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Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices

An Erratum to this article was published on 30 September 2014


Self-similarity and fractals have fascinated researchers across various disciplines. In graphene placed on boron nitride and subjected to a magnetic field, self-similarity appears in the form of numerous replicas of the original Dirac spectrum, and their quantization gives rise to a fractal pattern of Landau levels, referred to as the Hofstadter butterfly. Here we employ capacitance spectroscopy to probe directly the density of states (DoS) and energy gaps in this spectrum. Without a magnetic field, replica spectra are seen as pronounced DoS minima surrounded by van Hove singularities. The Hofstadter butterfly shows up as recurring Landau fan diagrams in high fields. Electron–electron interactions add another twist to the self-similar behaviour. We observe suppression of quantum Hall ferromagnetism, a reverse Stoner transition at commensurable fluxes and additional ferromagnetism within replica spectra. The strength and variety of the interaction effects indicate a large playground to study many-body physics in fractal Dirac systems.

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Figure 1: Capacitance spectroscopy of graphene superlattices.
Figure 2: Magnetocapacitance oscillations and quantized density of states.
Figure 3: Hofstadter butterfly in graphene superlattices.
Figure 4: Interactions in Hofstadter minibands.


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This work was supported by the European Research Council, the Royal Society, Graphene Flagship, Science and Innovation Award from the EPSRC (UK) and EuroMagNET II (EU Contract 228043).

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Authors and Affiliations



J.S.T., A.V.K., Y.C., R.J. and F.W. designed and fabricated the devices. A.M. and G.L.Y. carried out the measurements. B.A.P. and M.P. helped with high-field experiments. X.C. and V.I.F. provided theoretical support. K.W. and T.T. provided hBN crystals. R.V.G. devised the fabrication technology for graphene capacitors. A.M. developed the on-chip capacitance bridge. A.M., V.I.F. and A.K.G. analysed the results. A.K.G. together with V.I.F. wrote the manuscript. K.S.N., I.V.G., L.A.P. and D.C.E. helped with experiments and/or writing the paper. Section 4 of the Supplementary Information was written by V.I.F. All authors contributed to discussions.

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Correspondence to V. I. Fal’ko, A. K. Geim or A. Mishchenko.

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The authors declare no competing financial interests.

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Yu, G., Gorbachev, R., Tu, J. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nature Phys 10, 525–529 (2014).

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