Raman and Infra-red properties and layer dependence of the phonon dispersions in multi-layered graphene

The symmetry group analysis is applied to classify the phonon modes of $N$-stacked graphene layers (NSGL's) with AB- and AA-stacking, particularly their infra-red and Raman properties. The dispersions of various phonon modes are calculated in a multi-layer vibrational model, which is generalized from the lattice vibrational potentials of graphene to including the inter-layer interactions in NSGL's. The experimentally reported red shift phenomena in the layer number dependence of the intra-layer optical C-C stretching mode frequencies are interpreted. An interesting low frequency inter-layer optical mode is revealed to be Raman or Infra-red active in even or odd NSGL's respectively. Its frequency shift is sensitive to the layer number and saturated at about 10 layers.