Edge states and flat bands in graphene nanoribbons with arbitrary geometries

Abstract

We prescribe general rules to predict the existence of edge states and zero-energy flat bands in graphene nanoribbons and graphene edges of arbitrary shape. No calculations are needed. For the so-called minimal edges, the projection of the edge translation vector into the zigzag direction of graphene uniquely determines the edge bands. By adding nodes to minimal edges, arbitrary modified edges can be obtained; their corresponding edge bands can be found by applying hybridization rules of the extra states with those belonging to the original edge. Our prescription correctly predicts the localization and degeneracy of the zero-energy bands at one of the graphene sublattices, confirmed by tight-binding and first-principles calculations. It also allows us to qualitatively predict the existence of E = 0 bands appearing in the energy gap of certain edges and nanoribbons.

van der Waals interaction in magnetic bilayer graphene nanoribbons

“…11 Other edge terminations are possible, but they can be mapped onto three basic types, the armchair being the only one without edge states. 12,13 These localized states close to the Fermi energy are responsible for the magnetic and transport properties of zigzag graphene ribbons, and they are the origin of defect-related interface bands in graphene junctions. 14 Within a simple tightbinding model, armchair graphene nanoribbons (AGNRs) can be either metallic or semiconducting depending on their width, 15 whereas zigzag graphene nanoribbons (ZGNRs) are metallic with edge states.…”

Electronic properties of twisted bilayer nanoribbons

“…In pure zigzag nanoribbons, zero-energy states are strongly localized at these atoms [29,30]. In chiral geometries, edge states are also related to the presence of zigzag edge atoms, although they present remarkable size effects [28]. Indeed, physical properties of chiral GNRs and general edges can have a strong dependence on chirality [31,32].…”

“…The type of zero-energy spectrum of a given periodic chiral edge with edge translation vector T = (p,q) (with p q) is determined by its zigzag component (p − q,0) [28]. The edge spectrum is obtained by folding p − q times the (1,0) edge band, which corresponds to a pure zigzag edge.…”

“…We first recall that minimal periodic graphene edges can be classified into four groups, depending on the position of the Dirac points in the one-dimensional (1D) Brillouin zone and on the wave vector span and degeneracies of the edge bands [28]. Armchair edges are the only case without edge states; all the rest do have zero-energy localized edge states due to the presence of zigzag edge atoms.…”

“…It has been theoretically established that all edges, with the exception of the pure armchair, present zero-energy localized states with a predominant * eric.suarez@usm.cl weight at the edge atoms [27,28]. In fact, in minimal [27] edges, zero-energy states stem from the zigzag terminations.…”