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

TLDR: It is demonstrated 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.

Abstract: Moire 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. 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 moire pattern superstructures. Electronic transport measurements on the moire 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 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 moire 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.