Multi-scale lattice relaxation in general twisted trilayer graphenes

TL;DR

This paper investigates the lattice relaxation and electronic structures in non-symmetric twisted trilayer graphene, revealing complex moiré patterns, topological boundary states, and distinct electronic behaviors depending on stacking configurations.

Contribution

It provides a comprehensive theoretical analysis of lattice relaxation and electronic properties in general twisted trilayer graphene, highlighting the formation of moiré-of-moiré domains and topological boundary states.

Findings

Formation of patchwork moiré-of-moiré domains with layer-dependent patterns

Presence of highly one-dimensional electron bands localized at domain boundaries

Coexistence of flat bands and Dirac cones in alternating trilayer configurations

Abstract

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.