Symmetry Preservation in Commensurate Twisted Bilayers

TL;DR

This paper investigates how symmetry is preserved or altered in commensurate twisted bilayer materials, revealing that initial stacking influences symmetry properties, with specific focus on Kagome and graphene bilayers.

Contribution

It demonstrates that the symmetry of twisted Kagome bilayers is preserved at all angles and shows how initial stacking affects symmetry in twisted bilayer graphene.

Findings

Symmetry in Kagome bilayers is preserved for all commensurate angles.

Initial stacking determines symmetry properties in twisted bilayer systems.

The method recovers known results for twisted bilayer graphene with a 3-fold symmetric twist center.

Abstract

Symmetry plays a key role in materials hosting Dirac electrons and underpins our ability to completely flatten the Dirac cone through the tuning of physical parameters such as twisting in van der Waals heterostructures. The emergent moir\'{e} patterns in twisted bilayers at small twist angles appear, at first glance, to be independent of the initial stacking order and hence are only shifted when one layer is translated with respect to the other. However, when the twist angle is large, differences can be seen at the level of both the lattice and electronic structure in the case of twisted bilayer graphene. In this work, we first address the problem of twisted Kagome bilayers and show that the rotational and dihedral symmetry of high-symmetry Kagome bilayers is preserved for all commensurate twist angles with a 6-fold symmetric twist centre. Hence, we demonstrate that the exact symmetry of small twist angle systems depends upon the initial stacking of the bilayer. We further apply the principles of our method to twisted bilayer graphene with a 3-fold symmetric twist centre to recover the results of [E. J. Mele, Phys. Rev. B 81, 161405 (2010)].