Tunable incommensurability, complete structural phase diagrams and peculiar electronic properties in trilayer moiré-of-moiré systems

In this talk, we present a comprehensive structural analysis and the unique electronic properties of twisted trilayer graphene (TTG), a minimal multilayer configuration beyond twisted bilayer graphene. We demonstrate a complete catalog of reconstructed moiré-of-moiré structural phases by manipulating combinations of two twist angles. Our findings reveal cascades of spontaneous symmetry breaking as a function of the twist angles, resulting in a diverse array of large-scale moiré lattices, including triangular, kagome, and corner-shared hexagram-shaped domain patterns. Furthermore, our analysis emphasizes the crucial role of long-range interactions across entire layers, alongside the well-known contributions of twist angles and strain between adjacent layers, in realizing various domain lattices. Building on our results for TTG, we apply our methods to other twisted trilayer (TTL) systems made from two-dimensional semiconductors, demonstrating how external fields can control domain lattices. The diverse tessellation of distinct domains, whose topological network can be adjusted by modifying the twist angles, positions TTL systems as a platform for exploring the interplay between emerging quantum properties and controllable nontrivial lattices.