System-size convergence of point defect properties: The case of the silicon vacancy

TL;DR

This study systematically analyzes the convergence of point defect properties in silicon, demonstrating faster transition level convergence with Gamma-point sampling and validating approaches for accurate defect energy calculations.

Contribution

It introduces a comprehensive supercell approach for defect calculations, compares k-point sampling effects, and proposes two methods for valence band offset determination.

Findings

Transition levels converge faster with Gamma-point sampling.

Calculated transition levels agree with experimental data.

MLWFs elucidate defect bonding and verify the Watkins model.

Abstract

We present a comprehensive study of the vacancy in bulk silicon in all its charge states from 2+ to 2-, using a supercell approach within plane-wave density-functional theory, and systematically quantify the various contributions to the well-known finite size errors associated with calculating formation energies and stable charge state transition levels of isolated defects with periodic boundary conditions. Furthermore, we find that transition levels converge faster with respect to supercell size when only the Gamma-point is sampled in the Brillouin zone, as opposed to a dense k-point sampling. This arises from the fact that defect level at the Gamma-point quickly converges to a fixed value which correctly describes the bonding at the defect centre. Our calculated transition levels with 1000-atom supercells and Gamma-point only sampling are in good agreement with available experimental results. We also demonstrate two simple and accurate approaches for calculating the valence band offsets that are required for computing formation energies of charged defects, one based on a potential averaging scheme and the other using maximally-localized Wannier functions (MLWFs). Finally, we show that MLWFs provide a clear description of the nature of the electronic bonding at the defect centre that verifies the canonical Watkins model.