Multi-scale lattice relaxation in general twisted trilayer graphenes

We present comprehensive theoretical studies on the lattice relaxation and the electronic structures in general non-symemtric twisted trilayer graphenes. By using an effective continuum model, we show that the relaxed lattice structure forms a patchwork of moir\'e-of-moir\'e domains, where a moir\'e pattern given by layer 1 and 2 and another pattern given by layer 2 and 3 become locally commensurate. The atomic configuration inside the domain exhibits a distinct contrast between chiral and alternating stacks, which are determined by the relative signs of the two twist angles. In the chiral case, the electronic band calculation reveals a wide energy window ($>$ 50 meV) with low density of states, featuring sparsely distributed highly one-dimensional electron bands. These one-dimensional states exhibit a sharp localization at the boundaries between super-moir\'e domains, and they are identified as a topological boundary state between distinct Chern insulators. The alternating trilayer exhibits a coexistence of the flat bands and a monolayer-like Dirac cone, and it is attributed to the formation of moir\'e-of-moir\'e domains equivalent to the mirror-symmetric twisted trilayer graphene.