Quantum size effects in armchair graphene nanoribbons (AGNRs) with hydrogen termination are investigated via density functional theory (DFT) in the Kohn-Sham formulation. “Selection rules” are formulated, which allow extraction (approximately) the electronic structure of the AGNR bands starting from the four graphene dispersion sheets. In analogy with the case of carbon nanotubes, a threefold periodicity of the excitation gap with the ribbon width ($N$; number of carbon atoms per carbon slice) is predicted, which is confirmed by ab initio results. While traditionally such a periodicity would be observed in electronic response experiments, the DFT analysis presented here shows that it can also be seen in the ribbon geometry: the length of a ribbon with $L$ slices approaches the limiting value for a very large width, $1\ll N$ (keeping the aspect ratio low, $N\ll L$), with $1/N$ oscillations that display electronic selection rules. The oscillation amplitude is so strong that the asymptotic behavior is nonmonotonous, i.e.; wider ribbons exhibit a stronger elongation than narrower ones.

• Received 11 November 2011

DOI:https://doi.org/10.1103/PhysRevB.85.125433