One of our objectives when we started this study was to determine whether the stability of the crystal is correlated or not with the topological structure. The lattice energy, which measures the difficulty in breaking apart the crystal into its isolated ions, is dominated by the electrostatic interactions and follows the simple Q/a Madelung law. Grouping together the crystals that have the same M(+2) cation, we may also observe, as a secondary effect, that the lattice energy is proportional to the 1/3 power of the value of the electronic density at the M-X bond CP.
On a more fundamental basis, the stability of the crystal should be addressed with respect to: (a) solid state reactions producing compounds of different stoichiometry; and (b) the existence of crystals with the same AMX3 stoichiometry but a different crystalline structure. To our knowledge there is neither experimental nor theoretical data regarding the thermodynamic stability of perovskites but for a handful of compounds.
The relation between the topology of the energetic hypersurfaces connecting several crystalline structures has been previously investigated for the alkali halides (Martín Pendás et al. 1995). Based on the definition of topological ionic radii we were able to establish a single criterion that exactly predicted the stability of the B1 and B2 structures, defined in terms of the properties of the Gibbs free energy. The relation between phase transitions and topological structure of the perovskites, albeit very interesting, is beyond the scope of this work.
On the other hand, perovskites are usually prepared by melting a mixture of MX2 and AX crystals in 1:1 stoichiometric proportions and slowly cooling whereas the AMX3 crystal is simultaneously grown and extracted according to the Czochralski procedure. Even though, to the better of our knowledge, there is no systematic study on the energetics of the above reaction, most of the 120 AMX3 compounds have never been referred to in the literature, which suggests that many of the perovskites may be unstable with respect to MX2+AX.
To address this problem we have determined the energy of the reaction:
MX2(s) + AX(s) ---> AMX3(s)
E = E(AMX3) - E(MX2) - E(AX)
where E(compound) is the aiPI total energy. The fluorite structure [Cubic, Fm3m, M(+2) at (0,0,0), X(-1) at (1/4,1/4,1/4)] has been assumed for the MX2 compounds, and the rock-salt structure has been assumed for the AX compounds. Even though perovskite, fluorite and rock-salt are not necessarily the most stable structures of the three types of compounds, the difference should not be very large nor it should modify the general trends for the whole set of crystals.
Fig. 5.7 represents the reaction energy together with the topological scheme for the 120 AMX3 compounds. The first thing to notice is that the crystal stability is dominated by the identity of the M(+2) cation: Ba, Sr, and most Ca perovskites are unstable, whereas most Zn and Mg, and all Be perovskites are stable. For a given M(+2), stability is favored by having large A(+1) ions. The role of X(-1) is more complex, but it is seen that fluorides are different to the other halides, either more or less stable, the chlorides, bromides and iodides behaving rather similarly.
It is very interesting to observe in Fig. 5.7 that all crystals having R9 and B10, and most crystals having C8 topological schemes, are unstable. Contrarily, all C12 and most R13 crystals are stable. It appears that the existence of bizarre features, like wings, in the attraction basins works against the stability of the crystal. Alternatively, we can argue that the formation of X-X bonds is important to stabilize the structure. This sheds new light to the question of what is simple and what is complex in relation to the topological structure of the electronic density. The simple enumeration of critical points would suggests that those topological schemes having smallest values for T are the simplest ones. The shape of the attraction basins is simpler, however, in the schemes of large T and, as the analysis of the above reaction indicates, those schemes are more favored in energetic terms.
The above results refer to the difference between quantum mechanical total energies. It corresponds, thermodynamically, to null temperature and pressure. We have investigated the role of thermal effects, up to about 900 K, on the perovskite stability and have found interesting but small effects that do not modify the general trends discussed above.