Trigonal Quasicrystalline States in $30^\circ$ Rotated Double Moir\'{e} Superlattices

TL;DR

This paper investigates the electronic structure of a double moiré superlattice with a dodecagonal quasicrystalline configuration, revealing trigonal quasicrystalline states arising from resonant interactions of graphene's Bloch states.

Contribution

It uncovers the formation of trigonal quasicrystalline states in a double moiré superlattice, a novel phenomenon distinct from traditional moiré wave mixing effects.

Findings

Identification of 0 to 4 quasicrystalline configurations depending on lattice mismatch.

Discovery of resonant interactions hybridizing twelve graphene Bloch states.

Observation of trigonal quasicrystalline order near graphene's charge neutrality point.

Abstract

We study the lattice configuration and electronic structure of a double moir\'{e} superlattice, which is composed of a graphene layer encapsulated by two other layers in a way such that the two hexagonal moir\'{e} patterns are arranged in a dodecagonal quasicrystalline configuration. We show that there are between 0 and 4 such configurations depending on the lattice mismatch between graphene and the encapsulating layer. We then reveal the resonant interaction, which is distinct from the conventional 2-, 3-, 4-wave mixing of moir\'{e} superlattices, that brings together and hybridizes twelve degenerate Bloch states of monolayer graphene. These states do not fully satisfy the dodecagonal quasicrystalline rotational symmetry due to the symmetry of the wave vectors involved. Instead, their wave functions exhibit trigonal quasicrystalline order, which lacks inversion symmetry, at the energies much closer to the charge neutrality point of graphene.