A Convergent Understanding of Charged Defects

Abstract

Metrics & MoreArticle Recommendations * sı Supporting Information CONSPECTUS: 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, ΔG d ⊖ . ΔG d ⊖ 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 CeO 2−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 Mg 3 Sb 2 , 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.

Shapes of phases in isothermal phase diagrams: what is wrong with the Thermo-Calc logo

“…The statistical treatment of defects describes the equilibrium concentration of defects, n, as a function of its defect formation energy or defect energy (Δ E d ). In the dilute limit it is expressed as 14,28,29 n = n 0 exp(−Δ E d / k B T )Here n 0 represents the concentration of available sites for the defect in the solid.…”

“…The defect energy depends not only on the energy needed to disrupt the solid by removing or adding atoms but also on the chemical potential of the species added or removed. For example, the substitution of element C for B in a compound AB, which can be described by the chemical reactionAB + δ C → AB 1− δ C δ + δ Bhas a defect energy (Δ E C B ) described by 14 where E AB 1− δ C δ − E AB can be calculated by DFT. Here we emphasize, , is the defect energy at standard elemental reference (SER) with the superscript .…”

“…1) can also greatly influence thermal and electrical properties, [9][10][11] even when electrically neutral. 12,13 Charged defects also have profound effects on electronic and ionic properties of semiconductors, 14 insulators used in transparent conducting oxides, 15 battery materials 16,17 and thermoelectrics. [18][19][20] The solubility of defects not only determines the range of compositions that are possible but often the solubility limits are used to engineer the desired concentration of defects as it is often easier to control processing conditions (temperature, equilibrium phases, vapor pressure) of a multi-phase sample than precisely control the exact atomic composition of a single phase material.…”

“…1) can also greatly influence thermal and electrical properties, 9–11 even when electrically neutral. 12,13 Charged defects also have profound effects on electronic and ionic properties of semiconductors, 14 insulators used in transparent conducting oxides, 15 battery materials 16,17 and thermoelectrics. 18–20…”

VTAnDeM: A Python Toolkit for Simultaneously Visualizing Phase Stability, Defect Energetics, and Carrier Concentrations of Materials

“…[6][7][8][9][10][11][12][13][14] The phase stability, defect formation energies, and carrier concentrations are interconnected properties of a semiconductor that are influenced by the thermodynamic environment. 15,16 In particular, a thermodynamic state is defined by a fixed set of elemental chemical potentials that are determined by the impurity phases with which a material is in equilibrium. For a material in thermodynamic equilibrium, chemical potentials can vary in well-defined ranges leading to variations in material stoichiometry.…”