Universal geometric classification of armchair graphene nanoribbons by their properties in a staggered sublattice potential

We demonstrate the topological properties of the band-gap of armchair graphene nanoribbons in a spatially varying staggered sublattice potential. Several general scaling laws are presented to quantify the band gap variation. It is found that all armchair nanoribbons are described by one of three distinct classes depending on their width, one of which is the well known massless Dirac condition, and the other two we call potentially gapless, and gapless-superlattice. We construct an effective theory which faithfully reproduces these results, and makes explicit the nature of the competing masses and overlap integrals across a particular sample. Finally we propose several systems on which these results should shed considerable light, and which have all already been experimentally realized.