Defect equilibria

The equilibrium between the defects present can be described in terms of chemical equations  

Creation and Elimination of Schottky Defects These defects can form at the crystal surface or vanish by diffusing to the surface. The equation describing this and the associated equilibrium constant is 

Creation and Elimination of Electronic Defects These are the normal intrinsic electrons and holes present in a semiconductor. Electrons can combine with holes to be eliminated from the crystal thus  

Composition Change This comes about by interaction with the gas phase to produce cation or anion vacancies. For oxidation the equation is 

Electroneutrality At all times the crystal must remain electrically neutral. Equations (7.9) and (7.10) define the formation of charged species, so that the appropriate electroneutrality equation is 

The defect structure of these materials is far from simple, with a number of coincident defect equilibria contributing to the observed behavior. First, we must consider the intrinsic defect processes for a hypothetical mixed conductor Lai xSrxBOsi, which undergoes Schottky disorder together with intrinsic electronic disorder. Following an analysis given in [16]  

Equation (5.11) can also be viewed as the dissociation of the effectively neutral B cation (+3) into two charge states (+2 and -i-4)  

For perovskites with variable valent B cations, the redox processes that occur in the lattice must also be taken into account. Oxidation of the lattice can lead to oxygen excess material in which cation vacancies form [17]. 

In this oxygen excess region, the formation of cation vacancies will have the effect of strongly decreasing the oxygen vacancy concentration through the Schottky equilibrium. 

Compensation of the acceptor can occur either by electronic or by a vacancy mechanism, or perhaps more importantly for these materials, by a combination of the two. 

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Another heterotype example is CdCl2 containing Sb3+. Here, we can write at least three different equations involving defect equilibria ... 

Here, we have expressed the concentration as the ratio of defects to the number of M- atom sites (this has certain advantages as we will see). We can than rewrite the defect equilibria equations of Table 3-3 and 3-4 in terms of numbers of intrinsic defect concentrations, shown as follows ... 

Point defect populations profoundly affect both the physical and chemical properties of materials. In order to describe these consequences a simple and self-consistent set of symbols is required. The most widely employed system is the Kroger-Vink notation. Using this formalism, it is possible to incorporate defect formation into chemical equations and hence use the powerful methods of chemical thermodynamics to treat defect equilibria. 

Antisite defect equilibria can be treated in the same way as the other point defect equilibria. The creation of a complementary pair of antisite defects consisting of an A atom on a B atom site, AB, and a B atom on an A atom site, BA, can then be written ... 

Although the formula of this phase is more complex than the binary compounds described above, the procedure is exactly the same. It is only necessary to identify the likely defect equilibria that pertain to the experimental situation. 

Frenkel and Schottky defect equilibria are temperature sensitive and at higher temperatures defect concentrations rise, so that values of Ks and Kv, increase with temperature. The same is true of the intrinsic electrons and holes present, and Kc also increases with temperature. This implies that the defect concentrations in the central part of a Brouwer diagram will move upward at higher temperatures with respect to that at lower temperatures, and the whole diagram will be shifted vertically. 

It is important to know that the defect equilibria that apply to the pure material, and the associated equilibrium constants, also apply to the doped material. The only additional information required is the nature and concentration of the dopant. To illustrate the construction of a diagram, an example similar to that given in Chapter 7 will be presented, for a nonstoichiometric phase of composition MX, nominally containing M2+ and X2- ions, with a stoichiometric composition MXl 0. In this example, it is assumed that the relevant defect formation equations are the same as those given in Chapter 7 ... 

Non-stoichiometric compounds Mass action law treatment of defect equilibria... 

This type of defect equilibrium treatment has been used extensively to model the defect chemistry and non-stoichiometry of inorganic substances and has the great advantage that it easily takes several simultaneous defect equilibria into account [22], On the other hand, the way the mass action laws are normally used they are focused on partial thermodynamic properties and not on the integral Gibbs energy. The latter is often preferred in other types of thermodynamic analyses. In such cases the following solid solution approach is an alternative. 

Stemming from a comparison between calculated and experimental methods, the proposed calculations appear quite precise and thus open up new perspectives in polarization calculations for natural crystalline phases. As we will see in chapter 4, polarization energy is of fundamental importance in the evaluation of defect equilibria and consequent properties. 

Intrinsic disorder is observed in conditions of perfect stoichiometry of the crystal. It is related to two main defect equilibria Schottky defects and Frenkel defects. 

Stability of a Crystalline Compound in the Presence of Defect Equilibria Fayalite as an Example... 

Table 10.3 Defect equilibria affecting REE solubility in pyrope and their effects in bulk garnet/melt REE distribution (Morlotti and Ottonello, 1982)...

Harrison W. J. and Wood B. J. (1980). An experimental investigation of the partitioning of REE between garnet and liquid with reference to the role of defect equilibria. Contrib. Mineral. Petrol., 72 145-155. 

Some earlier thermodynamic studies on rutile reported expressions involving simple idealized quasi-chemical equilibrium constants for point defect equilibria (see, e.g., Kofstad 1972) by correlating the composition x in TiOx with a function of AGm (O2), which is the partial molar free energy of oxygen. However, the structural effects were not accounted for in these considerations. Careful measurements of AGm (O2) in the TiOjc system (Bursill and Hyde 1971) have indicated that complete equilibrium is rarely achieved in non-stoichiometric rutile. 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

Table 5.1. Defect equilibria in an MX crystal exhibiting Schottky defects"...

Analysis of defect equilibria in KBr is a typical example (Kroger, 1974). Since K, and Ki are 3 x 10 and 8 x 10 respectively. Region II in Fig. 5.3 extends over 43 decades (equation 5.7) and hence the partial pressure of Brj, which is proportional to R (see equation 5.5 ), does not affect the defect concentrations. The deviation from stoichiometry, 5, given by — 2[Vx] is also rather small ( 10 ) over the same... 

Solid-solid reactions are as a rule exothermic, and the driving force of the reaction is the difference between the free energies of formation of the products and the reactants. A quantitative understanding of the mechanism of solid-solid reactions is possible only if reactions are studied under well-defined conditions, keeping the number of variables to a minimum. This requires single-crystal reactants and careful control of the chemical potential of the components. In addition, a knowledge of point-defect equilibria in the product phase would be useful. 

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