On the forbidden wave numbers in the out-of-plane optical phonon (ZO) branch in carbon fullerenes

A fundamental question in nanoscience is when and how the transition from molecular properties to the bulk behavior occurs. In particular, the convergence of the vibrational properties as a function of the diameter of fullerenes to the bulk material (i.e. graphene) remains as an unsolved challenge. Here we address this issue by studying fullerene’s vibrational density of states (VDOS) as a function of the size (going from 20→720 atoms) and its convergence to graphene’s VDOS using density functional theory. In general, a smooth convergence with the size is observed in most the regions of the VDOS, nevertheless a fundamental difference between the vibrational modes in the fullerene family and the phonon bands in graphene appears: The lack of out-of-plane optical band (ZO-modes) in fullerenes. The higher part of the ZO phonon branch is forbidden in fullerenes due to the presence of pentagonal faces, which restricts the symmetries of the vibrational modes allowed in fullerenes. The analysis and results in this work can be extrapolated to other structures containing pentagonal rings either as a closed shell structure or as pentagonal defects in graphene.