Managing the supercell approximation for charged defects in semiconductors: finite size scaling, charge correction factors, the bandgap problem and the ab initio dielectric constant

TL;DR

This paper analyzes finite size errors in ab initio defect calculations in semiconductors, demonstrating effective scaling methods, evaluating correction techniques, and addressing the bandgap problem for more accurate defect energy predictions.

Contribution

It introduces reliable finite size scaling procedures and evaluates correction methods, improving the accuracy of charged defect energies in semiconductor supercell calculations.

Findings

Finite size errors are inverse linear and cubic in supercell size.

Scaling over supercells yields defect energies within ±0.05 eV.

A scissors correction scheme aligns theoretical and experimental band gaps.

Abstract

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.