We present a unified theory of the phonon dispersions and elastic properties of graphene, graphite, and graphene multilayer systems. Starting from a fifth-nearest-neighbor force-constant model derived from full in-plane phonon dispersions of graphite [Mohr et al., Phys. Rev. B 76, 035439 (2007)], we use Born’s long-wave method to calculate the tension and bending coefficients of graphene. Extending the model by interplanar interactions, we study the phonon dispersions and the elastic constants of graphite, and the phonon spectra of graphene multilayers. We find that the inner displacement terms due to sublattice shifts between inequivalent C atoms are quantitatively important in determining the elastomechanical properties of graphene and of graphite. The overall agreement between theory and experiment is very satisfactory. We investigate the evolution from graphene to graphite by studying the increase in the ${\omega }_{B_{2g_1}}^{\left(N\right)}$ rigid plane optical mode as a function of the number of layers $N$. At $N=10$ the graphite value ${\omega }_{B_{2g_1}}\approx 127{\text{cm}}^{-1}$ is attained within a few percent.

• Received 25 April 2008

DOI:https://doi.org/10.1103/PhysRevB.78.085424