Bending modes, anharmonic effects, and thermal expansion coefficient in single-layer and multilayer graphene

摘要

We present a simple analytical approach to study anharmonic effects in single-layer, bilayer, and multilayer graphene. The coupling between in-plane and out-of-plane modes leads to negative Griineisen coefficients and negative thermal expansion. The value of the thermal expansion coefficient depends on the coupling to the substrate. The bending rigidity in bilayer graphene shows a crossover between a long wavelength regime where its value is determined by the in-plane elastic properties, and a short wavelength regime where its value approaches twice that of a single layer.