performed the measurements and analysed the data. K.W. and T.T. grew the bulk hBN crystals. Y.Z., Z.X., K.F.M. and J.S. designed the scientific objectives, oversaw the project and cowrote the manuscript. All authors discussed the results and commented on the manuscript.

Corresponding authors

Correspondence to Jie Shan or Kin Fai Mak.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Materials thanks David Ruiz-Tijerina and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Reflectance contrast (RC) spectra.

a,b, Representative RC spectra of the sensor 2s exciton (a) and the fundamental moiré exciton in WS2 (b) of device S1 with lattice filling (νt, νb). The spectral line shape is given by the optical interference effect in the multiple layer structure on the Si/SiO2 substrate. We use the peak-to-peak amplitude of the RC between 1.836 and 1.853 eV (R2s) as an indicator of the 2s exciton spectral weight (a), and between 1.95 and 2.05 eV (RMX) for the moiré exciton (b).

Extended Data Fig. 2 Optical reflectance contrast spectrum of device S1.

a–c, Gate voltage dependence of the reflectance contrast spectrum covering the 1 s and 2s resonances of the sensor and the WS2 moiré excitons. Electrons are located in the bottom lattice solely (a), the top lattice solely (b) and equally in two lattices (c). The total filling factors ν = 0, 1 and 2 are determined according to the sensor 2s exciton resonance.

Extended Data Fig. 3 Magnetic-field dependence.

Filling dependence of R2s at representative perpendicular magnetic fields. The curves are displaced vertically for clarity. The perpendicular electric field is 2 mV/nm. The black dashed lines mark total filling factor ν = 0, 2/3, 1 and 2, at which insulating states are observed in the absence of the magnetic field. These states do not disperse with magnetic field and are therefore non-topological. The orange line at each magnetic field denotes the peak positions if the state were topological with Chern number 1.

Extended Data Fig. 4 Determination of band alignment in the heterostructure.

a, Left: Simplified band diagram of natural bilayer WSe2. While the valence bands are from the K/K’ valley, the conduction band minimum is from the Q valley. The indirect optical gap is 1.565 eV as determined by the emission peak in the PL spectrum (right). b, Left: Type-II band alignment in the dual-moiré heterostructure. The dashed and solid lines represent spin-down and spin-up bands from the K valley of the monolayer TMDs. The energy scales are determined by PL measurements. Right: PL spectrum of the heterostructure showing the interlayer exciton emission at 1.43 eV and the bilayer WSe2 indirect optical gap at 1.56 eV.

Extended Data Fig. 5 Additional sample regions of device S1.

a,b, Dependence of R2s on the total filling factor and electric field at two additional regions of the same device as in the main figures. Electrons are in both moiré lattices in the region enclosed by the dashed lines. The insulating state at ν = 1 is robust in all regions. Fewer and less robust insulating states are observed here particularly in a. The features appear curved at certain electric fields because the large contact resistance causes nonlinear gating effects. The electric field offset at νt = νb is likely caused by the layer asymmetry in these regions.

Extended Data Fig. 6 Additional devices.

a,b, Dependence of R2s on the total filling factor and electric field for device S2 (a) and S3 (b). Electrons are in both moiré lattices in the region enclosed by the dashed lines. The insulating state at ν = 1 is robust in both devices. The insulating states at total fractional fillings are observed only in device S3, including an insulating state at \(\nu = \frac{1}{2}\). The features appear curved at certain electric fields because the large contact resistance causes nonlinear gating effects.

Extended Data Fig. 7 Exciton density waves in a particle–hole transformation picture.

a,–c, Schematic representation of the correlated insulating states in coupled moiré lattices at total filling ν = 1 (a), \(\nu = \frac{1}{3}\) (b) and \(\nu = \frac{2}{3}\) (c). Top, electrons in the top lattice (red) and bottom lattice (blue). Electrons in the top lattice are bound to the empty sites in the bottom lattice directly below them to minimize the Coulomb interactions. Middle, particle–hole transformation performed on the bottom lattice generates interlayer excitons (red-blue circles) and excess holes (empty blue circles). Bottom, the excess holes form generalized Wigner crystals (or charge-ordered states) because of the hole-hole repulsion V. The excitons are guided to the channels defined by the hole Wigner crystals by the exciton–hole repulsion V′. The exciton density distribution shows an exciton fluid at ν = 1 and exciton density waves at \(\nu = \frac{1}{3},\frac{2}{3}\). The latter breaks the translational symmetry of the lattice.

Source data

Source Data Fig. 1

Statistical source data.

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Statistical source data.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeng, Y., Xia, Z., Dery, R. et al. Exciton density waves in Coulomb-coupled dual moiré lattices. Nat. Mater. 22, 175–179 (2023). https://doi.org/10.1038/s41563-022-01454-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41563-022-01454-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing