Energy Gaps in Graphene Nanoribbons

TL;DR

This paper presents first-principles scaling rules for the energy gaps in graphene nanoribbons with armchair and zigzag edges, revealing the origins of gaps due to quantum confinement, edge effects, and magnetization.

Contribution

It introduces ab initio calculations and analytical models to accurately predict energy gaps in GNRs, surpassing simple tight-binding approaches.

Findings

Both armchair and zigzag GNRs have energy gaps.

Gaps in armchair GNRs arise from quantum confinement and edge effects.

Gaps in zigzag GNRs originate from edge magnetization and sublattice potential.

Abstract

Based on a first-principles approach, we present scaling rules for the band gaps of graphene nanoribbons (GNRs) as a function of their widths. The GNRs considered have either armchair or zigzag shaped edges on both sides with hydrogen passivation. Both varieties of ribbons are shown to have band gaps. This differs from the results of simple tight-binding calculations or solutions of the Dirac's equation based on them. Our {\it ab initio} calculations show that the origin of energy gaps for GNRs with armchair shaped edges arises from both quantum confinement and the crucial effect of the edges. For GNRs with zigzag shaped edges, gaps appear because of a staggered sublattice potential on the hexagonal lattice due to edge magnetization. The rich gap structure for ribbons with armchair shaped edges is further obtained analytically including edge effects. These results reproduce our {\it ab initio} calculation results very well.