Band topology and quantum spin Hall effect in bilayer graphene

TL;DR

This paper investigates the topological properties of bilayer graphene with spin orbit coupling, revealing doubled edge states compared to monolayer and classifying it as a weak topological insulator with specific stability characteristics.

Contribution

It provides a detailed analysis of the Chern numbers and topological phases of bilayer graphene, including numerical and analytical confirmation of edge states and stability under perturbations.

Findings

Bilayer graphene has twice as many edge states as monolayer.

Bilayer graphene is classified as a weak topological insulator.

The topological state is sensitive to spin-mixing perturbations.

Abstract

We consider bilayer graphene in the presence of spin orbit coupling, to assess its behavior as a topological insulator. The first Chern number $n$ for the energy bands of single and bilayer graphene is computed and compared. It is shown that for a given valley and spin, $n$ in a bilayer is doubled with respect to the monolayer. This implies that bilayer graphene will have twice as many edge states as single layer graphene, which we confirm with numerical calculations and analytically in the case of an armchair terminated surface. Bilayer graphene is a weak topological insulator, whose surface spectrum is susceptible to gap opening under spin-mixing perturbations. We also assess the stability of the associated topological bulk state of bilayer graphene under various perturbations. Finally, we consider an intermediate situation in which only one of the two layers has spin orbit coupling, and find that although individual valleys have non-trivial Chern numbers, the spectrum as a whole is not gapped, so that the system is not a topological insulator.