Atomic geometry and stability of mono-, Di-, and trivacancies in graphene

Stability and atomic geometry of mono-, di-, and trivacancies in graphene sheets are studied by using first-principles calculations. We find that the atomic relaxation substantially contributes to the stability of the vacancies. The monovacancy is found to have a nonplanar structure, i.e., its symmetry is C-1h, while the ideal monovacancy has D-3h symmetry. The divacancy is found to have a 5-8-5 membered ring structure. The trivacancy is also found to have two five membered rings. The energetics of these vacancies are not explained by the conventional dangling-bond counting model, which does not include lattice relaxation. Our calculations show that the divacancy is very stable and is thus expected to be detected under some experimental conditions.