Charged defects in two-dimensional semiconductors of arbitrary thickness and geometry: Formulation and application to few-layer black phosphorus

Energy evaluation of charged defects is tremendously important in two-dimensional (2D) semiconductors for the industrialization of 2D electronic devices because of its close relation with the corresponding type of conductivity and its strength. Although the method to calculate the energy of charged defects in single-layer one-atom-thick systems of equilateral unit-cell geometry has recently been proposed, few-layer 2D semiconductors are more common in device applications. As it turns out, one may not apply the one-layer formalism to multilayer cases without jeopardizing accuracy. Here, we generalize the approach to 2D systems of arbitrary cell geometry and thickness and use few-layer black phosphorus to illustrate how defect properties, mainly group-VI substitutional impurities, are affected. Within the framework of density functional theory, we show that substitutional Te (TeP) is the best candidate for n-type doping, and as the thickness increases, the ionization energy is found to decrease monotonically from 0.67 eV (monolayer) to 0.47 eV (bilayer) and further to 0.33 eV (trilayer). Although these results show the ineffectiveness of the dielectric screening at the monolayer limit, they also show how it evolves with increasing thickness whereby setting a new direction for the design of 2D electronics. The proposed method here is generally suitable to all the 2D materials regardless of their thickness and geometry.