Finite-width effects for the localized edge modes in zigzag graphene nanoribbons

Abstract

A matrix formalism is used to derive the analytical Green's functions describing correlations between any two atomic sites on a zigzag (ZZ) graphene nanoribbon, incorporating modified electronic hopping values between edge sites that may be distinct from the hopping between interior sites. An analysis of the poles of our Green's functions shows two distinct types of localized edge modes in the electronic spectrum. The first of these, the ``zero'' mode, is a topologically induced mode arising from the bipartite honeycomb lattice structure of graphene and is always present along ZZ edges. The second type of localized edge mode is present at edges when the edge-to-bulk hopping ratio deviates significantly from unity. The correlations between edge sites are found to exhibit strikingly different features when mediated by the zero edge mode compared with mediation by the ``modified'' edge mode. In particular, the zero-mode spectral intensity for correlations between two atomic sites along opposite edges can be comparable in strength with that between two sites on the same edge of a finite-width ribbon, before it eventually tends to zero as the ribbon width tends to infinity. This remarkable behavior shows a strong dependence on the sublattice labels of the sites and is in contrast with properties of the modified hopping edge modes. The explicit form of our analytical expressions for the electronic spectrum enables us to predict the zero-mode properties (including frequency, spatial attenuation, and intensity) when the hopping values along ZZ edges are modified.