Extrinsic Defect Concentrations

In solid solutions or alloys the atoms on the lattice sites are replaced by the atoms of the dissolved species, the solute. Apart from these substitutional solutions, interstitial solutions can also occur if the atoms or ions of the solute are small and can be accommodated in the interstices of the host lattice. The solubilities vary. The phase diagrams of spinel, lithium aluminum silicate (LAS), Ta/C, and Ti/N show to what extent dissolution is possible. Apart from having the dopants in the lattice, these alloys often have other defects that are the result of a difference in charge between the ions being replaced in the lattice and those replacing them. 

Doping or replacing atoms in a host crystal by foreign atoms with another valence, e.g., doping magnesium oxide by replacing Mg with Li or with Sc, is 

The LiJ g defects are charged and need positive counterions to maintain electroneutrality. This charge compensation is provided by the positive oxygen vacancies. 

If scandium oxide is dissolved in MgO, Sc replaces Mg and the defect Scijg needs a negative countercharge, which is a metal vacancy in this case. The dissolution reaction is 

doping scandium in magnesium oxide increases the magnesium vacancies. 

---

Heat capacities at high temperatures, T > 1000 K, are most accurately determined by drop calorimetry [23, 24], Here a sample is heated to a known temperature and is then dropped into a receiving calorimeter, which is usually operated around room temperature. The calorimeter measures the heat evolved in cooling the sample to the calorimeter temperature. The main sources of error relate to temperature measurement and the attainment of equilibrium in the furnace, to evaluation of heat losses during drop, to the measurements of the heat release in the calorimeter, and to the reproducibility of the initial and final states of the sample. This type of calorimeter is in principle unsurpassed for enthalpy increment determinations of substances with negligible intrinsic or extrinsic defect concentrations... 

Since the data were obtained in the transition region where intrinsic and extrinsic defects are contributing to the total defect concentration, the calculation of an enthalpy of motion cannot be made in a simple way because the temperature dependence of the Frenkel constant is not known. However, Ail. probably increases with temperature while the extrinsic defect concentration decreases with temperature. If, to a first approximation, these two trends cancel, then the enthalpy of motion is just the experimentally determined activation energy. Using this value from (16) and the defect concentration shown in Table VII, the preexponential constant Dq and hence the diffusion coefficient can be determined. 

The intrinsic defect concentration (at equilibrium) is calculated from the enthalpies and entropies of the defect formation reactions and the concentrations are strongly temperature-dependent. The extrinsic defect concentrations are strongly dependent on the quantity of dopant or impurity dissolved in the lattice and not by the temperature. 

Extrinsic Defects Extrinsic defects occur when an impurity atom or ion is incorporated into the lattice either by substitution onto the normal lattice site or by insertion into interstitial positions. Where the impurity is aliovalent with the host sublattice, a compensating charge must be found within the lattice to pre-serve elec-troneutality. For example, inclusion of Ca in the NaCl crystal lattice results in the creation of an equal number of cation vacancies. These defects therefore alter the composition of the solid. In many systems the concentration of the dopant ion can vary enormously and can be used to tailor specific properties. These systems are termed solid solutions and are discussed in more detail in Section 25.1.2. 

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) ... 

If majority point defect concentrations depend on the activities (chemical potentials) of the components, extrinsic disorder prevails. Since the components k are necessarily involved in the defect formation reactions, nonstoichiometry is the result. In crystals with electrically charged regular SE, compensating electronic defects are produced (or annihilated). As an example, consider the equilibrium between oxygen and appropriate SE s of the transition metal oxide CoO. Since all possible kinds of point defects exist in equilibrium, we may choose any convenient reaction between the component oxygen and the appropriate SE s of CoO (e.g., Eqn. (2.64))... 

Both the intrinsic and extrinsic defects in these materials can affect the properties of interest for applications. An example of this is the observed decrease in the damage susceptibility or photorefractivity of LiNb03 with additions of H, Li or Mg to Li-deficient congruently grown crystals (Table I).(26) The additions produce a reduction in the octahedral vacancy concentrations in the crystals. Therefore, at least some photorefractive optical damage in LiNbC>3 is believed to be related to the concentration of octahedral vacancies present in crystals. 

Nonequihbrium concentrations of point defects can be introduced by materials processing (e.g. rapid quenching or irradiation treatment), in which case they are classified as extrinsic. Extrinsic defects can also be introduced chemically. Often times, nonstoichiometry results from extrinsic point defects, and its extent may be measmed by the defect concentration. Many transition metal compounds are nonstoichiometric because the transition metal is present in more than one oxidation state. For example, some of the metal ions may be oxidized to a higher valence state. This requires either the introduction of cation vacancies or the creation of anion interstitials in order to maintain charge neutrality. The possibility for mixed-valency is not a prerequisite for nonstoichiometry, however. In the alkah hahdes, extra alkah metal atoms can diffuse into the lattice, giving (5 metal atoms ionize and force an equal number... 

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. 

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. 

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, . 

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 ... 

In general, this dependence can be more complex since it can be influenced by the relative amounts of extrinsic defects and impurities, as well as by association between defects. In this case, we should use the effective concentration of acceptors which is -... 

---