Letter | Published:

Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices

Nature volume 497, pages 598602 (30 May 2013) | Download Citation


Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum1. Known as Hofstadter’s butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields1,2,3,4,5,6,7,8,9,10, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely11,12,13,14,15,16,17. Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1.

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

  2. 2.

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

  3. 3.

    Quantised Hall effect in a two-dimensional periodic potential. J. Phys. C 15, L1299–L1303 (1982)

  4. 4.

    , , & Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)

  5. 5.

    Landau-level subband structure of electrons on a square lattice. Phys. Rev. B 28, 6713–6717 (1983)

  6. 6.

    & Hall plateau diagram for the Hofstadter butterfly energy spectrum. Phys. Rev. B 73, 155304 (2006)

  7. 7.

    & Hofstadter butterflies of bilayer graphene. Phys. Rev. B 75, 201404(R) (2007)

  8. 8.

    & Moiré butterflies in twisted bilayer graphene. Phys. Rev. B 84, 035440 (2011)

  9. 9.

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

  10. 10.

    , & Topological phases in a two-dimensional lattice: magnetic field versus spin-orbit coupling. Phys. Rev. B 86, 075118 (2012)

  11. 11.

    , & Magnetoresistance oscillations in a grid potential: indication of a Hofstadter-type energy spectrum. Phys. Rev. B 43, 5192–5195 (1991)

  12. 12.

    , & Novel magneto-resistance oscillations in laterally modulated two-dimensional electrons with 20nm periodicity formed on vicinal GaAs (111)B substrates. Physica E 2, 944–948 (1998)

  13. 13.

    et al. Fermiology of two-dimensional lateral superlattices. Phys. Rev. Lett. 83, 2234–2237 (1999)

  14. 14.

    , , & Landau subbands generated by a lateral electrostatic superlattice: chasing the Hofstadter butterfly. Semicond. Sci. Technol. 11, 1582–1585 (1996)

  15. 15.

    et al. Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance. Phys. Rev. Lett. 86, 147–150 (2001)

  16. 16.

    et al. Detection of Landau band coupling induced rearrangement of the Hofstadter butterfly. Physica E 25, 227–232 (2004)

  17. 17.

    et al. Laterally modulated 2D electron system in the extreme quantum limit. Phys. Rev. Lett. 92, 036802 (2004)

  18. 18.

    Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874–878 (1955)

  19. 19.

    , & New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980)

  20. 20.

    & Microwave realization of the Hofstadter butterfly. Phys. Rev. Lett. 80, 3232–3235 (1998)

  21. 21.

    & Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. N. J. Phys. 5, 56 (2003)

  22. 22.

    et al. Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nature Mater. 10, 282–285 (2011)

  23. 23.

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

  24. 24.

    et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nature Phys. 8, 382–386 (2012)

  25. 25.

    et al. Boron nitride substrates for high-quality graphene electronics. Nature Nanotechnol. 5, 722–726 (2010)

  26. 26.

    et al. Multicomponent fractional quantum Hall effect in graphene. Nature Phys. 7, 693–696 (2011)

  27. 27.

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

  28. 28.

    , , , & Generic miniband structure of graphene on a hexagonal substrate. Preprint at (2012)

  29. 29.

    , , , & Substrate-induced band gap in graphene on hexagonal boron nitride: ab initio density functional calculations. Phys. Rev. B 76, 073103 (2007)

  30. 30.

    , & Tunable band gaps in bilayer graphene-BN heterostructures. Nano Lett. 11, 1070–1075 (2011)

Download references


We thank A. MacDonald for discussions. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by US National Science Foundation cooperative agreement no. DMR-0654118, the State of Florida and the US Department of Energy. This work is supported by AFOSR MURI. J.K. and M.I. were supported by the US National Science Foundation under grant no. 0955625. K.L.S. was supported by DARPA under Office of Naval Research contract N00014-1210814. P.K. and F.G. acknowledge sole support from the US Department of Energy (DE-FG02-05ER46215).

Author information


  1. Department of Physics, The City College of New York, New York, New York 10031, USA

    • C. R. Dean
  2. Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA

    • L. Wang
    • , Y. Gao
    •  & J. Hone
  3. Department of Physics, Columbia University, New York, New York 10027, USA

    • P. Maher
    • , C. Forsythe
    • , F. Ghahari
    •  & P. Kim
  4. Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, Florida 32816-2385, USA

    • J. Katoch
    •  & M. Ishigami
  5. Department of Physics, Tohoku University, Sendai 980-8578, Japan

    • P. Moon
    •  & M. Koshino
  6. National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

    • T. Taniguchi
    •  & K. Watanabe
  7. Department of Electrical Engineering, Columbia University, New York, New York 10027, USA

    • K. L. Shepard


  1. Search for C. R. Dean in:

  2. Search for L. Wang in:

  3. Search for P. Maher in:

  4. Search for C. Forsythe in:

  5. Search for F. Ghahari in:

  6. Search for Y. Gao in:

  7. Search for J. Katoch in:

  8. Search for M. Ishigami in:

  9. Search for P. Moon in:

  10. Search for M. Koshino in:

  11. Search for T. Taniguchi in:

  12. Search for K. Watanabe in:

  13. Search for K. L. Shepard in:

  14. Search for J. Hone in:

  15. Search for P. Kim in:


C.R.D., P. Maher, L.W., C.F., F.G. and Y.G. performed device fabrication and transport measurements. J.K. and M.I. performed AFM measurements. P. Moon and M.K. provided theoretical support. K.W. and T.T. synthesized the hBN samples. K.L.S., J.H. and P.K. advised on experiments. C.R.D., P. Maher, P. Moon, M.K., J.H. and P.K. wrote the manuscript in consultation with all other authors.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to P. Kim.

Supplementary information

PDF files

  1. 1.

    Supplementary Information

    This file contains Supplementary Text and Data 1-7, Supplementary Figures 1-6 and additional references.

About this article

Publication history






Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.