First-principles density-functional-theory calculations of formation and dissociation energies in germanene multivacancies

Stabilities and atomic geometries in germanene multivacancies has been studied. We carry out eight configurations, mono-, di-, tri-, tetra-, penta-, hexa-, hepta-, and octavacancies by using the density-functional-theory calculations (DFT). We calculate formation and dissociation energies to analyze the stability by finding the magic numbers. Magic numbers are the number of vacancy belonging to stable configurations. We find that di-, tetra-, hexavacancy, and octavacancies (2, 4, 6, and 8) are the magic numbers of gemanene multivacancies. Therefore, we propose that the magic numbers are 2n(n = 1, 2, 3, …). These results are in agreement with those of the DBCM (dangling-bond counting model). We also analize symmetries of the stable configurations and find that the di- and hexavacancies have C2h and D6h symmetries, respectively, while the tetra- and ocatavacancies belong to a symmetry.