# Dodecagonal bilayer graphene quasicrystal and its approximants

## TL;DR

This study investigates the electronic and optical properties of dodecagonal bilayer graphene quasicrystals using large-scale calculations and proposes periodic approximants to accurately model their unique non-periodic structure.

## Contribution

The paper introduces a series of periodic approximants for graphene quasicrystals and demonstrates their effectiveness in reproducing quasicrystal properties through band-unfolding and tight-binding calculations.

## Key findings

- Emergence of new Dirac points and band gap at M point
- Agreement of effective band structure with recent experiments
- Lattice mismatch identified as key factor for approximant accuracy

## Abstract

Dodecagonal bilayer graphene quasicrystal has 12-fold rotational order but lacks translational symmetry which prevents the application of band theory. In this paper, we study the electronic and optical properties of graphene quasicrystal with large-scale tight-binding calculations involving more than ten million atoms. We propose a series of periodic approximants which reproduce accurately the properties of quasicrystal within a finite unit cell. By utilizing the band-unfolding method on the smallest approximant with only 2702 atoms, the effective band structure of graphene quasicrystal is derived. Novel features, such as the emergence of new Dirac points (especially the mirrored ones), the band gap at M point and the Fermi velocity are all in agreement with recent experiments. The properties of quasicrystal states are identified in the Landau level spectrum and optical excitations. Importantly, our results show that the lattice mismatch is the dominant factor determining the accuracy of layered approximants. The proposed approximants can be used directly for other layered materials in honeycomb lattice, and the design principles can be applied for any quasi-periodic incommensurate structures.