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


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.

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Figure 1: Moiré superlattice.
Figure 2: Emergence of anomalous quantum Hall states.
Figure 3: Fractal gaps.
Figure 4: Recursive structure.


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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).

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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.

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Correspondence to P. Kim.

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

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