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. The 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.

十二边形双层石墨烯准晶体具有12倍的旋转顺序，但缺乏平移对称性，这妨碍了能带理论的应用。在本文中，我们通过涉及超过一千万个原子的大规模紧密结合计算研究了石墨烯准晶体的电子和光学性质。我们提出了一系列周期近似值，可以精确地再现有限晶胞内准晶体的性质。通过利用仅具有2702个原子的最小近似上的能带展开方法，得出了石墨烯准晶体的有效能带结构。特征，例如出现新的狄拉克点（尤其是镜像点），\（M \）处的带隙点和费米速度都与最近的实验一致。在Landau能级谱和光学激发中鉴定出准晶态的性质。重要的是，我们的结果表明，晶格失配是决定分层近似值准确性的主要因素。拟议的近似值可以直接用于蜂窝格中的其他层状材料，并且设计原理可以应用于任何准周期不相称的结构。