Correcting charged supercell defect calculations in low-dimensional semiconductors

Accurate predictions of the nature of defects in low-dimensional materials are essential to enable and promote their applications in micro- and optoelectronic devices. However, when dealing with charged defects in low-dimensional semiconductors, conventional first-principles calculations based on the jellium model encounter the divergence problem, where the formation energy diverges with the vacuum layer thickness. Two promising approaches, the supercell extrapolation method based on the jellium model [Wang et al., Phys. Rev. Lett. 114, 196801 (2015)] and the real charge model [Xiao et al., Phys. Rev. B 101, 165306 (2020)], have been proposed to address this divergence issue. However, these two models disagree when extrapolated to an infinitely large supercell following the first-order Makov-Payne correction. Whether they are unified or physically different remains unknown. In this work, we analyze the scaling behaviors of charged defect formation energy in these two computational models. We find that when considering the real charge distribution of ionized charge in two-dimensional systems, an extra quadratic finite-size correction ${L}_{S}^{\ensuremath{-}2}$ (${L}_{S}$ is the lateral supercell size) is required for the formation energy calculations. After this correction, the results of these two models can be unified when extrapolated to infinity following the derived scaling behaviors. Meanwhile, based on the understanding of the differences between these two models, we propose a simple and convenient approach to correct the jellium model error for charged defect calculations in low-dimensional semiconductors. Our work thus provides a practical understanding of the charged defect calculations in low-dimensional semiconductors.