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We present a new semi-empirical potential for graphene, which includes alsoan out-of-plane energy term. This novel potential is developed from densityfunctional theory (DFT) calculations for small numbers of atoms, and can beused for configurations with millions of atoms. Our simulations show thatbuckling caused by typical defects such as the Stone-Wales (SW) defect extendsto hundreds of nanometers. Surprisingly, this long-range relaxation lowers thedefect formation energy dramatically - by a factor of $2$ or $3$ - implyingthat previously published DFT-calculated defect formation energies suffer fromlarge systematic errors. We also show the applicability of the novel potentialto other long-range defects including line dislocations and grain boundaries,all of which exhibit pronounced out-of-plane relaxations. We show that theenergy as a function of dislocation separation diverges logarithmically forflat graphene, but converges to a constant for free standing buckled graphene.A potential in which the atoms are attracted to the 2D plane restores thelogarithmic behaviour of the energy. Future simulations employing thispotential will elucidate the influence of the typical long-range buckling andrippling on the physical properties of graphene.

Strong long-range relaxations of structural defects in graphene simulated using a new semi-empirical potential