Electronic structure of twisted graphene flakes

We study the electronic structure of bilayer graphene flakes in which the constituent layers are mutually rotated by some angle θ. The large system sizes involved (up to 10⁵ carbon atoms) necessitate the use of a tight-binding approach in conjunction with Lanczos diagonalization. We find that a single moir\'e spot is sufficient for the low-energy density of states to resemble closely that of the periodic analog of such flakes, the graphene twist bilayer, implying that the low-energy physics in this system is well described as that of a ``moir\'e quantum well'' trapping low-energy graphene electrons. Furthermore, a graphene twist flake consisting of a single moir\'e unit cell leads already to electron localization on the AA ``moir\'e spot,'' in agreement with this moir\'e quantum well picture. The electron density fluctuations induced by the moir\'e lattice in twist graphene flakes are significant, being an order of magnitude greater than those generated by the rippling of suspended graphene. Finally, we determine the electronic properties of such flakes in the presence of an external magnetic field, finding a ``zero-mode'' structure and Landau states that exhibit an electron current well described as a charge flow on a torus situated at the AA regions of the moir\'e lattice.