Effect of edges on the dynamics and (de)localization in a tight-binding nanolattice

A square tight-binding lattice model, where the hopping integrals to of surface (edge or boundary) sites are different from the hopping integrals t(1) of interior (bulk) sites, is used to show the effect of edges on dynamics and (de)localization. The quantum propagation dynamics of a particle, or the time dependence of a state vector initially localized on a surface site (for example, on the first site (1,1)), and the probability distribution averaged over time are studied. For each lattice, there exists a value of t(0) = t(c) at which the highest degree of delocalization of the propagation occurs. We show that the propagation of a particle initially localized on a surface site undergoes a transition from being localized over the surface layer to being delocalized over the whole lattice as t(0) changes. This transition is described by an exponential law (y = ae(-x/b) + y(0)) when t(0) < t(c), but by a power law when t(0) > t(c).