Phonon dispersions and vibrational properties of monolayer, bilayer, and trilayer graphene: Density-functional perturbation theory

The phonon dispersions of monolayer and few-layer graphene ($AB$ bilayer, and $ABA$ and $ABC$ trilayers) are investigated using the density-functional perturbation theory. Compared with the monolayer, the optical phonon E2g mode at Γ splits into two and three doubly degenerate branches for bilayer and trilayer graphene, respectively, due to the weak interlayer coupling. These modes are of various symmetries and exhibit different sensitivities to either Raman or infrared measurements (or both). The splitting is found to be 50.3em0excm^-1 for bilayer and 2--50.3em0excm^-1 for trilayer graphene. The interlayer coupling is estimated to be about 20.3em0excm^-1. We found that the highest optical modes at K move up by about 120.3em0excm^-1 for bilayer and 180.3em0excm^-1 for trilayer relative to monolayer graphene. The atomic displacements of these optical eigenmodes are analyzed.