A number of defect formation energies were calculated by introducing a single isolated defect into the lattice. Isolated vacancies were introduced at an La3+ site and at each of the inequivalent F- sites. Similarly, F- interstitials were incorporated at a variety of sites chosen by inspection from a model that was built for that purpose. Plan views of a unit cell of LaF3 for each space group are provided in
Figures 5.14 and 5.15.
The formation energies of the isolated point defects and the accompanying unbound Schottky quartets or Frenkel pairs for equilibrium structures are reported in
Tables 5.13 and 5.14
for the P63cm space97
group and in
Tables 5.15 and 5.16
for the alternative P3c1. The vacancy formation energy is defined as the energy required to remove a lattice ion from the perfect lattice to infinity, leaving a vacant site. For interstitials, the formation energy is the energy to bring an interstitial ion from infinity into the perfect lattice. Schottky energies, Es , are evaluated according to the expression:Es = E . + 3 E + (5.2)
s vu vF
where Ev is the vacancy formation energy for a particular defect, Elat is the crystal lattice energy. A full evaluation of the formation energies for Schottky quartets is given in
Appendix II.
Anion-Frenkel pair energies, Ep, are calculated from:EP = E + E . (5.3)
F Vp F,
where EF|» is the interstitial formation energy.
On the basis of these results two general points can be made: first, Frenkel pairs are predicted to be the favoured type of disorder and, second, vacancies are most likely to form on sites on the A sub-lattice. A more detailed discussion on trends intra and inter space groups follows.
Given that LaF3 consists of a number of inequivalent F' sites, it is reasonable to expect that structural site differences will lead to differences in vacancy formation energy and this is borne out in Tables 5.13 and 5.15. Such results are also found for Li+ sites in the superionic conductor Li3N (Walker and Catlow, 1981). These LaF3 results also confirm experimental observations that, at low temperature (T < 450K), mobile ions show sub lattice dependency, although which is still to be agreed. None-the-less these calculated energies are consistent with there being a significant difference
98
in the potential well depth at most sites and can thus help clarify the favoured mobile sub-lattice.
It is predicted that ions on a type sites on the A sub-lattice, i.e. F3/F4 or F3, only depending on choice of space group, form vacancies more readily and require almost identical energies to do so ~ 2.6 eV. In both cases ions F1/F2 or y/p type vacancies on the B sub-lattice are more stable to formation by between 0.4 - 0.7 eV. A comparison of formation energies for F1/F2 vacancies between groups does, at first sight, show an anomaly in that F1 for P63cm is closer to F2 for P3c1 (2.98 v 3.07 eV), and F2 for P63cm is closer to F1 for P3c1 (3.27 v 3.25 eV). Although F1 and F2 occupy crystallographically different sites they are generally taken as being equivalent in terms of defects. A comparison of coordinates for F1 and F2 atom types, between groups, shows similarities in x and y values, but some differences in the z coordinates, (Tables 5.11 and 5.12). Therefore, if taken collectively, there are no significant energy differences.
La3+ vacancy formation energies are almost identical in both groups, and considerably higher than for the anions. This general behavioural difference arises because of the greater relative charge of one of the species, in this case the cation, thus ensuring that it is more difficult to remove the cation from its regular site, where there is a favourable Madelung potential.
This discussion of vacancy formation shows that: firstly, both groups follow the same trends in that there are no significant differences in Schottky formation energies for comparative F- sites and, secondly, that should Schottky defects form, those incorporating a type vacancies will do so preferentially by between 1.1 - 2.0 eV.
Table 5.14 shows that, for P63cm, it is more difficult to accommodate an interstitial at the first site, whilst the rest are exothemic, two being identical. The values in Table 5.16, for P3c1 are, however, all favourable
and no particular site stands out as being preferential. Comparing interstitial formation energies shows that the first site in P3c1 is ~0.6 eV more stable than its equivalent in P63cm, whilst the other site differences are an order of
magnitude smaller. Combining these values with those for vacancy formation leads to the Frenkel pair energies for which a comparison between groups is given in Table 5.17. These show that particularly for site 1, Frenkel pairs form more easily in P3c1, as a consequence of the higher interstitial formation energy in P63cm. For the other sites there in no clear preference.
The results of this defect study, therefore, point to an anion Frenkel model as the favoured type of disorder. Schottky formation is about 2 - 2.5 times more costly in energy terms, irrespective of the space group used. This view is, however, contrary to experiment where, as reported in Chapter 4, Schottky defects have, to date, been assumed on the basis of the dilatation experiments of Sher et al., (1966). A Schottky interpretation has led to experimental estimates for Schottky quartet formation energies of 2.3 +/- 0.3 eV (Roos, 1983) and 2.08 eV (Chadwick et al., 1979). The latter authors have not, however, discounted the possibility of a Frenkel model with a predicted formation energy of 1.06 eV (Chadwick etal., 1980). Whatever the interpretation, the calculated energies are rather higher than those predicted by experiment: the lowest calculated Schottky energy being ~ 5.5 eV versus an experimental value ~ 2.2 eV and similarly for the Frenkel pair being ~ 2.2 eV versus 1.06 eV. The differences in Frenkel energies could, however, be partially explained if the sites chosen for the interstitial were not the true sites. Despite these differences, however, the calculations can be used to predict qualitatively that Frenkel disorder is preferential.