Anharmonic effects in the optical and acoustic bending modes of graphene

The out-of-plane fluctuations of carbon atoms in a graphene sheet have been studied by means of classical molecular dynamic simulations with an empirical force-field as a function of temperature. The Fourier analysis of the out-of-plane fluctuations often applied to characterize the acoustic bending mode of graphene is extended to the optical branch, whose polarization vector is perpendicular to the graphene layer. This observable is inaccessible in a continuous elastic model of graphene but it is readily obtained by the atomistic treatment. Our results suggest that the long-wavelength limit of the acoustic out-of-plane fluctuations of a free layer without stress is qualitatively similar to that predicted by a harmonic model under a tensile stress. This conclusion is a consequence of the anharmonicity of both in-plane and out-of-plane vibrational modes of the lattice. The most striking anharmonic effect is the presence of a linear term, $\omega_{A}=v_{A}k$, in the dispersion relation of the acoustic bending band of graphene at long wavelengths ($k\rightarrow0$). This term implies a strong reduction of the amplitude of out-of-plane oscillations in comparison to a flexural mode with a $k^{2}$-dependence in the long-wavelength limit. Our simulations show an increase of the sound velocity associated to the bending mode, as well as an increase of its bending constant, $\kappa,$ as the temperature increases. Moreover, the frequency of the optical bending mode, $\omega_{O}(\Gamma)$, also increases with the temperature. Our results are in agreement with recent analytical studies of the bending modes of graphene using either perturbation theory or an adiabatic approximation in the framework of continuous layer models.