Many-body effects on out-of-plane phonons in graphene

We study the properties of out-of-plane phonons in the framework of the many-body theory of graphene. We investigate, in particular, the way in which the coupling to electron-hole excitations renormalizes the dispersion of the acoustic branch of out-of-plane phonons. We show that the effect of the charge polarization cuts off the quadratic dispersion at low energies, implying the absence of long-wavelength flexural phonons. This result holds in the low-energy Dirac theory of graphene, and it is confirmed by an analysis of the corrections to the interaction vertex beyond the random phase approximation (RPA). Furthermore, we show that the acoustic branch of out-of-plane phonons presents near the K point a strong Kohn anomaly, which is much more pronounced than in the case of the in-plane phonons. The origin of the strong softening of the dispersion lies in the singular behaviour of the intervalley polarization at the threshold of electron-hole formation. This leads to a new branch of hybrid modes below the electron-hole continuum, with the potential to induce significant effects in the transport properties of graphene in the low-temperature regime.