Letter | Published:

Correlated insulator behaviour at half-filling in magic-angle graphene superlattices

Nature volume 556, pages 8084 (05 April 2018) | Download Citation


A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked two-dimensional building blocks held together by the van der Waals forces between the layers. This design means that the properties of van der Waals heterostructures can be engineered precisely, even more so than those of two-dimensional materials1. One such property is the ‘twist’ angle between different layers in the heterostructure. This angle has a crucial role in the electronic properties of van der Waals heterostructures, but does not have a direct analogue in other types of heterostructure, such as semiconductors grown using molecular beam epitaxy. For small twist angles, the moiré pattern that is produced by the lattice misorientation between the two-dimensional layers creates long-range modulation of the stacking order. So far, studies of the effects of the twist angle in van der Waals heterostructures have concentrated mostly on heterostructures consisting of monolayer graphene on top of hexagonal boron nitride, which exhibit relatively weak interlayer interaction owing to the large bandgap in hexagonal boron nitride2,3,4,5. Here we study a heterostructure consisting of bilayer graphene, in which the two graphene layers are twisted relative to each other by a certain angle. We show experimentally that, as predicted theoretically6, when this angle is close to the ‘magic’ angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling. These flat bands exhibit insulating states at half-filling, which are not expected in the absence of correlations between electrons. We show that these correlated states at half-filling are consistent with Mott-like insulator states, which can arise from electrons being localized in the superlattice that is induced by the moiré pattern. These properties of magic-angle-twisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic many-body quantum phases in two dimensions in the absence of a magnetic field. The accessibility of the flat bands through electrical tunability and the bandwidth tunability through the twist angle could pave the way towards more exotic correlated systems, such as unconventional superconductors and quantum spin liquids.

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  1. 1.

    & Van der Waals heterostructures. Nature 499, 419–425 (2013)

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

    , & Electron interactions and gap opening in graphene superlattices. Phys. Rev. Lett. 111, 266801 (2013)

  6. 6.

    & Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011)

  7. 7.

    , , & Flat bands and Wigner crystallization in the honeycomb optical lattice. Phys. Rev. Lett. 99, 070401 (2007)

  8. 8.

    , , , & Superconducting transitions in flat-band systems. Phys. Rev. B 90, 094506 (2014)

  9. 9.

    , , & Interaction-driven topological and nematic phases on the Lieb lattice. New J. Phys. 17, 055016 (2015)

  10. 10.

    Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204 (1989)

  11. 11.

    Exact ground states for the Hubbard model on the kagome lattice. J. Phys. A 25, 4335–4345 (1992)

  12. 12.

    & Heavy fermions and quantum phase transitions. Science 329, 1161–1166 (2010)

  13. 13.

    et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016)

  14. 14.

    , , , & Flat bands in slightly twisted bilayer graphene: tight-binding calculations. Phys. Rev. B 82, 121407 (2010)

  15. 15.

    , & Continuum model of the twisted graphene bilayer. Phys. Rev. B 86, 155449 (2012)

  16. 16.

    & Electronic structure theory of weakly interacting bilayers. Phys. Rev. B 93, 235153 (2016)

  17. 17.

    et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. USA 114, 3364–3369 (2017)

  18. 18.

    , & Numerical studies of confined states in rotated bilayers of graphene. Phys. Rev. B 86, 125413 (2012)

  19. 19.

    et al. Observation of van Hove singularities in twisted graphene layers. Nat. Phys. 6, 109–113 (2010)

  20. 20.

    et al. Single-layer behavior and its breakdown in twisted graphene layers. Phys. Rev. Lett. 106, 126802 (2011)

  21. 21.

    et al. Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis. Phys. Rev. Lett. 109, 196802 (2012)

  22. 22.

    et al. van der Waals heterostructures with high accuracy rotational alignment. Nano Lett. 16, 1989–1995 (2016)

  23. 23.

    et al. Single-electron capacitance spectroscopy of discrete quantum levels. Phys. Rev. Lett. 68, 3088–3091 (1992)

  24. 24.

    et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004)

  25. 25.

    et al. Giant intrinsic carrier mobilities in graphene and its bilayer. Phys. Rev. Lett. 100, 016602 (2008)

  26. 26.

    , , , & Temperature-dependent transport in suspended graphene. Phys. Rev. Lett. 101, 096802 (2008)

  27. 27.

    Metal-Insulator Transitions (Taylor and Francis, 1990)

  28. 28.

    , & Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998)

  29. 29.

    , & Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006)

  30. 30.

    , & Mott transition and magnetism of the triangular-lattice Hubbard model with next-nearest-neighbor hopping. Phys. Rev. B 95, 075124 (2017)

  31. 31.

    Density Waves In Solids (Westview Press, 2009)

  32. 32.

    Spin liquids in frustrated magnets. Nature 464, 199–208 (2010)

  33. 33.

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

  34. 34.

    & Energy spectrum and quantum Hall effect in twisted bilayer graphene. Phys. Rev. B 85, 195458 (2012)

  35. 35.

    & Lattice relaxation and energy band modulation in twisted bilayer graphenes. Phys. Rev. B 96, 075311 (2017)

  36. 36.

    et al. Charge inversion and topological phase transition at a twist angle induced van Hove singularity of bilayer graphene. Nano Lett. 16, 5053–5059 (2016)

  37. 37.

    Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976)

  38. 38.

    , A result not dependent on rationality for Bloch electrons in a magnetic field. Phys. Status Solidi b 88, 757–765 (1978)

  39. 39.

    , , & Measurement of the quantum capacitance of graphene. Nat. Nanotechnol. 4, 505–509 (2009)

  40. 40.

    , , & Carrier statistics and quantum capacitance of graphene sheets and ribbons. Appl. Phys. Lett. 91, 092109 (2007)

  41. 41.

    The band theory of graphite. Phys. Rev. 71, 622–634 (1947)

  42. 42.

    & in Dirac Matter (eds et al.) 25–53 (Springer, 2017)

  43. 43.

    & Dirac point metamorphosis from third-neighbor couplings in graphene and related materials. Phys. Rev. B 83, 115404 (2011)

  44. 44.

    An equivalence between monolayer and bilayer honeycomb lattices. Eur. Phys. J. B 85, 375 (2012)

  45. 45.

    & The electronic properties of bilayer graphene. Rep. Prog. Phys. 76, 056503 (2013)

  46. 46.

    & Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006)

  47. 47.

    , & Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances. J. Phys. Soc. Jpn 74, 1674–1677 (2005)

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We acknowledge discussions with L. Levitov, P. Lee, S. Todadri, B. I. Halperin, S. Carr, Z. Alpichshev, J. Y. Khoo and N. Staley. This work was primarily supported by the National Science Foundation (NSF; DMR-1405221) and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4541 for device fabrication, transport measurements and data analysis (Y.C., J.Y.L., J.D.S.-Y. and P.J.H.), with additional support from the NSS Program, Singapore (J.Y.L.). Capacitance work by R.C.A., A.D. and S.L.T. and theory work by S.F. was supported by the STC Center for Integrated Quantum Materials, NSF grant number DMR-1231319. Data analysis by V.F. was supported by AFOSR grant number FA9550-16-1-0382. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by MEXT, Japan and JSPS KAKENHI grant numbers JP15K21722 and JP25106006. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the NSF (DMR-0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS-0335765). E.K. acknowledges support by ARO MURI award W911NF-14-0247. R.C.A. acknowledges support by the Gordon and Betty Moore Foundation under grant number GBMF2931.

Author information


  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Yuan Cao
    • , Valla Fatemi
    • , Ahmet Demir
    • , Spencer L. Tomarken
    • , Jason Y. Luo
    • , Ray C. Ashoori
    •  & Pablo Jarillo-Herrero
  2. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • Shiang Fang
    • , Javier D. Sanchez-Yamagishi
    •  & Efthimios Kaxiras
  3. National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan

    • Kenji Watanabe
    •  & Takashi Taniguchi
  4. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

    • Efthimios Kaxiras


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Y.C., J.Y.L. and J.D.S.-Y. fabricated the devices and performed transport measurements. Y.C. and V.F. performed data analysis. P.J.-H. supervised the project. S.F. and E.K. provided numerical calculations. S.L.T., A.D. and R.C.A. measured capacitance data. K.W. and T.T. provided hexagonal boron nitride devices. Y.C., V.F. and P.J.-H. wrote the paper with input from all authors.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Pablo Jarillo-Herrero.

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