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 $E_{2g}$ mode at $\Gamma$ 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 $5{\mathrm{cm}}^{-1}$ for bilayer and $2–5{\mathrm{cm}}^{-1}$ for trilayer graphene. The interlayer coupling is estimated to be about $2{\mathrm{cm}}^{-1}$. We found that the highest optical modes at $K$ move up by about $12{\mathrm{cm}}^{-1}$ for bilayer and $18{\mathrm{cm}}^{-1}$ for trilayer relative to monolayer graphene. The atomic displacements of these optical eigenmodes are analyzed.

• Received 24 August 2007

DOI:https://doi.org/10.1103/PhysRevB.77.125401