Dispersion of Flexural Modes in Graphene

Using the Born–von Karman model, we have constructed a simple analytic theory of dispersion of flexural vibrations in graphene, which makes it possible to take into account any number of configuration spheres. In the framework of this theory, the quadratic dispersion of flexural acoustic phonons in graphene has been analyzed. It is shown that such a dispersion of flexural vibrations in graphene is the interaction of each atom with not only nearest neighbors, but also with more distant atoms. The signs of effective force constants corresponding to different coordination spheres must be various, preserving the system stability. We have obtained analytic relations between elastic constants, for which flexural vibrations can propagate in the plane of graphene. In the “critical” relation derived in this study, negative elastic constants of the second and third coordination sphere compensate the positive contribution from the first coordination sphere almost completely. As a result, the propagation of low-frequency flexural acoustic waves with conventional linear dispersion in a narrow interval of wavevector values near k = 0 turns out to be impossible. We have determined the conditions in which the dispersion relation of flexural modes in graphene becomes quadratic, which is typical of flexural vibrations in thin macroscopic membranes.