Multi-moir\'{e} trilayer graphene: lattice relaxation, electronic structure, and magic angles

We systematically explore the structural and electronic properties of twisted trilayer graphene systems. In general, these systems are characterized by two twist angles, which lead to two incommensurate moir\'{e} periods. We show that lattice relaxation results in the formation of domains of periodic single-moir\'{e} structures only for twist angles close to the simplest fractions. For the majority of other twist angles, the incommensurate moir\'{e} periods lead to a quasicrystalline structure. We identify experimentally relevant magic angles at which the electronic density of states is sharply peaked and strongly correlated physics is most likely to be realized.