Elastic properties of graphene flakes: Boundary effects and lattice vibrations

We present a phenomenological theory together with explicit calculations of the electronic ground-state energy, the surface contribution, and the elastic constants (``Lam\'e parameters,'' i.e., Poisson ratio, Young's modulus) of graphene flakes on the level of the density-functional theory employing different standard functionals. We observe that the Lam\'e parameters in small flakes can differ from the bulk values by 30% for hydrogenated zigzag edges. The change results from the edge of the flake that compresses the interior. When including the vibrational zero-point motion, we detect a decrease in the bending rigidity, κ, by ~26%. The vibrational frequencies flow with growing N due to the release of the edge-induced compression. We calculate the corresponding Gr\"uneisen parameters and find good agreement with previous authors.