Edge States of Monolayer and Bilayer Graphene Nanoribbons

TL;DR

This paper investigates the edge states in monolayer and bilayer graphene nanoribbons, exploring how edge modifications, spin-orbit coupling, and bias voltage influence their electronic properties and topological phases.

Contribution

It introduces new edge states via edge-hopping modulation and analyzes the effects of SOC and bias voltage on the topological phases of GNRs.

Findings

Edge-hopping modulation creates new edge states in monolayer GNRs.

Intralayer Rashba SOC reduces the energy gap linearly with its strength.

Bias voltage can diminish the intrinsic SOC-induced gap and destroy topological phases.

Abstract

On the basis of tight-binding lattice model, the edge states of monolayer and bilayer graphene nanoribbons (GNRs) with different edge terminations are studied. The effects of edge-hopping modulation, spin-orbital coupling (SOC), and bias voltage on bilayer GNRs are discussed. We observe the following: (i) Some new extra edge states can be created by edge-hopping modulation for monolayer GNRs. (ii) Intralayer Rashba SOC plays a role in depressing the band energy gap $E_g$ opened by intrinsic SOC for both monolayer and bilayer GNRs. An almost linear dependent relation, i.e., $E_g\sim \lambda_R$, is found. (iii) Although the bias voltage favors a bulk energy gap for bilayer graphene without intrinsic SOC, it tends to reduce the gap induced by intrinsic SOC. (iv) The topological phase of the quantum spin Hall effect can be destroyed completely by interlayer Rashba SOC for bilayer GNRs.