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

• Published in: Nature (London) Vol. 497; no. 7451; pp. 598 - 602
• Main Authors: Dean, C. R., Wang, L., Maher, P., Forsythe, C., Ghahari, F., Gao, Y., Katoch, J., Ishigami, M., Moon, P., Koshino, M., Taniguchi, T., Watanabe, K., Shepard, K. L., Hone, J., Kim, P.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 30.05.2013; Nature Publishing Group
• Subjects: 639/301/357/918/1052; 639/766/119/2794; 639/766/119/995; Boron; Condensed matter: electronic structure, electrical, magnetic, and optical properties; Crystal lattices; Electric properties; Electronic structure and electrical properties of surfaces, interfaces, thin films and low-dimensional structures; Electronic transport in interface structures; Electronic transport in multilayers, nanoscale materials and structures; Exact sciences and technology; Fractals; Graphene; Humanities and Social Sciences; letter; Magnetic fields; Measurement; multidisciplinary; Physics; Properties; Quantum Hall effect; Quantum hall effect (including fractional); Science; Studies; Superlattices as materials; Two-dimensional systems; Graphene; Superlattices; Anomalous Hall effect; Magnetic field effects; Moire fringes; Boron nitride; Quantum Hall effect; Fractals; Recursion method; Electrostatic potential; Periodic potential
• ISSN: 0028-0836; 1476-4687; 1476-4687
• DOI: 10.1038/nature12186

Moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic potential modulation on a length scale ideally suited to studying the fractal features of the Hofstadter energy spectrum in large magnetic fields. Hofstadter's butterfly emerges in graphene superlattices In 1976 Douglas Hofstadter predicted that electrons in a lattice subjected to electrostatic and magnetic fields would show a characteristic energy spectrum determined by the interplay between two quantizing fields. The expected spectrum would feature a repeating butterfly-shaped motif, known as Hofstadter's butterfly. The experimental realization of the phenomenon has proved difficult because of the problem of producing a sufficiently disorder-free superlattice where the length scales for magnetic and electric field can truly compete with each other. Now that goal has been achieved — twice. Two groups working independently produced superlattices by placing ultraclean graphene (Ponomarenko et al .) or bilayer graphene (Kim et al .) on a hexagonal boron nitride substrate and crystallographically aligning the films at a precise angle to produce moiré pattern superstructures. Electronic transport measurements on the moiré superlattices provide clear evidence for Hofstadter's spectrum. The demonstrated experimental access to a fractal spectrum offers opportunities for the study of complex chaotic effects in a tunable quantum system. 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 spectrum 1 . Known as Hofstadter’s butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields 1 , 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 completely 11 , 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.