A Convergent Understanding of Charged Defects

Historically, defects in semiconductors and ionic conductors have been studied using very different approaches. In the solid-state ionics community, nonstoichiometry and defect thermochemistry are often probed directly through experiments. The dependency of defect concentrations on chemical conditions (typically oxygen pressure) are modeled using a physical chemistry framework and compactly represented by the well-known Brouwer diagrams. In contrast, defects in electronic materials are now studied primarily with computational approaches-often density functional theory (DFT)-based on semiconductor physics in which the energy of defect formation also has an explicit dependence on the Fermi level, making the defect energy diagram multidimensional. As charged defects begin to attract the attention of experts from both schools of thought for applications in thermoelectrics, solar cells, batteries, fuel cells, and other electrochemical devices, a consistent understanding of charged defects addressing the apparent gaps in the two approaches is necessary. This Account reviews both methods using a common notation and thermodynamics to clarify misunderstandings between the fields. We demonstrate the equivalence between the Brouwer diagrams obtained from DFT calculated defect energy diagrams and those constructed using the simple analytical theory described in physical chemistry textbooks. We show how the explicit Fermi level dependence of defect energy in semiconductor physics appears as an electron concentration in the mass action law using a constant defect energy defined in a standard state, Delta G(d)(circle minus) center dot Delta G(d)(circle minus) can also be visualized on a defect energy diagram. Furthermore, we develop the utility of a Brouwer band diagram to compactly map defect and charge concentration as well as important electronic dopability information in compound semiconductors over a multidimensional chemical potential space into a single 2-dimensional plot. Semiconductors and ion conductors often have distinct mechanisms to compensate for the additional charge introduced by extrinsic (or impurity) doping with aliovalent species. Whereas such extrinsic doping in ionic conductors (e.g., Gd doped CeO2-x) results in the formation of intrinsic ionic defects (e.g., V-O), in the case of traditional semiconductors (e.g., P doped Si) free electronic charge carriers are formed. Using the example of thermoelectric Mg3Sb2, which can exhibit both these mechanisms depending on chemical conditions, we explain charge compenstaion of extrinsic dopants (e.g., doping efficiency) using the simple mass action laws for intrinsic defect reactions.