Vibrational properties of graphene quantum dots: Effects of confinement, geometrical structure, and edge orientation

We perform ab initio density functional theory calculations to study the lattice vibrations of graphene quantum dots (GQDs) with triangular, quadrate, and hexagonal shapes in lateral confinement of 1 to 3 nm and terminated in both zigzag and armchair orientations. We see the vibrational properties of GQD transform from molecular type into bulk-like type with increasing dot size and the features of the vibrational density of state also highly depend on the symmetry of the GQD. We find that the out-of-plane vibrated standing waves induced by lateral confinement instead of the in-plane vibrations dominate the lattice vibrations of small GQDs. By projecting the vibrational eigenvectors of GQDs on those of single-layer graphene, we see the mixture of the out-of-plane and in-plane vibrational characters in GQDs as the result of the strong lateral confinement. We identify coherent acoustic phonon modes in GQDs and find that the size dependence of coherent acoustic phonon frequency increases with increasing the isotropy of nanostructures. Moreover, different confinement effects on lattice vibrations along zigzag and armchair edge orientations are identified from ab initio calculations. We describe the Raman intensity of GQDs and observe a blue shift of the G-mode position in GQDs comparing to that of the single-layer graphene. Combining the bond length distribution at the edge of GQDs with the positive Gr\"uneisen parameters, we link such blue shift to the surface effect of GQDs.