Symmetry constraints on phonon dispersion in graphene

We calculate the phonon dispersion for graphene with interactions between the first, second, and third nearest neighbors in the framework of the Born–von Karman model taking into account the constraints imposed by the lattice symmetry. Analytical expressions give the nonzero sound velocity for the out-of-plane (bending) mode. The dispersion of four in-plane modes is determined by coupled equations. Values of the force constants are found in fitting with frequencies at critical points and elastic constants measured on graphite.