Quasi-periodic Hamiltonians have a fractal density of states with a hierarchical structure of bands. One pathway to controllably creating quasi-periodic Hamiltonian is stacking layers of graphene...Show moreQuasi-periodic Hamiltonians have a fractal density of states with a hierarchical structure of bands. One pathway to controllably creating quasi-periodic Hamiltonian is stacking layers of graphene twisted with incommensurate angles. Nevertheless, due to the weakness of the inter-layer coupling, the fractality in these type of systems remains to be observed. We investigate the fractality of graphene quasicrystals numerically. To achieve this, we develop an algorithm to compute the fractal dimension of the density of states. Using this approach, we demonstrate that increasing the value of inter-layer coupling of dodecagonal graphene does generate fractal density of states. Finally, we propose a four-layer system that utilizes the flat band appearing in the twisted bilayer graphene at the magic twist angles TBLG together with the dodecagonal stacking as a pathway towards observing fractal density of states.Show less
