Leopold Infeld Colloquium

I will present a general prescription, which allows to predict, without performing any calculations, the existence of edge states and zero-energy flat bands in graphene nanoribbons (GNR) with edges of arbitrary shape. First, one need to define the so-called minimal edges, i.e., those having minimum number of edge nodes and dangling bonds per translation period. For GNRs with such edges, the spectrum of the zero-energy edge bands is obtained by folding n-times the spectrum of the simple zigzag GNR, where n is uniquely determined by the projection of the edge translation vector into the zigzag direction. By adding extra nodes to minimal edges, arbitrary modified edges can be obtained. The edge bands of GNRs with modified edges can be found by applying hybridization rules of the extra atoms with the ones belonging to the original edge. The entire prescription reduces to simple diagrams, which additionally predict the localization and degeneracy of the zero-energy bands at one of the graphene sublattices. This is confirmed by the tight-binding and first-principle calculations. The presented rules also allows us to qualitatively predict the existence of E0 bands appearing in the energy gap of certain nanoribbons.