Ripples in isotropically compressed graphene

An isotropic compression of graphene is shown to induce a structural deformation on the basis of Density Functional Perturbation Theory. Static instabilities, indicated by imaginary frequency phonon modes, are induced in the high symmetry Γ-K (zigzag) and Γ-M (armchair) directions by an isotropic compressive strain of the graphene sheet. The wavelength of the unstable modes (ripples) is directly related to the magnitude of the strain and remarkably insensitive to the direction of propagation in the 2D lattice. These calculations further suggest that the formation energy of the ripple is isotropic for lower strains and becomes anisotropic for larger strains. This is a result of graphene's elastic property, which is dependent on direction and strain. Within the quasi-harmonic approximation this is combined with the observation that molecular adsorption energies depend strongly on curvature to suggest a strategy for generating ordered overlayers in order to tune the functional properties of graphene.