Edge States and Flat Bands of Graphene Nanoribbons with Edge Modification

We study the electronic states of graphene nanoribbons with modified edge structures by attaching Klein's bearded bonds as a minimal model of edge modification. The partial attachment of Klein's bearded bonds to graphene nanoribbons gives rise to the partial flat bands at zero-energy even under the condition of vertical bar N-A - N-B vertical bar = 0, where N-A (N-B) is the number of A (B)-sublattice sites. Using transfer matrix method, we successfully derive the analytic representation of edge states for modified zigzag edge. The modification of armchair edges causes the complete flat bands, where the wavefunction has the character of valley polarization. We also applied the density functional theory to optimize the lattice structure and estimate the spin density. Our results indicate that the chemical and structural modification of graphene edge will serve to design and stabilize the spin polarized edge states.