Quasicrystalline 30$^{\circ}$ twisted bilayer graphene: Fractal patterns and electronic localization properties

TL;DR

This paper investigates the electronic properties of 30° twisted bilayer graphene, revealing a mix of extended and localized fractal states due to its quasicrystalline structure, using a novel real-space numerical approach.

Contribution

It introduces an efficient real-space method to study large incommensurate multilayer 2D materials respecting their symmetry and Dirac electron nature, applied to 30° twisted bilayer graphene.

Findings

Electronic spectrum includes extended and localized states.

Localized states display fractal patterns matching quasicrystal tiling.

Method effectively handles large, incommensurate systems.

Abstract

The recently synthesized 30$^\circ$ twisted bilayer graphene (30$^\circ$-TBG) systems are unique quasicrystal systems possessing dodecagonal symmetry with graphene's relativistic properties. We employ a real-space numerical atomistic framework that respects both the dodecagonal rotational symmetry and the massless Dirac nature of the electrons to describe the local density of states of the system. The approach we employ is very efficiency for systems with very large unit cells and does not rely on periodic boundary conditions. These features allow us to address a broad class of multilayer two-dimensional crystal with incommensurate configurations, particularly TBGs. Our results reveal that the 30$^\circ$-TBG electronic spectrum consist of extended states together with a set of localized wave functions. The localized states exhibit fractal patterns consistent with the quasicrystal tiling.