Statistics of Crystal Defects from Dilute Limit to Alloys, Excluded Volume Exchange Interaction and Effective Energetics of Defect Ensembles

Point defects determine the properties of otherwise-perfect semiconductors and insulators, thus computing their concentrations accurately is of prime importance. Typically, the dilute limit is assumed leading to Boltzmann statistics, however it is obvious that multiple defects or defect complexes cannot occupy the same site at the same time. Herein, we present closed-form expressions for point defect statistics applicable for all concentrations; thus unifying defect and alloy theories. These expressions prevent unphysical prediction of more defects than available sites when either a) any defect of type j in chargestate q has zero or negative formation energy ΔE_j^q (e.g. charged defects at certain Fermi energies) or more interestingly b) when multiple defects and/or chargestates have small ΔE_j^q compared to the thermal energy k_BT. An important insight is that different statistics arise if 1) host atom or 2) site conservation is assumed: Case 1 corresponds to a finite crystal with N_o sites and moving displaced host-crystal atoms to extra unit cells on the surface, while Case 2 corresponds to an infinite crystal with N_o sites per volume and moving displaced host-crystal atoms into external reservoirs (e.g. the case corresponding to modern DFT supercell calculations).<br/>Analogously to quantum Fermions but classical in origin, the mutual exclusion between defects of any type can be conceptualized as an exchange interaction energy. Additionally, we demonstrate that multiple chargestates of a defect, multiple configurations of a chargestate, or other arbitrary subgroups of defects and complexes can be gathered together into objects having effective formation energy. This allows, for example, plotting the finite-temperature formation enthalpy of a defect having multiple chargestates as a continuous function of Fermi level or using an effective chemical potential to enforce constraints on subgroups of defects, amongst other applications.<br/>These fundamental tools add flexibility and accuracy to quantitative defect calculations and join smoothly to thermodynamics of solutions and alloys while adding only minor computational costs.