From graphene sheets to graphene nanoribbons: Dimensional crossover signals in structural thermal fluctuations

We study the dimensional crossover from two dimensional to one dimensional (1D) in the structure of the graphene honeycomb lattice when one of the dimensions of the layer is reduced by analyzing the thermally excited rippling. Through a joint study, by Monte Carlo (MC) simulations using a quasiharmonic potential and by analytical calculations, we find that the normal-normal correlation function does not change its power law behavior in the long wavelength limit. However, the dependency of the square of the out-of-plane displacement (< h(2)>) against the system size changes its scaling behavior when going from a layer to a nanoribbon. We show that a new scaling law appears which corresponds to a truly 1D behavior and we estimate the ratio of the sample dimensions where the crossover takes place as R-2D <-> 1D approximate to 1.609. Having explored a wide number of realistic system sizes, we conclude that narrow ribbons present stronger corrugations than the square graphene layers and we discuss the implications for the electronic properties of free-standing graphene samples.