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.

  • Article
  • Published:

Hofstadter states and re-entrant charge order in a semiconductor moiré lattice

Nature Physics volume 19, pages 1861–1867 (2023)Cite this article

Subjects

Abstract

The emergence of moiré materials with flat bands provides a platform to systematically investigate and precisely control the correlated electronic phases. Here we report on a rich phase diagram of interpenetrating Hofstadter states—also called Chern insulators—and electron solids in a twisted WSe2/MoSe2 heterobilayer using local electronic compressibility measurements. We show that this reflects the presence of both flat and dispersive moiré bands whose relative energies, and therefore occupations, are tuned by the density and magnetic field. At low density, the competition between moiré bands leads to a transition from the commensurate arrangements of singlets at doubly occupied sites to triplet configurations at high fields. Hofstadter states are generally favoured at high density as dispersive bands are populated, but are suppressed by an intervening region of re-entrant charge-ordered states in which holes originating from multiple bands cooperatively crystallize. Our results reveal the key microscopic ingredients that favour distinct correlated ground states in semiconductor moiré systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.

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: Competing Hofstadter and charge-ordered states.
Fig. 2: Tuning between flat and dispersive moiré bands.
Fig. 3: Magnetic-field dependence of charge-ordered states.
Fig. 4: Interpenetrating Hofstadter states.

Similar content being viewed by others

Data availability

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

Code availability

The codes that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Kravchenko, S. V. & Sarachik, M. P. Metal-insulator transition in two-dimensional electron systems. Rep. Prog. Phys. 67, 1–44 (2003).

    ADS  Google Scholar 

  2. Sarma, S. D. & Pinczuk, A. Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-Dimensional Semiconductor Structures (John Wiley & Sons, 2008).

  3. Gross, C. & Bakr, W. S. Quantum gas microscopy for single atom and spin detection. Nat. Phys. 17, 1316–1323 (2021).

    Google Scholar 

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

    Google Scholar 

  5. Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).

    ADS  Google Scholar 

  6. Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. H. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  11. Li, T. et al. Charge-order-enhanced capacitance in semiconductor moiré superlattices. Nat. Nanotechnol. 16, 1068–1072 (2021).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Google Scholar 

  14. Liu, E. et al. Excitonic and valley-polarization signatures of fractional correlated electronic phases in a WSe2/WS2 moiré superlattice. Phys. Rev. Lett. 127, 037402 (2021).

    ADS  Google Scholar 

  15. Chu, Z. et al. Nanoscale conductivity imaging of correlated electronic states in WSe2/WS2 moiré superlattices. Phys. Rev. Lett. 125, 186803 (2020).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  17. Li, T. et al. Continuous Mott transition in semiconductor moiré superlattices. Nature 597, 350–354 (2021).

    ADS  Google Scholar 

  18. Chang, M.-C. & Niu, Q. Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands. Phys. Rev. B 53, 7010–7023 (1996).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  20. Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Negative compressibility of interacting two-dimensional electron and quasiparticle gases. Phys. Rev. Lett. 68, 674–677 (1992).

    ADS  Google Scholar 

  21. Skinner, B. & Shklovskii, B. I. Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas. Phys. Rev. B 82, 155111 (2010).

    ADS  Google Scholar 

  22. Foutty, B. A. et al. Tunable spin and valley excitations of correlated insulators in Γ-valley moiré bands. Nat. Mater. 22, 731–736 (2023).

  23. Dultz, S. C. & Jiang, H. W. Thermodynamic signature of a two-dimensional metal-insulator transition. Phys. Rev. Lett. 84, 4689–4692 (2000).

    ADS  Google Scholar 

  24. Feldman, B. E. et al. Fractional quantum Hall phase transitions and four-flux states in graphene. Phys. Rev. Lett. 111, 076802 (2013).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  26. Zhou, H. et al. Half- and quarter-metals in rhombohedral trilayer graphene. Nature 598, 429–433 (2021).

    ADS  Google Scholar 

  27. Zhou, H. et al. Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene. Science 375, 774–778 (2022).

    ADS  Google Scholar 

  28. Yu, J. et al. Correlated Hofstadter spectrum and flavour phase diagram in magic-angle twisted bilayer graphene. Nat. Phys. 18, 825–831 (2022).

    Google Scholar 

  29. Shabani, S. et al. Deep moiré potentials in twisted transition metal dichalcogenide bilayers. Nat. Phys. 17, 720–725 (2021).

    Google Scholar 

  30. Nieken, R. et al. Direct STM measurements of R-type and H-type twisted MoSe2/WSe2. APL Mater. 10, 031107 (2022).

    ADS  Google Scholar 

  31. Lieb, E. & Mattis, D. Theory of ferromagnetism and the ordering of electronic energy levels. Phys. Rev. 125, 164–172 (1962).

    ADS  MATH  Google Scholar 

  32. Zhang, Y., Yuan, N. F. Q. & Fu, L. Moiré quantum chemistry: charge transfer in transition metal dichalcogenide superlattices. Phys. Rev. B 102, 201115 (2020).

    ADS  Google Scholar 

  33. Padhi, B., Chitra, R. & Phillips, P. W. Generalized Wigner crystallization in moiré materials. Phys. Rev. B 103, 125146 (2021).

    ADS  Google Scholar 

  34. Zhang, S.-S., Zhu, W. & Batista, C. D. Pairing from strong repulsion in triangular lattice Hubbard model. Phys. Rev. B 97, 140507 (2018).

    ADS  Google Scholar 

  35. Davydova, M., Zhang, Y. & Fu, L. Itinerant spin polaron and metallic ferromagnetism in semiconductor moiré superlattices. Phys. Rev. B 107, 224420 (2023).

    ADS  Google Scholar 

  36. Zhang, Y. & Fu, L. Pseudogap metal and magnetization plateau from doping moiré Mott insulator. Preprint at https://arxiv.org/abs/2209.05430 (2022).

  37. Morera, I. et al. High-temperature kinetic magnetism in triangular lattices. Preprint at https://arxiv.org/abs/2209.05398 (2022).

  38. Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

    ADS  Google Scholar 

  39. Saito, Y. et al. Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene. Nat. Phys. 17, 478–481 (2021).

    Google Scholar 

  40. Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Flavour Hund’s coupling, Chern gaps and charge diffusivity in moiré graphene. Nature 592, 43–48 (2021).

    Google Scholar 

  41. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).

    ADS  Google Scholar 

  42. Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021).

    ADS  Google Scholar 

  43. Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

    ADS  Google Scholar 

  44. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

    ADS  Google Scholar 

  45. Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

    ADS  Google Scholar 

  46. Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  47. Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

    ADS  Google Scholar 

  48. Zhao, W. et al. Gate-tunable heavy fermions in a moiré Kondo lattice. Nature 616, 61–65 (2023).

    ADS  Google Scholar 

  49. Liu, X. et al. Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening. Science 371, 1261–1265 (2021).

    ADS  Google Scholar 

  50. Matty, M. & Kim, E.-A. Melting of generalized Wigner crystals in transition metal dichalcogenide heterobilayer Moiré systems. Nat. Commun. 13, 7098 (2022).

    ADS  Google Scholar 

  51. Paul, N., Crowley, P. J., Devakul, T. & Fu, L. Moiré Landau fans and magic zeros. Phys. Rev. Lett. 129, 116804 (2022).

    ADS  Google Scholar 

  52. Crépel, V. & Fu, L. Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides. Phys. Rev. B 107, L201109 (2023).

    ADS  Google Scholar 

  53. Arovas, D. P., Berg, E., Kivelson, S. A. & Raghu, S. The Hubbard model. Annu. Rev. Condens. Matter Phys. 13, 239–274 (2022).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  55. Kim, H., Park, H., Lee, G. & Kim, J. Intimate ohmic contact to two-dimensional WSe2 via thermal alloying. Nanotechnology 30, 415302 (2019).

    Google Scholar 

  56. Movva, H. C. P. et al. High-mobility holes in dual-gated WSe2 field-effect transistors. ACS Nano 9, 10402–10410 (2015).

    Google Scholar 

Download references

Acknowledgements

We thank T. Heinz, A. O’Beirne and H. B. Ribeiro for their assistance with the second-harmonic generation measurements, and T. P. Devereaux and A. A. Zibrov for helpful discussions. Experimental work was primarily supported by NSF-DMR-2103910. B.E.F. acknowledges an Alfred P. Sloan Foundation Fellowship and a Cottrell Scholar Award. The work at MIT was funded by the Air Force Office of Scientific Research (AFOSR) under award FA9550-22-1-0432. Y.Z. was supported by the start-up fund at the University of Tennessee. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant nos. 20H00354 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. B.A.F. acknowledges a Stanford Graduate Fellowship. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822.

Author information

Author notes

  1. These authors contributed equally: Carlos R. Kometter, Jiachen Yu, Trithep Devakul.

Authors and Affiliations

  1. Geballe Laboratory for Advanced Materials, Stanford, CA, USA

    Carlos R. Kometter, Jiachen Yu, Benjamin A. Foutty & Benjamin E. Feldman

  2. Department of Physics, Stanford University, Stanford, CA, USA

    Carlos R. Kometter, Benjamin A. Foutty & Benjamin E. Feldman

  3. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Jiachen Yu

  4. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Trithep Devakul, Aidan P. Reddy & Liang Fu

  5. Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USA

    Yang Zhang

  6. Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, USA

    Yang Zhang

  7. Research Center for Electronic and Optical Materials, National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe

  8. Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan

    Takashi Taniguchi

  9. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    Benjamin E. Feldman

Authors
  1. Carlos R. Kometter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiachen Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Trithep Devakul
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aidan P. Reddy
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Benjamin A. Foutty
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kenji Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Takashi Taniguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Liang Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Benjamin E. Feldman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

C.R.K. and J.Y. conducted the scanning SET measurements. C.R.K., J.Y. and B.E.F. designed the experiment. T.D., A.P.R., Y.Z. and L.F. conducted the theoretical calculations. C.R.K. fabricated the sample. K.W. and T.T. provided the hBN crystals. B.E.F. and L.F. supervised the project. All authors participated in the data analysis and writing of the manuscript.

Corresponding author

Correspondence to Benjamin E. Feldman.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Brian LeRoy and Eric Spanton 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 Device image.

Optical microscope image of the MoSe2/WSe2 device. Platinum contacts to the sample, gold ‘contact’ gates to locally dope the heterobilayer, and the back gate electrode are all indicated. Scale bar: 10 μm.

Supplementary information

Supplementary Information

Supplementary Figs. 1–12 and discussion.

Source data

Source Data Fig. 2

Gap source data.

Source Data Fig. 3

Compressibility and gap 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

Kometter, C.R., Yu, J., Devakul, T. et al. Hofstadter states and re-entrant charge order in a semiconductor moiré lattice. Nat. Phys. 19, 1861–1867 (2023). https://doi.org/10.1038/s41567-023-02195-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-02195-0

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