Interactions between defects

The above examples showed that ideal mass action laws require correction in a variety of cases. Ideal mass action laws for defects only apply when there are random distributions and, hence, at very low defect concentrations. A high disorder — consider, for instance, a solution of 5% SrO in lanthanum copper oxide — can certainly no longer be treated as a simple structural perturbation rather for accurate analysis, individual treatment of the mixed phases is necessary. Then, of course, tractable models lose their general character, and the reader s expectations of the correction possibiUties should not be too high. In this section the intention is — with just one exception — to discuss simple concepts, that are valid, when the ideal mass action laws have just lost their validity , that is, we are still limiting ourselves, in principle, to more or less low defect concentrations. In many cases, however, such concepts qualitatively indicate the right direction, beyond the region of strict vahdity. 

In a purely formal manner it is possible to correct the Boltzmann approach by replacing concentration by activity and, hence, inserting an activity coefficient term  

Let us consider the Frenkel reaction in an ideally pure crystal. If we gradually reduce the temperature, the interstitial ion will eventually fall back into the vacancy (cf. Kp — 0 for T- 0). In the same way, the semiconductor s conduction electrons will, via annihilation of holes, become valence electrons (Kb — 0). In a similar sense, in strongly Cd-doped AgCl, there is a Coulomb attraction between Cd g and the counterdefect V g leading to a mutual trapping and thus to a deviation from random distribution at low temperatures. This can be described approximately by an exothermal production of associates [203,204] of the form 

The associate formed corresponds to a mutual neutralization the complex formed possesses (approximately ) a zero effective charge and does not contribute to conductivity . Structurally the association means that the vacancy is to be found in the immediate neighbourhood of the dopant ion, from which it cannot be released without perceptible effort. 

Apart from this, the mobility will be also negligible owing to the low Cd-mobiUty. 

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The thermodynamic functions of non-stoichiometric solids at very high deviations from stoichiometry are strongly affected by defect clusters and molecularities. The detailed theoretical description of the interactions between defects and the lattice as well... 

In this chapter non-stoichiometric compounds derived from point defects have been reviewed mainly from a thermodynamical point of view, and many examples have been presented for the purpose of understanding the nature of non-stoichiometry. As mentioned above it is not necessary to take the interaction between defects for vacancies) into consideration if the... 

Both equations (7.2.16) and (7.2.18) have the same dependence on the relative diffusion coefficient, D — D + D, but different dependence on the elastic interaction between defects. However, in both cases the stronger similar defect attraction, the lower is the critical dose rate. In the mesoscopic approach this effect is less pronounced (logarithmic vs. linear dependence) and here pc is considerably higher. It seems that this approach is able to detect only those mesoscopic-size aggregates which are already well-developed - unlike the microscopic formalism able to detect even the marginal aggregation effects. 

A variable activity coefficient, should be added as a multiplier of the bulk concentration, C, if the defect concentrations are so large that there is an interaction between defects). The chemical potential at an arbitrary position x within the layer (refered to the chemical potential at x = 0) can similarly be written as... 

Clustering of Defects. In a crystal of almost ideal composition, vacant sites or interstitial ions are relatively widely separated A random distribution implies that they are as far apart as possible. It might be adjudged, at first sight, that the interaction between defects—due to their real or virtual charges, or to the strain induced in the surrounding crystal lattice—was inherently repulsive. 

On this view, interaction between defects determines the concentration at which a phase becomes saturated with defects, and also the manner in which defects aggregate or order. It can be foreseen that stoichiometrically defined intermediate phases are likely to undergo an order-disorder transition at higher temperatures and that, in general, some detectable stoichiometric range is likely before the disordering temperature is reached. 

A number of studies have treated the effects of impurities on phase transitions from a theoretical perspective (Halperin and Varma 1976, Hock et al. 1979, Levanyuk et al. 1979, Weyrich and Siems 1981, Lebedev et al. 1983, Bulenda et al. 1996, ScWabl and Tauber 1996). By and large, however, theoreticians have focused on the way in which local interactions between defect fields and the order parameter produce an anomalous central peak in neutron scattering cross-sections of impure ferroelectrics up to 65°C above the critical temperature (Shirane and Axe 1971, Shapiro et al. 1972, Muller 1979). 

RGS wafers have the advantage of a more cost-effective fabrication due to the high-production speed. The expectation is that, even if the efficiency is somewhat lower, the introduction of RGS would further reduce the costs per Wp of PV modules. Improvements in wafer quality and an increased understanding of the interaction between defects and solar cell processing are necessary to reach higher efficiency values. 

In the absence of interactions between defects, the following reactions determine the defect concentrations in oxides with the fluorite structure. 

After the altered catalytic activity has been attributed to identified defects, so far as possible, there may remain a final ambiguity as to the nature of the interaction between defect and substrate that brings about the alteration. This ambiguity arises from the dual natmre of many defects and from the possible influence of geometric as well as of electronic parameters upon catalysis. 

Numerical calculations of dislocation pair interactions have been carried out for systems of particles with / [89] and LJ [90] potentials. For the potential, Fisher et al. [89] find that the elastic dislocation interaction potential is accurate for dislocation separations as small as 3 lattice spacings, while Joos and Duesbery [90] find that separations of 30 lattice spacings are necessary to reach the asymptotic elastic limit. The adequacy of the continuum elastic approximation in describing the short-range interactions between defects is thus still something of an open question, and may depend on the range of the interparticle potential. 

To properly estimate the entropy and calculate the structural properties of the liquid phase, we must incorporate long-range interactions between defects into our model. We are currently studying a melting model formulated in real space in an attempt to take into account the long-range interactions, and will report on the results of these studies in a future publication. 

It is already dear that the equilibrium behavior of materials having gross defects is not predicted by theories which are a simple extension of the classical point defect models. This is because strong interactions between defects in parent structures yield extended defects that become regular structural features of textured intermediate phases, which when ordered leave very low concentrations of point defects. The most studied of these structural features is the Wadsley defect resulting from crystallographic shear in typical... 

Millot F, Lorin JC, Klossa B, Niu Y, Tarento JR (1997) Oxygen self-diffusion in Fes04 An experimental example of interactions between defects. Ber Bunsenges Phys Chem 101 1351-1354 Mills GA (1940) Oxygen exchange between water and inorganic oxyanions. J Am Chem Soc 62 ... 

Niu Y, Millot F (1999) Oxygen self-diffusion and interactions between defects in Fc304. Acta Metallurgica Sinica 12 137-142... 

The electronic conductivity of nanocrystalline ScSZ dominates the electrical transport at P02 values below lO atm, while for YSZ, becomes significant only at oxygen partial pressure less than lO atm. This observation strongly suggests the presence on nonequivalent positions of oxygen vacancies distributed in Zr02 matrix when different acceptors are introduced. Also there may be an interaction between defects resulting in defect association. 

One such effect which has so far been omitted from our discussion is the electrostatic interaction between defects. This effect, which was considered in detail by Ons er Dupuis (i960) and is also treated by Jaccard (1964) and by Dougherty (1965), becomes important when one type of carrier is in considerable excess and the motion of the minority carriers is being studied, for the majority carriers can then shield the other species electrostatically. Since these effects do not cause any very appreciable change in the final conclusions in the case of simple conductivity or dielectric measurements, we shall not consider them further here. 

A universal representation describing the interactions between defects, faults, and failures of a system is shown in Figure 5.1 (16) ... 

Interactions between defects within the clusters can themselves lead to an increase in the defect concentration [78]. 

Quasi-coupKng between the ionic and electronic currents is sometimes introduced by adding a term LieVAe to Eq. (1) and a term LeiVAi to Eq. (2). However, it was shown that these corrections are not really necessary (except perhaps in metals [33]) if the current density of all mobile species, as well as interactions between defects are all taken into consideration [36]. 

Obviously, computed properties are required to converge with the supercell size. This internal consistency check is important for estimating the interaction between defects in neighboring cells. In fact, two kinds of limitations in the model exist, which correspond to two different levels of complexity ... 

We shall just mention here that the simple association theory may be extended by considering the interactions between defects in solution in a medium with dielectric constant a [14]. This is analogous to the Debye-Huckel theory of electrolytic solutions. As a result, the mole fractions of charged point defects of sort i in the mass action laws have to be replaced by their corresponding activities which according to Debye-Huckel are of the form... 

With respect to standard molecular-cluster techniques, this approach has some attractive features explicit reference is made to the HF LCAO periodic solution for the unperturbed (or perfect) host crystal. In particular, the self-embedding-consistent condition is satisfied, that is, in the absence of defects, the electronic structure in the cluster region coincides with that of the perfect host crystal there is no need to saturate dangling bonds the geometric constraints and the Madelung field of the environment are automatically included. With respect to the supercell technique, this approach does not present the problem of interaction between defects in different supercells, allows a more flexible definition of the cluster subspace, and permits the study of charged defects. The perturbed-cluster approach is implemented in the computer code EMBEDOl [703] and applied in the calculations of the point defects both in the bulk crystal, [704] and on the surface [705]. The difficulties of this approach are connected with the lattice-relaxation calculations. 

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