Quasicrystalline electronic states in 30${}^{\circ }$ rotated twisted bilayer graphene

Abstract

The recently realized bilayer graphene system with a twist angle of $30^{\circ }$ offers a new type of quasicrystal which unites the dodecagonal quasicrystalline nature and graphene's relativistic properties. Here, we introduce a concise theoretical framework that fully respects both the dodecagonal rotational symmetry and the massless Dirac nature, to describe the electronic states of the system. We find that the electronic spectrum consists of resonant states labeled by 12-fold quantized angular momentum, together with the extended relativistic states. The resulting quasi-band structure is composed of the nearly flat bands with spiky peaks in the density of states, where the wave functions exhibit characteristic patterns which fit to the fractal inflations of the quasicrystal tiling. We also demonstrate that the 12-fold resonant states appear as spatially-localized states in a finite-size geometry, which is another hallmark of quasicrystal. The theoretical method introduced here is applicable to a broad class of "extrinsic quasicrystals" composed of a pair of two-dimensional crystals overlaid on top of the other with incommensurate configurations.

9 pages, 6 figures

References in corpus (6)

• Flat Bands in Slightly Twisted Bilayer Graphene
• Numerical studies of confined states in rotated bilayers of graphene
• Electronic properties of graphene hexagonal boron nitride moiré superlattice
• Quantum critical state in a magnetic quasicrystal
• Interlayer interaction in general incommensurate atomic layers
• Incommensurate double-walled carbon nanotubes as one-dimensional moiré crystals

Cited by in corpus (5)

• Interlayer Decoupling in 30° Twisted Bilayer Graphene Quasicrystal
• Emergent Localization in Dodecagonal Bilayer Quasicrystals
• Electrical band flattening, valley flux, and superconductivity in twisted trilayer graphene
• Topological charge pumping in quasiperiodic systems characterized by Bott index
• Construction of optimized tight-binding models using \textit{ab initio} Hamiltonian: Application to monolayer $2H$-transition metal dichalcogenides