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We study the dimensional crossover from 2D to 1D type behavior, which takesplace in the thermal excited rippling of a graphene honeycomb lattice, when oneof the dimensions of the layer is reduced. Through a joint study, by MonteCarlo (MC) atomistic simulations using a quasi-harmonic potential andanalytical calculations, we find that the normal-normal correlation functiondoes not change its power law behavior in the long wavelength limit. Howeverthe system size dependency of the square of out of plane displacement $ <h^2>$changes its scaling behavior when going from a layer to a nanoribbon. We showthat a new scaling law appears which corresponds to a truly 1D behavior and weestimate the ratio of the sample dimensions where the crossover takes place as$R_{2D \leftrightarrow 1D}\approx 1.609$. Having explored a wide number ofrealistic systems sizes, we conclude that narrow ribbons present strongercorrugations than the square graphene sheets and we discuss the implicationsfor the electronic properties of freestanding graphene systems.

From the 2D graphene honeycomb lattice to 1D nanoribbons: dimensional crossover signals in the structural thermal fluctuations