Localized states in graphene nanoribbons

We present spatially localized states in graphene nanoribbons using the simple tight-binding model with nearest neighbor interactions, which correctly describes the electronic properties of graphene close to the Fermi level. By monitoring the time evolution of initially localized wave packets, we identify different final states depending on edge geometry and initial condition. For armchair nanoribbons, we find, both numerically and analytically, flat band states that remain strictly localized across the nanoribbon width instead of spreading in the infinite periodic direction. For zigzag nanoribbons, we find partially flat band states at the Fermi level, which are localized both in the transverse and longitudinal directions, different from the well-known localized edge states that extend along the whole length of the zigzag edge and decay to zero in the transverse direction. The effects of nonlinearity induced by interactions on these states and on wave packet spreading in general are examined within the discrete nonlinear Schrödinger equation model in and out of the self-trapping regime. We also examine the effects of disorder by introducing random on-site energies to find that all wave packets evolve to exponentially localized states, as expected. These localization phenomena with different origin, from edge geometry to nonlinearity and disorder, should affect wave propagation and transport in atomically thin two-dimensional nanostructures and should be observed in honeycomb lattice systems in photonics, cold atoms, and other physical contexts, opening new directions toward the targeted transfer of relevant excitations.