First-principles calculations on third-order elastic constants and internal relaxation for monolayer graphene

The method of homogeneous deformation is combined with first-principles total-energy calculations on determining third-order elastic constants and internal relaxation for monolayer graphene. We employ density functional theory (DFT) within generalized-gradient-approximation (GGA). The elastic constants are obtained from a polynomial fitted to the calculations of strain-energy and strain-stress relations. Our results agree well with recent calculations by DFT calculations, tight-binding atomistic simulations, and experiments with an atomic force microscope. The internal relaxation displacement has also been determined from ab initio calculations. The details of internal lattice relaxation by first principles are basically consistent with the previous molecular dynamics (MD) simulation. But for tiny deformation, there is an anomalous region in which the behavior of internal relaxation is backward action. In addition, we have also demonstrated that the symmetry of the relationship between the internal displacement and the infinitesimal stains can be satisfied.