Solids, defect concentrations

Conventional physical descriptions of materials in the solid state are concerned with solids in which properties are controlled or substantially influenced by the crystal lattice. When defects are treated in typical solid state studies, they are considered to modify and cause local perturbations to bonding controlled by lattice properties. In these cases, defect concentrations are typically low and usually characterized as either point, linear, or higher-order defects, which are seldom encountered together. 

We have already discussed diffusion in solids to some degree. While bulk properties such as heat capacity are not sensitive to defect concentration, many other properties such as conductivity are. Thus, the method of preparation becomes important if one wishes to obtain a conductive or... 

These limitations are largely eliminated in sophisticated defect calculations described in the following section. This approach can also include more sophisticated site exclusion rules, which allow defects to either cluster or keep apart from each other. Nevertheless, the formulas quoted are a very good starting point for an exploration of the role of defects in solids and do apply well when defect concentrations are small and at temperatures that are not too high. 

Defect populations and physical properties such as electronic conductivity can be altered and controlled by manipulation of the surrounding atmosphere. To specify the exact electronic conductivity of such a material, it is necessary to specify its chemical composition, the defect types and populations present, the temperature of the crystal, and the surrounding partial pressures of all the constituents. Brouwer diagrams display the defect concentrations present in a solid as a function of the partial pressure of one of the components. Because the defect populations control such properties as electronic and ionic conductivity, it is generally easy to determine how these vary as the partial pressure varies. 

Brouwer diagrams plot the defect concentrations in a solid as a function of the partial pressure of the components of the material and are a convenient way of displaying electronic properties (Sections 7.6-7.9). These can be readily extended to include the effects of doping by acceptors or donors. 

A number of factors must be taken into account when the diagrammatic representation of mixed proton conductivity is attempted. The behavior of the solid depends upon the temperature, the dopant concentration, the partial pressure of oxygen, and the partial pressure of hydrogen or water vapor. Schematic representation of defect concentrations in mixed proton conductors on a Brouwer diagram therefore requires a four-dimensional depiction. A three-dimensional plot can be constructed if two variables, often temperature and dopant concentration, are fixed (Fig. 8.18a). It is often clearer to use two-dimensional sections of such a plot, constructed with three variables fixed (Fig. 8.18h-8.18<7). 

Figure 8.19 Schematic representation of the variation of the defect concentrations in BaPrj- YbjOs-s as a function of dopant concentration, assuming fixed temperature, water, and oxygen pressure. The electroneutrality equation used is [h ] = [YbPr]. [Adapted from data in S. Mimuro, S. Shibako, Y. Oyama, K. Kobayashi, T. Higuchi, S. Shin, and S. Yamaguchi, Solid State Ionics, 178, 641-647 (2007).]...

The most important application to be considered under this heading is the calculation of intrinsic defect concentrations in dilute solid solutions. If the solution is so dilute that only the leading terms in the various cluster expansions need be retained then the results required are slight generalizations of those above and follow at once from the notation for the general results. For example, the equilibrium concentration of vacancies in a dilute solution of a single solute, s, is found from Eqs. (74a) and (75) to be... 

The defect concentration comes from thermodynamics. While we will discuss thermodynamics of solids in more detail in Chapter 2, it is useful to introduce some of the concepts here to help us determine the defect concentrations in Eq. (1.43). The free energy of the disordered crystal, AG, can be written as the free energy of the perfect crystal, AGq, plus the free energy change necessary to create n interstitials and vacancies (n, =n-o = n), Ag, less the entropy increase in creating the interstitials ASc at a temperature T ... 

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... 

Figure 5.3 Plots of defect concentrations in a solid MX against the partial pressure of X2(X > K,).

The concepts required for a quantitative treatment of the reactivity of solids were now clear, except for one important issue. According to the foregoing, point defect energies should be on the same order as lattice energies. Since the distribution of point defects in the crystal conforms to Boltzmann statistics, one was able to estimate their concentrations. It was found that the calculated defect concentrations were orders of magnitude too small and therefore could not explain the experimentally observed effects which depended on defect concentrations (e.g., conductivity, excess volume, optical absorption). Jost [W. Jost (1933)] provided the correct solution to this problem. Analogous to the fact that NaCl can be dissolved in H20... 

Defect thermodynamics is more complicated when applied to binary (or multi-component) compound crystals. For binary systems, there is one more independent thermodynamic variable to control. In the case of extended binary solid solutions, one would normally choose a composition variable for this purpose. For compounds with very narrow ranges of homogeneity (i.e., point defect concentrations), however, the composition is obviously not a convenient variable. The more natural choice is the chemical potential of one of the two components of the compound crystal. In practice one will often use the vapor pressure ( activity) of this component. 

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. 

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. 

For elemental solids and stoichiometric compound crystals, the primary influence of irradiation on their kinetic behavior is due to the increase in Acv(s Ac,). We would expect the enhancement in the component diffusion to be in proportion to the increase in the (average) defect concentrations, thus influencing all homogeneous, inhomogeneous, and heterogeneous solid state reactions. 

The influence of plastic deformation on the reaction kinetics is twofold. 1) Plastic deformation occurs mainly through the formation and motion of dislocations. Since dislocations provide one dimensional paths (pipes) of enhanced mobility, they may alter the transport coefficients of the structure elements, with respect to both magnitude and direction. 2) They may thereby decisively affect the nucleation rate of supersaturated components and thus determine the sites of precipitation. However, there is a further influence which plastic deformations have on the kinetics of reactions. If moving dislocations intersect each other, they release point defects into the bulk crystal. The resulting increase in point defect concentration changes the atomic mobility of the components. Let us remember that supersaturated point defects may be annihilated by the climb of edge dislocations (see Section 3.4). By and large, one expects that plasticity will noticeably affect the reactivity of solids. 

Let us -assert, however, that the input of mechanical energy into solids in the sense of tribochemistry always results in a change of their kinetic behavior. The change in point defect concentration, dislocation or crack density, and structure influences the transport coefficients and reactive properties (e.g., catalytic activity, nucleation rate, etc.). 

Chemical Equilibrium. Although CVD is a nonequilibrium process controlled by chemical kinetics and transport phenomena, equilibrium analysis is usefiil in understanding the CVD process. The chemical reactions and phase equilibria determine the feasibility of a particular process and the final state attainable. Equilibrium computations with intentionally limited reactants can provide insights into reaction mechanisms, and equilibrium analysis can be used also to estimate the defect concentrations in the solid phase and the composition of multicomponent films. 

Thermodynamic analysis can be useful also in predicting the effect of gas-phase composition on defect concentrations in the solid and, implicitly, on the electrical properties of the deposited film (I, 95, 96). This technique has been used to predict the concentration and change in electrical carriers from electrons to holes in PbS with increasing sulfur pressure over the PbS crystal (96). 

To our knowledge there have been no reported measurements of equilibrium defect concentrations in soft-sphere models. Similarly, relatively few measurements have been reported of defect free energies in models for real systems. Those that exist rely on integration methods to connect the defective solid to the perfect solid. In ab initio studies the computational cost of this procedure can be high, although results have recently started to appear, most notably for vacancies and interstitial defects in silicon. For a review see Ref. 109. 

In solids the free positron lifetime r lies in the approximate range 100-500 ps and is dependent upon the electron density. Following implantation, the positrons are able to diffuse in the solid by an average distance L+ = (D+t)1//2, where D+ is the diffusion coefficient. This quantity is usually expressed in cm2 s-1 and is of order unity for defect-free metallic moderators at 300 K (Schultz and Lynn, 1988). The requirement of very low defect concentration arises because the value of D+ is otherwise dramatically reduced owing to positron trapping at such sites. 

---