Bending rigidity, sound propagation and ripples in flat graphene

Many of the applications of graphene rely on its uneven stiffness and high thermal conductivity, but the mechanical properties of graphene—and, in general, of all two-dimensional materials—are still not fully understood. Harmonic theory predicts a quadratic dispersion for the out-of-plane flexural acoustic vibrational mode, which leads to the unphysical result that long-wavelength in-plane acoustic modes decay before vibrating for one period, preventing the propagation of sound. The robustness of quadratic dispersion has been questioned by arguing that the anharmonic phonon–phonon interaction linearizes it. However, this implies a divergent bending rigidity in the long-wavelength regime. Here we show that rotational invariance protects the quadratic flexural dispersion against phonon–phonon interactions, and consequently, the bending stiffness is non-divergent irrespective of the temperature. By including non-perturbative anharmonic effects in our calculations, we find that sound propagation coexists with a quadratic dispersion. We also show that the temperature dependence of the height fluctuations of the membrane, known as ripples, is fully determined by thermal or quantum fluctuations, but without the anharmonic suppression of their amplitude previously assumed. These conclusions should hold for all two-dimensional materials.