Bending rigidity, sound propagation and ripples in flat graphene

TL;DR

This paper demonstrates that rotational symmetry preserves the quadratic flexural dispersion in graphene, ensuring finite bending rigidity and coexistence with sound propagation, while clarifying the origin of ripples in 2D materials.

Contribution

It provides a non-perturbative analysis showing that phonon interactions do not linearize flexural modes, reconciling theory with experimental observations in 2D materials.

Findings

Quadratic flexural dispersion is protected by rotational symmetry.

Bending rigidity remains finite at all temperatures.

Ripples are driven by thermal and quantum fluctuations, not anharmonic effects.

Abstract

Despite many of the applications of graphene rely on its uneven stiffness and high thermal conductivity, the mechanical properties of graphene, and in general of all 2D materials, are still elusive. The harmonic theory predicts a quadratic dispersion for the flexural acoustic vibrational mode, which leads the unphysical result that long wavelength in-plane acoustic modes decay before vibrating one period, preventing the propagation of sound. The robustness of the 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 limit not reproduced experimentally. Here we show that rotational symmetry protects the quadratic flexural dispersion against phonon-phonon interactions and that, consequently, the bending stiffness is non-divergent irrespective of the temperature. Our non-perturbative anharmonic calculations also determine 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. The universality of our conclusions reconcile experimental evidence and theory not just in graphene, but all 2D materials.