Multiscale lattice relaxation and topological 1D state in twisted trilayer graphene

We present comprehensive theoretical studies on the lattice relaxation and the electronic structures in general nonsymmetric twisted trilayer graphenes. By using an effective continuum model, we show that the relaxed lattice structure forms a patchwork of moiré-of-moiré domains, where a moiré 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 supermoiré 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 monolayerlike Dirac cone, and it is attributed to the formation of moiré-of-moiré domains equivalent to the mirror-symmetric twisted trilayer graphene.