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Formation energies of charged point defects in semiconductors are calculatedusing periodic supercells, which entail a divergence arising from long-rangeCoulombic interactions. The divergence is typically removed by the so-calledjellium approach. Recently, Wu, Zhang and Pantelides [WZP, Phys. Rev. Lett.119, 105501 (2017)] traced the origin of the divergence to the assumption thatcharged defects are formed by physically removing electrons from or addingelectrons to the crystal, violating charge neutrality, a key principle ofstatistical mechanics that determines the Fermi level. An alternative theorywas constructed by recognizing that "charged" defects form by trading carrierswith the energy bands, whereby supercells are always charge-neutral so that nodivergence is present and no ad-hoc procedures need to be adopted forcalculations. Here we give a more detailed exposition of the foundations ofboth methods and show that the jellium approach can be derived from thestatistical-mechanics-backed WZP definition by steps whose validity cannot beassessed a priori. In particular, the divergence appears when the chargedensity of band carriers is dropped, leaving a supercharged crystal. In thecase of charged defects in two-dimensional (2D) materials, unphysical fieldsappear in vacuum regions. None of these pathological features are present inthe reformulated theory. Finally, we report new calculations in both bulk and2D materials. The WZP approach yields formation energies that differ fromjellium values by up to ~1 eV. By analyzing the spatial distribution of wavefunctions and defect potentials, we provide insights into the inner workings ofboth methods and demonstrate that the failure of the jellium approach toinclude the neutralizing electron density of band carriers, as is the case inthe physical system, is responsible for the numerical differences between thetwo methods.

Formation energies of charged defects in two-dimensional materials -- resolution of long-standing difficulties