Flat bands and Moiré phases in Bilayer Quasicrystals

The discovery in 2018 of superconducting state in Twisted Bilayer Graphene opens a new way to study heterostructures and correlated states, introducing the Moiré phases. The superconducting state was also found in multiple twisted layers of graphene. This is attributed to the emergence of flat bands near the Fermi surface. Almost 40 years has passed since the discovery of Quasicrystals (QCs). Their structure was characterized, the stability has been studied, and the superconducting states have been found. Also, the finding of naturally occurring quasicrystals has been added to the list. The decagonal quasicrystals are a type of QCs that are periodic in one direction and quasiperiodic in two directions. The Penrose tessellation is the two dimensional most used model to study QCs due to presenting the same diffraction pattern as QCs. In this work, we find flat bands in Bilayer Penrose Lattices due to rotation and translation operations. This system allows us to study the electronic structure in grain boundary and interfaces in QCs. The twisting and translation of the Bilayer Penrose lattices forms Moiré patterns. Using the tight binding Hamiltonian, we find the Density of states and study the electronic wavefunction. It is observed electronic localization in the regions of destructive interference of the Moiré pattern.