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Dominant 1/3-filling correlated insulator states and orbital geometric frustration in twisted bilayer graphene

Nature Physics volume 20, pages 1407–1412 (2024)Cite this article

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Abstract

Geometric frustration occurs in a lattice system when not all the interactions can be satisfied simultaneously. The simplest example is antiferromagnetically coupled spins on a triangular lattice. Frustrated systems are characterized by having many nearly degenerate ground states, leading to non-trivial phases, such as spin ice and spin liquids. To date, most studies have looked at geometric frustration of spins whereas orbital geometric frustration has been much less explored. For electrons in twisted bilayer graphene, when the electronic bands are filled to a fraction with denominator 3, Coulomb interactions and the Wannier orbital shapes are predicted to strongly constrain spatial charge ordering, leading to geometrically frustrated ground states that produce a new class of correlated insulating states. Here we report the observation of dominant, denominator 3, fractional-filling, insulating states in large-angle twisted bilayer graphene. These states persist in magnetic fields and display magnetic ordering signatures and tripled unit cell reconstruction. These results are in agreement with a strong-coupling theory for symmetry-breaking in geometrically frustrated fractional states.

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Fig. 1: Temperature-dependent data from device D1 with θ = 1.32°.
Fig. 2: Real-space model.
Fig. 3: Magnetotransport of device D1 at T = 300 mK.
Fig. 4: Magnetotransport of device D2 at T = 1.5 K.

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Data availability

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

Code availability

The code that supports the findings of this study is available from the corresponding authors upon reasonable request.

References

  1. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  Google Scholar 

  2. Padhi, B., Setty, C. & Phillips, P. W. Doped twisted bilayer graphene near magic angles: proximity to Wigner crystallization, not Mott insulation. Nano Lett. 18, 6175–6180 (2018).

    Article  Google Scholar 

  3. Padhi, B. & Phillips, P. W. Pressure-induced metal-insulator transition in twisted bilayer graphene. Phys. Rev. B 99, 205141 (2019).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Zhang, K., Zhang, Y., Fu, L. & Kim, E.-A. Fractional correlated insulating states at one-third filled magic-angle twisted bilayer graphene. Commun. Phys. 5, 250 (2022).

    Article  Google Scholar 

  9. Zhou, Y. et al. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 595, 48–52 (2021).

    Article  Google Scholar 

  10. Lau, C. N., Bockrath, M. W., Mak, K. F. & Zhang, F. Reproducibility in the fabrication and physics of moiré materials. Nature 602, 41–50 (2022).

    Article  Google Scholar 

  11. Streda, P. Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C 15, L1299 (1982).

    Article  ADS  Google Scholar 

  12. MacDonald, A. Landau-level subband structure of electrons on a square lattice. Phys. Rev. B 28, 6713 (1983).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  14. Cheng, B. et al. Fractional and symmetry-broken Chern insulators in tunable moiré superlattices. Nano Lett. 19, 4321–4326 (2019).

    Article  Google Scholar 

  15. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).

    Article  Google Scholar 

  16. Rickhaus, P. et al. Correlated electron-hole state in twisted double-bilayer graphene. Science 373, 1257–1260 (2021).

    Article  Google Scholar 

  17. Li, Q. et al. Tunable quantum criticalities in an isospin extended Hubbard model simulator. Nature 609, 479–484 (2022).

    Article  Google Scholar 

  18. Shen, C. et al. Dirac spectroscopy of strongly correlated phases in twisted trilayer graphene. Nat. Mater. 22, 316–321 (2023).

    Article  Google Scholar 

  19. Mao, D., Zhang, K. & Kim, E.-A. Fractionalization in fractional correlated insulating states at n ±1/3 filled twisted bilayer graphene. Phys. Rev. Lett. 131, 106801 (2023).

  20. Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

    Google Scholar 

  21. Koshino, M. et al. Maximally localized Wannier orbitals and the extended Hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).

    Google Scholar 

  22. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. 8, 031089 (2018).

    Article  Google Scholar 

  23. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419 (2012).

    Article  ADS  Google Scholar 

  24. Lu, X. et al. Superconductors, orbital magnets, and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

  25. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Article  Google Scholar 

  26. Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).

    Article  ADS  Google Scholar 

  27. Codecido, E. et al. Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle. Sci. Adv. 5, eaaw9770 (2019).

    Article  ADS  Google Scholar 

  28. 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  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  31. Stepanov, P. et al. Competing zero-field Chern insulators in superconducting twisted bilayer graphene. Phys. Rev. Lett. 127, 197701 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  34. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. 20, 488–494 (2021).

    Google Scholar 

  35. Das, I. et al. Symmetry-broken Chern insulators and Rashba-like Landau-level crossings in magic-angle bilayer graphene. Nat. Phys. 17, 710–714 (2021).

    Article  Google Scholar 

  36. Krishna Kumar, R. et al. High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices. Science 357, 181–184 (2017).

    Article  Google Scholar 

Download references

Acknowledgements

We thank F. Zhang for helpful discussions. The experiments were supported by the Basic Energy Sciences division of the Department of Energy (Grant No. DE-SC0020187). Devices were fabricated using a nanofabrication facility supported by the Materials Research Science and Engineering Center of the National Science Foundation (NSF; Grant No. DMR-2011876). D.M. and E-A.K. are supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative (Grant No. GBMF10436). K.Z. is supported by the NSF (Grant No. EAGER OSP-136036) and the Natural Sciences and Engineering Research Council of Canada (Grant No. PGS-D-557580-2021). E.-A.K. acknowledges support from the Ewha Frontier 10-10 Research Grant and the Simons Fellowship in Theoretical Physics (Award 920665). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the NSF (Grant No. NSF/DMR-1644779) and the State of Florida. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Nos. 21H05233 and 23H02052) and World Premier International Research Center Initiative, MEXT, Japan.

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Author notes

  1. These authors contributed equally: Haidong Tian, Emilio Codecido, Dan Mao.

Authors and Affiliations

  1. Department of Physics, The Ohio State University, Columbus, OH, USA

    Haidong Tian, Emilio Codecido, Shi Che, Marc Bockrath & Chun Ning Lau

  2. Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA

    Dan Mao, Kevin Zhang & Eun-Ah Kim

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

    Kenji Watanabe

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

    Takashi Taniguchi

  5. National High Magnetic Field Laboratory, Tallahassee, FL, USA

    Dmitry Smirnov

  6. Department of Physics, Ewha Womans University, Seoul, South Korea

    Eun-Ah Kim

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  1. Haidong Tian
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Contributions

C.N.L. and M.B. conceived the project. H.T. and E.C. fabricated devices with assistance from S.C. H.T., E.C. and D.S. performed the measurements. K.W. and T.T. provided the hBN crystals. D.M. and K.Z. performed the theoretical calculations under the supervision of E.-A.K. C.N.L and M.B. analysed the data. C.N.L., M.B., E.-A.K. and D.M. interpreted the data and wrote the paper. All authors read and commented on the paper.

Corresponding authors

Correspondence to Eun-Ah Kim, Marc Bockrath or Chun Ning Lau.

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Supplementary Figs. 1–10, Discussion A–E and Tables 1–5.

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Tian, H., Codecido, E., Mao, D. et al. Dominant 1/3-filling correlated insulator states and orbital geometric frustration in twisted bilayer graphene. Nat. Phys. 20, 1407–1412 (2024). https://doi.org/10.1038/s41567-024-02546-5

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