Defects equilibrium” defects

A plot of this equation (Fig. 2.1 b) closely resembles Figure 2.1a. The minimum in the curve gives the equilibrium number of vacancies present and confirms that vacancies exist in all crystals at temperatures above 0 K. For this reason these defects cannot be removed by thermal treatment but are always present in a crystal. Such defects are thus intrinsic defects. At equilibrium, AGy will be equal to zero and the minimum in the AGy versus tiy curve is given by... 

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. 

Let us imagine equilibrating a fayalite crystal (Fe2Si04) in an atmosphere sufficiently oxidizing to allow a defect equilibrium of the following type to proceed toward the right ... 

The equilibrium concentration of intrinsic defects in a structure depends on temperature. For the Schottky defect, the equilibrium constant K for the defect-generation reaction is... 

Let us first examine what happens to a crystal when we remove, add, or displace an atom in the lattice. We will then describe how a different atom, called an impurity (regardless of whether or not it is beneficial), can fit into an established lattice. As shown by Eq. (1.36), point defects have equilibrium concentrations that are determined by temperature, pressure, and composition. This is not true of all types of dimensional defects that we will study. 

Defect Reaction Equilibrium Constants. Recall that a Frenkel disorder is a self interstitial-vacancy pair. In terms of defect concentrations, there should be equal concentrations of vacancies and interstitials. Frenkel defects can occur with metal... 

In the discussion of defect equilibrium we discussed neutral defects such as V corresponds to an anion vacancy which has captured an electron. Such vacancies give rise to interesting optical phenomena. Defects associated with electrons or holes lead to colouration of the crystals and are known as colour centres. The term colour centre also includes impurity centres such as Tl, which are responsible for absorption and... 

Oxidation of zinc to zinc oxide is another example whose kinetics have been interpreted in terms of the Wagner model (Wagner Grunewald, 1938). At 670 K, the reaction has been found to be independent of oxygen pressure between 0.02 and 1 atm. ZnO is a n-type semiconductor, having a stoichiometric excess of zinc accommodated as interstitials the defect equilibrium could be represented as... 

The conductivity measurements show that equilibrium (1) sets in rapidly at temperatures as low as 500°C. Since the melting point of zinc oxide is about 2100°C. and accordingly its Tamman temperature about 900°C., the process under consideration cannot possibly involve the bulk of the crystal because defects could not diffuse rapidly enough through the lattice at such low temperatures. Except at very high temperatures, the defect equilibrium is realized only at the surface of the crystal, that is, in a layer of a few unit cells thickness. 

Let us consider the defect equilibrium, with the condition Pqi < 1 0 ° at fixed temperature, where Pq denotes the equilibrium oxygen pressure at <5 = 0. In this condition, oxygen escapes from the solid phase by the following reactions... 

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

In crystals, non-steady state component transport locally alters the number, and sometimes even the kind, of point defects (irregular SE s). As a consequence, the relaxation of defect concentrations takes place continuously during chemical interdiffusion and solid state reactions. The rate of these relaxation processes determines how far local defect equilibrium can be established during transport. 

The Kirkendall effect in metals shows that during interdiffusion, the relaxation time for local defect equilibration is often sufficiently short (compared to the characteristic time of macroscopic component transport) to justify the assumption of local point defect equilibrium. In those cases, the (isothermal, isobaric) transport coefficients (e.g., Dh bj) are functions only of composition. Those quantitative methods introduced in Section 4.3.3 which have been worked out for multicomponent diffusion can then be applied. 

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

If local point defect equilibrium prevails and space charge effects can be neglected, one finds from the condition of electroneutrality that... 

Self-diffusion of Ag cations in the silver halides involves Frenkel defects (equal numbers of vacancies and interstitials as seen in Fig. 8.116). In a manner similar to the Schottky defects, their equilibrium population density appears in the diffusivity. Both types of sites in the Frenkel complex—vacancy and interstitial— may contribute to the diffusion. However, for AgBr, experimental data indicate that cation diffusion by the interstitialcy mechanism is dominant [4]. The cation Frenkel pair formation reaction is... 

In thermal equilibrium, some ionic crystals at a temperature above absolute zero enclose a certain number of Schottky pair defects, that is, anion and cation vacancies in the structure (see Section 5.7.1) [13]. Since the concentration of Schottky pair defects at equilibrium at an absolute temperature, T, obeys the mass action law, then [16]... 

With respect to the electronic charge carriers in ionic solids, it should be mentioned that semiconductor-like band structures with relatively wide band gaps (for example ca. 3.1 eV in SrTiC>3, 7.3 eV in NaCl and 3.2 eV in AgCl [76-78]) can often be found. In these solids, electrons in the conduction band (CB), and holes in the valence band (VB), can be regarded as electronic defects in equilibrium via the reaction... 

Fig. 6.16. Diagram of the doping dependence of the defect gap state distribution predicted by the defect equilibrium model.

The phenomenon of metastability is closely related to the defect equilibrium properties. Some external excitation - illumination, charge, current flow, energetic particles, etc. - induces defects (or dopants) which are subsequently removed by annealing. The metastable... 

Figure 1. Native-defect equilibrium in ZnS at 1000 K points andB correspond to saturated Zn and S2 vapors, respectively. Other designations are explained in text.

In the enumeration of the types of defect in 1.2 some effort was made to produce a rational classification. The placing of a defect solid in one or other class is undoubtedly somewhat arbitrary, particularly as one solid of definite composition may well contain defect systems of different kinds and these may interact. Moreover, some types of defect have temperature-dependent concentrations and would disappear in the true thermodynamic equilibrium state at 0 i they may be referred to as thermal, indicating their origin. Defects of other kinds that are inherent in the particular solid would be present in the equilibrium state at they have been referred to as biographical defects. Defects of all types... 

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