Emergent fractals in hBN-encapsulated graphene based supermoir\'e structures and their experimental signatures

TL;DR

This paper reveals that supermoiré structures in hBN-encapsulated graphene can form emergent fractals, providing new insights into their low-energy electronic properties and experimental signatures.

Contribution

It introduces the concept of emergent fractals in supermoiré structures and develops a theoretical framework to analyze their properties and experimental detection.

Findings

Emergent fractals occur in specific supermoiré parameter regimes.

Fractality enables accurate calculation of low-energy band counts.

Proposed experimental methods to verify fractals via ARPES and STM.

Abstract

Supermoir\'e structures (SMS), formed by overlapping moir\'e-patterns in van der Waals heterostructures, display complex behaviour that lacks a comprehensive low-energy theoretical description. We demonstrate that these structures can form emergent fractals under specific conditions and identify the parameter space where this occurs in hexagonal trilateral SMS. This fractality enables a reliable calculation of low-energy band counts, which are crucial for understanding both single-particle and correlation effects. Using an effective Hamiltonian that includes in- and out-of-plane lattice relaxation, we analyze SMS in hBN-encapsulated single and bilayer graphene. We prescribe methods to experimentally verify these fractals and extract their fractal dimension through angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM).