Periodic Behavior of Topology in Graphene with Nanohole Array

TL;DR

This paper presents a method to diagnose the topological properties of graphene with nanohole arrays directly from the lattice constant, revealing periodic topological behaviors linked to array symmetry.

Contribution

It introduces a symmetry-based approach to identify topological phases in graphene nanohole structures, highlighting periodicity with respect to array size.

Findings

Topological phases appear periodically with array size m.

Nontrivial topology occurs with period two for triangular arrays.

Verification via Wannier centers and parity indices confirms the periodic behavior.

Abstract

We derive a way to diagnose band topology for graphene with triangular and/or honeycomb array of nanoholes directly from the lattice constant of superstructure $m\sqrt{3}\times m\sqrt{3}$ with integer $m$. Taking into account the $C_{6v}$ crystalline symmetry respected by nanoholes and their array, we demonstrate that nontrivial topology appears periodically with $m$ with period two (six) for triangular (honeycomb) array. These behaviors are verified by Wyckoff positions of Wannier centers and parity index of valence bands at high-symmetry points in Brillouin zone. The results provide a convenient guide for material design of topological electronic states based on graphene derivatives.