Structure and Electronic States of a Graphene Double Vacancy with an Embedded Si Dopant

Silicon represents a common intrinsic impurity in graphene, commonly bonding to either three or four carbon neighbors respectively in a single or double carbon vacancy. We investigate the effect of the latter defect (Si-C$_4$) on the structural and electronic properties of graphene using density functional theory (DFT). Calculations based both on molecular models and with periodic boundary conditions have been performed. The two-carbon vacancy was constructed from pyrene (pyrene-2C) which was then expanded to circumpyrene-2C. The structural characterization of these cases revealed that the ground state is slightly non-planar, with the bonding carbons displaced from the plane by up to $\pm$0.2 {\AA}. This non-planar structure was confirmed by embedding the defect into a 10$\times$8 supercell of graphene, resulting in 0.22 eV lower energy than the previously considered planar structure. Natural bond orbital (NBO) analysis showed sp$^3$ hybridization at the silicon atom for the non-planar structure and sp$^2$d hybridization for the planar structure. Atomically resolved electron energy loss spectroscopy (EELS) and corresponding spectrum simulations provide a mixed picture: a flat structure provides a slightly better overall spectrum match, but a small observed pre-peak is only present in the corrugated simulation. Considering the small energy barrier between the two equivalent corrugated conformations, both structures could plausibly exist as a superposition over the experimental timescale of seconds.