Point Defects and Diffusion in Nonstoichiometric Metal Oxides
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EMBER1991
librium that relates structural éléments of the crystal to one of the chemical potentials involved, and to a couple of thermal point defect formation equilibria. For ternary and higher compounds, relationships between the différent chemical potentials involved also enter in. In the following, for sake of simplicity, only binary ionic compounds of the type Me x _sO will be considered. Due to the Gibbs-Duhem relation, this reduces the number of independent chemical potentials of crystal components to one. An example for a reaction which relates structural éléments to a chemical potential of a crystal component is in this case the formation of cation vacancies: x(Me*$4+y + 1/2 0 2 (gas) ±5 XCVWCAH)^" + 2 h" + Me x O(surf).
(1)
The superscripts ", ', and ' indicate that the species are neutral or negatively or positively charged. Thermal point defect equilibria are the Frenkel disorder of cations (-» formation of cation vacancies and cation interstitials), the Schottky equilibrium (-» formation of cation and anion vacancies), the Frenkel disorder of anions (—> formation of anion vacancies and anion interstitials), and the recombination equilibrium ( -» formation of électrons and holes). In addition to the defects considered before, associâtes or even clusters of (ionic and electronic) point defects may form. The values of the equilibrium constants for the différent defect formation reactions vary with total pressure and température. For ternary and higher Systems, they are also a function of the system's composition. If the defects can occur in différent coordinations (sublattices), as for
example in spinels, in principle the defect formation reactions hâve to be formulated for ail occurring coordinations. For the quantitative treatment of point defect equilibria it is very convenient to work with concentrations per lattice molécule, i.e., concentrations relative to the formula unit Me x _ s O in the case under discussion hère. Such concentrations are denoted by rectangular brackets around the species considered, for example, [Me] = x — 8 and [O] = 1. In considering the constraint of electroneutrality one has in principle two différent possibilities: (1) to consider only relative charges, or (2) to consider only absolute charges. In most cases, the first approach is used because it is the easiest. However, the second approach must be used if it is not possible to unequivocally attribute a distinct charge to an ion occupying a certain lattice site, for example in spinels of the type Me 3 -s0 4 at high températures. For the following, it is assumed that Me x _ { 0 has only one type of lattice site for cations and anions, respectively. For the thermodynamic treatment of the point defect equilibria, the equilibrium constants hâve to be expressed in terms of activities of the reactants and products. Hère one encounters the problem of appropriately accounting for the (mainly) electrostatic interaction between charged point defects. In gênerai, one has to expect that this interaction increases with increasing poin
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