So far, we have ignored possible geometric prefer­ences between the constituent defects of a defect cluster. Of course, for oppositely charged defects, electrostatic considerations would drive the defects to sit as close as possible to one another, which would be described as a nearest neighbor configuration. However, as we saw in the previous section, defects can cause considerable lattice strain. Consequently, the most stable defect configuration will be dictated by a balance between electrostatic and strain effects.

To illustrate cluster geometry preference, we will consider simple defect pairs in the fluorite lattice, specifically in cubic ZrO2. These are formed between a trivalent ion, M3+, that has substituted for a tetra — valent lattice ion (i. e.,M’Zr) and its partially charge — compensating oxygen vacancy (i. e., VO*). This doping process produces a technologically important fast ion-conducting system, with oxygen ion transport via oxygen vacancy migration.2,

The lowest energy solution reaction that gives rise to the constituent isolated defects14 is

M2O3 + 2ZrXr! 2M’Zr + VO* + 2ZrO2 with the pair cluster formation following:

MZr + VO*! {M’zr + VO*}*

Figure 9 shows the options for the pair cluster geom­etry, in which, if we fix the trivalent substitutional ion at the bottom left-hand corner, the associated oxygen vacancy can occupy the first near neighbor, the sec­ond (or next) near neighbor, or the third near neigh­bor position.

Defect energy calculations have been used to pre­dict the binding energy of the pair cluster as a func­tion of the ionic radius15 of the trivalent substitutional

Figure 9 First, second, and third neighbor oxygen ion sites with respect to a substitutional ion (M3+).

ion.14 These suggest (see Figure 10) that there is a change in preference from the near neighbor configu­ration to the second neighbor configuration as the ionic radius of the substitutional ion increases. The change occurs close to the Sc3+ ion. Furthermore, the binding energy of the near neighbor cluster falls as a function of radius; conversely, the binding energy of the second neighbor cluster increases. Consequently, the change in preference occurs at a minimum in binding energy. The third neighbor cluster is largely independent of ionic radius. Interestingly, the minimum coincides with a maximum in the ionic conductivity, perhaps because the trapping of the oxygen vacancies as they move through the lattice is at a minimum.14

The change in preference for the oxygen vacancy to reside in a first or second neighbor site is a conse­quence of the balance of two factors: first, the Cou — lombic attraction between the vacancy and the dopant substitutional ion, which always favors the first neighbor site, and is largely independent of ionic radius, and second, the relaxation of the lattice, a crystallographic effect that always favors the second neighbor position. This is because, in the second neighbor configuration, the Zr4+ ion adjacent to the oxygen vacancy can relax away from the effectively positive vacancy without moving away from the effectively negative substitutional ion. Nevertheless, lattice relaxation in the first neighbor configuration contributes an important energy term. However, in the first neighbor configuration, the relaxation of oxygen ions is greatly hindered by the presence of larger trivalent cations, while small trivalent ions provide more space for relaxation. Thus, the relaxa­tion preference for the second neighbor site increases in magnitude as the ionic radius increases and conse­quently, the second neighbor configuration becomes more stable compared to the first.14

This example shows that even in a simple system such as a fluorite, which has a simple defect cluster, the factors that are involved in determining the clus­ter geometry become highly complex. Even so, we have so far only considered structural defects. Next, we investigate the properties of electronic defects.