Concentration of intrinsic defects

In this section we are concerned with the properties of intrinsic Schottky and Frenkel disorder in pure ionic conducting crystals and with the same systems doped with aliovalent cations. As already remarked in Section I, the properties of uni-univalent crystals, e.g. sodium choride and silver bromide which contain Schottky and cationic Frenkel disorder respectively, doped with divalent cation impurities are of particular interest. At low concentrations the impurity is incorporated substitutionally together with an additional cation vacancy to preserve electrical neutrality. At sufficiently low temperatures the concentration of intrinsic defects in a doped crystal is negligible compared with the concentration of added defects. We shall first mention briefly the theoretical methods used for such systems and then review the use of the cluster formalism. 

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... 

However, at the higher temperatures on the left-hand side of the graph, the concentration of intrinsic defects has increased to such an extent that it now is similar to, or greater... 

Table 7.1. Concentrations of intrinsic defects N (m 2) stabilized on the thermo (TSi), mechanically (MSi), and thermo chemically (RSi) activated SiOz surface [9, 10, 15, 18, 19, 52]...

The effects of deliberately added donors, such as titanium, and acceptors, such as iron and magnesium, on electrical conductivity have been studied. Doping with aliovalent ions affects the concentration of intrinsic defects and, in consequence, the diffusivity of A1 and O. In the case of variable-valency dopants, changes in p0l change the fraction of dopants in the aliovalent state and the nature and concentration of the defects. For example, the dopant Ti substitutes for A1 and, in the fully oxidized state, produces the defect TiA1, compensated by Va", so that... 

For diffusion by a vacancy mechanism, the temperature dependence of dilfusivity will depend on both the migration enthalpy A// and the energy required to form the vacancies if the latter are thermally activated i.e., the concentration of intrinsic defects is much greater than the concentration of extrinsic defects. If, however, A is fixed by doping, it becomes a constant independent of temperature. The activation energy for diffusion in the latter case will only depend on A/f, . 

MgO and CaO are so high, the concentration of intrinsic defects is going to be very much less than impurity concentrations (usually -100ppm or greater). Even close to the melting temperature we find that only one or two sites in one billion are vacant due to intrinsic effects. 

As for medium 2, nothing has changed since the previous situation it is now the X2 parameter that varies according to the carried charge value and the charge density, which is related to the concentration of intrinsic defects in the oxide, is constant. 

FIGURE 3.1 Schematic representation of the constitution of a carbon crystallite where Li and Lj is the thickness and sizes a crystallite, y is the angle characterizes concentration of intrinsic defects of graphite planes. (Taken from Carbon, 31, Bourrat, X., Electrically conductive grades of carbon black Structure and properties, 287-302, 1993. Copyright 1993, with permission from Elsevier.)... 

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 ... 

The same analysis can be applied to more complex situations. Suppose that cation vacancy diffusion is the predominant migration mechanism, in a sodium chloride structure crystal, of formula MX, which contains Schottky defects as the major type of intrinsic defects. The relevant defect concentration [ii] is [Eq. (2.11)]... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

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 explanation for the two slopes in the plot lies in the fact that even a very pure crystal of NaCl contains some impurities, and the line corresponding to low temperatures (on the right of the plot) is due to the extrinsic vacancies. At low temperatures, the concentration of intrinsic vacancies is so small that it can be ignored because it is dominated by the defects created by the impurity. For a particular amount of impurity, the number of vacancies present will be essentially constant, jj in this extrinsic region thus depends only on the cation mobility due to these extrinsic defects, whose temperature dependence is given by Equation (5.9) ... 

Let us first discuss intrinsic disorder types where the number of moles of the components is almost constant and independent of the component activities. Thus, the majority point defect concentrations are also (almost) independent of the component. activities. It follows that only two types of (intrinsic) defect formation reactions are allowed... 

Another contribution to variations of intrinsic activity is the different number of defects and amount of disorder in the metallic Cu phase. This disorder can manifest itself in the form of lattice strain detectable, for example, by line profile analysis of X-ray diffraction (XRD) peaks [73], 63Cu nuclear magnetic resonance lines [74], or as an increased disorder parameter (Debye-Waller factor) derived from extended X-ray absorption fine structure spectroscopy [75], Strained copper has been shown theoretically [76] and experimentally [77] to have different adsorptive properties compared to unstrained surfaces. Strain (i.e. local variation in the lattice parameter) is known to shift the center of the d-band and alter the interactions of metal surface and absorbate [78]. The origin of strain and defects in Cu/ZnO is probably related to the crystallization of kinetically trapped nonideal Cu in close interfacial contact to the oxide during catalyst activation at mild conditions. A correlation of the concentration of planar defects in the Cu particles with the catalytic activity in methanol synthesis was observed in a series of industrial Cu/Zn0/Al203 catalysts by Kasatkin et al. [57]. Planar defects like stacking faults and twin boundaries can also be observed by HRTEM and are marked with arrows in Figure 5.3.8C [58],... 

A finite equilibrium concentration of intrinsic point defects can be found in any crystalline material because a small number of defects is thermodynamically favored. 

The number of Frenkel interstitials was obtained from a fit of the tensi-metric data to the observed pressure dependence behavior at 100 mm at 831°C. It has been assumed that the concentration of intrinsic interstitials is independent of pressure, whereas the concentration of extrinsic defects varies as the square root of the pressure. This is not strictly correct since they depend on the value of K. It should, however, be a good approximation at the lower temperatures. 

With the notable exception of transition metal oxides that generally exhibit wide deviations from stoichiometry, the concentration of intrinsic or nonstoichiometric defects in most ceramic compounds is so low that their defect concentrations are usually dominated by the presence of impurities. 

Note that when the defect concentration was intrinsically controlled, the activation energy for its formation appeared in the final expression for D [i.e., Eq. (7.20)], whereas when the concentration of the defects was extrinsically controlled, the final expression included only the energy of migration. How this fact is used to experimentally determine both A//, and A//5 is discussed in the following worked example. 

All the extrinsic defects modify the concentration of the intrinsic ones compared to the undoped ceria and therefore they modify the rate of the process. In order to get a quantitative model, the concentrations of point defects in ceria must be theoretically expressed as function of the oxygen partial pressure, the amount of foreign cation and physical constants such as equilibrium constants and diffusion coefficients[7,ll]. In the following, only two equilibrium constants will be considered ... 

The given effect is reversible, i.e. after the external loading was removed the equalization of concentration of point defects over the specimen s volume takes place. Later on it was shown by Kosevich [7] that the intrinsic defects of a crystal (vacancies and interstitials) can also diffuse in an inhomogeneous field of external stress, and the forces that act on them are equal accordingly to... 

The presence of a small number of Frenkel defects reduces the Gibbs energy of a crystal and so Frenkel defects are intrinsic defects. The formula for the equilibrium concentration of Frenkel defects in a crystal is similar to that for Schottky defects. There is one small difference compared with the Schottky defect equations the number of interstitial positions that are available to a displaced ion, N, need not be the same as the number of normally occupied positions, N, from which the ion moves. The number of Frenkel defects, np. present in a crystal of formula MX at equilibrium is given by ... 

There is another type of oxide which, although having quite close stoichiometry, shows relatively high electrical conductivity, which is independent of the oxygen partial pressure. Such behaviour is typical of so called intrinsic , or transitional semiconduction when the concentration of electronic defects far exceeds that of ionic defects and the equilibrium of the electronic defects may be represented by the excitation of an electron from the valence band to the conduction band, producing a quasi-free electron and an electron hole according to Equation (3.21) ... 

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