  The errors arising in ab initio density functional theory studies of
semiconductor point defects using the supercell approximation are analyzed. It
is demonstrated that a) the leading finite size errors are inverse linear and
inverse cubic in the supercell size, and b) finite size scaling over a series
of supercells gives reliable isolated charged defect formation energies to
around +-0.05 eV. The scaled results are used to test three correction methods.
The Makov-Payne method is insufficient, but combined with the scaling
parameters yields an ab initio dielectric constant of 11.6+-4.1 for InP. Gamma
point corrections for defect level dispersion are completely incorrect, even
for shallow levels, but re-aligning the total potential in real-space between
defect and bulk cells actually corrects the electrostatic defect-defect
interaction errors as well. Isolated defect energies to +-0.1 eV are then
obtained using a 64 atom supercell, though this does not improve for larger
cells. Finally, finite size scaling of known dopant levels shows how to treat
the band gap problem: in less than about 200 atom supercells with no
corrections, continuing to consider levels into the theoretical conduction band
(extended gap) comes closest to experiment. However, for larger cells or when
supercell approximation errors are removed, a scissors scheme stretching the
theoretical band gap onto the experimental one is in fact correct.
