Phonon dispersion of graphene revisited

The phonon dispersion of graphene is derived by using a simple mass spring model and considering up to the first, second, third, and fourth nearest-neighbor interactions. The results obtained from different nearest-neighbor interactions are compared and it is shown that the k ² dependence for the out-of-plane transverse acoustic mode obtained in other sophisticated methods as well as experiment occurs only after including the fourth nearest-neighbor interaction.