Supercell-size convergence of formation energies and gap levels of vacancy complexes in crystalline silicon in density functional theory calculations

Results for supercell-size convergence of formation energies and charge transition levels of vacancy complexes ${V}_{n}\phantom{\rule{0.28em}{0ex}}(1\ensuremath{\le}n\ensuremath{\le}11)$ in crystalline Si are reported for the ab initio density functional theory (DFT) with generalized gradient approximation (GGA) pseudopotentials. When extrapolated to the dilute limit, the formation energy of an uncharged vacancy becomes $3.74\ifmmode\pm\else\textpm\fi{}0.1\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$, and the binding energy of an uncharged divacancy becomes $1.9\ifmmode\pm\else\textpm\fi{}0.2\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$. Stable ${V}_{n}$ clusters are built on the basis of sixfold rings $(n\ensuremath{\le}6)$ and of octahedral voids $(n\ensuremath{\ge}7)$. If the well-known underestimate of the band gap by the DFT and the accuracy of extrapolations are taken into account, the extrapolated levels are in good agreement with experiment. We discuss the implications for simulation of vacancy clustering during thermal quenching, for interpretation of deep-level spectroscopy and electron paramagnetic resonance in irradiated Si, and for cell and Brillouin-zone sampling choice when DFT-related methods beyond local-density approximation or GGA are used.