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

Abstract

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. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, 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. 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.

Controlled Electrochemical Intercalation of Graphene/h-BN van der Waals Heterostructures

“…Upon intercalation of Li, the optical properties of few-layer graphene crystals (with thicknesses down to 1 nm) change significantly [4], Ca-intercalated few-layer-graphene is superconducting [5], and FeCl3 intercalated bilayer graphene showed a hint of ferromagnetism [6]. In addition, there have been theoretical predictions that heavy doping and proximity induced spin-orbit coupling from certain intercalants may induce exotic electronic properties in the graphene channel [7].Recently, it was demonstrated that one can stack different van der Waals (vdW) atomic layers to form vdW heterostructures, creating a new generation of few-atomic-layer functional heterostructures with emergent properties [8,9]. In particular, graphene encapsulated by h-BN, a layered insulator, forms a vdW heterostructure where the 2-dimensional (2D) graphene channel is well isolated from the environment [8].…”

Epitaxially Stabilized Oxide Composed of Twisted Triangular-Lattice Layers

“…The above result indicates that the new compound is composed of the triangular-lattice Rh In general, two triangular lattices coupled at a twist angle form a quasiperiodic structure that has no unit cell. [18][19][20][21] As evidenced by the in-plane x-ray and electron diffractions (Figures 2e and 3j), however, the new layered compound shows periodically modulated hexagonal patterns with the in-plane cell parameter of 8.05(±0.05) Å. The selected-area electron diffraction detects the single pattern from only one type of the two crystallographic domains, while their both patterns are observed in the entire-area in-plane XRD.…”

“…Actually, in the research field of atomic layer materials such as graphene or transition metal dichalcogenides, the twist angle has been recognized as a new important parameter for tuning electrical and optical properties of stacked layers. [18][19][20][21] In the following, we present a new compound with a twisted stack of oxide layers stabilized by epitaxy technique. For this purpose, we adopt a simple Bi-Rh-O system, because Bi with a relatively large ionic radius is expected to form a large triangular-lattice layer mismatched to the RhO 2 layer.…”

Moiré pattern as a magnifying glass for strain and dislocations in van der Waals heterostructures

“…The perturbation produced by the moiré superlattice results in the formation of mini- bands for Dirac electrons in graphene [13][14][15][16][17][18][19][20]. The latter can be found by numerical diagonalisation of the moiré superlattice Hamiltonian,…”

“…1. For the unstrained heterostructure (left) the spectrum exhibits a secondary Dirac point [13][14][15][17][18][19] in the valence band. Strain, as low as w ∼ 1 − 3%…”