Localized defect modes in graphene

We study the properties of localized vibrational modes associated with structural defects in a sheet of graphene. For the example of the Stone-Wales defects, one- and two-atom vacancies, many-atom linear vacancies, and adatoms in a honeycomb lattice, we demonstrate that the local defect modes are characterized by stable oscillations with the frequencies lying outside the linear frequency bands of an ideal graphene. In the frequency spectral density of thermal oscillations, such localized defect modes lead to the additional peaks from the right side of the frequency band of the ideal sheet of graphene. Thus, the general structure of the frequency spectral density can provide a fingerprint of its quality and the type of quantity of the structural defect a graphene sheet may contain.