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

摘要: Moire superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic potential modulation on length scale ideally suited studying the fractal features of Hofstadter energy spectrum large magnetic fields. In 1976 Douglas predicted that electrons lattice subjected electrostatic and fields would show characteristic determined by interplay between two quantizing The expected feature repeating butterfly-shaped motif, known as Hofstadter's butterfly. experimental realization phenomenon has proved difficult because problem producing sufficiently disorder-free superlattice where scales for electric field can truly compete with each other. Now goal been achieved — twice. Two groups working independently produced placing ultraclean (Ponomarenko et al.) or (Kim substrate crystallographically aligning films at precise angle produce moire pattern superstructures. Electronic transport measurements clear evidence spectrum. demonstrated access offers opportunities study complex chaotic effects tunable quantum system. Electrons moving through spatially develop quantized consisting discrete Bloch bands. dimensions, also spectrum, highly degenerate Landau levels. When subject both potential, two-dimensional systems exhibit self-similar recursive spectrum1. Known Hofstadter’s butterfly, this results from an lengths associated fields1,2,3,4,5,6,7,8,9,10, is one first fractals discovered physics. decades since its prediction, attempts effect have limited difficulties reconciling scales. Typical atomic lattices (with periodicities less than nanometre) require unfeasibly reach commensurability condition, artificially engineered structures greater about 100 nanometres) corresponding are too small overcome disorder completely11,12,13,14,15,16,17. Here we demonstrate ideal order ten nanometres, enabling unprecedented We confirm Hall gaps described integer topological numbers, report their structure. Observation means it possible investigate emergent behaviour within landscape system internal degrees freedom.