Flexural Phonons in Free-Standing Graphene

Rotation and reflection symmetries impose that out-of-plane (flexural) phonons of freestanding graphene membranes have a quadratic dispersion at long wavelength and can be excited by charge carriers in pairs only. As a result, we find that flexural phonons dominate the phonon contribution to the resistivity $\rho$ below a crossover temperature $T_x$ where we obtain an anomalous temperature dependence $\rho \propto T^{5/2}\ln T$. The logarithmic factor arises from renormalizations of the flexural-phonon dispersion due to coupling between bending and stretching degrees of freedom of the membrane.

Figure 1

Electron-phonon vertices. (a) Coupling of electrons to in-plane phonons. (b) Coupling of electrons to flexural phonons. Straight lines correspond to electrons, wavy (dashed) lines to in-plane (flexural) phonons.

Figure 2

One-loop correction to the bending rigidity due to the effective interaction between flexural modes.