Shear Structures and Non-stoicheiometry

BY J. S. ANDERSON

As solid-state theory developed from 1930 onwards, it appeared that the existence and the properties of non-stoicheiometric compounds could be interpreted in terms of point defects — vacancies or interstitial atoms. This outlook is no longer tenable without considerable modification, and this review examines the implications of recent work on one extensive group of substances: those in which the notion of a point defect must be replaced by that of an extended, planar singularity in the structure.

Before doing so, it is desirable to summarize briefly the grounds for modifying the theoretical outlook that has become generally accepted. The power and generality of the statistical thermodynamics of crystals are such that the basic concepts of point defect theory are beyond dispute; moreover, the transport properties of ionic and covalent crystals (self diffusion, chemical diffusion, and ionic conductivity) demand the existence of mobile defects. The native concentration of point defects in a stoicheiometric crystal is given by an expression of the form A exp Ed/2kT where Adepends upon the excess entropy of the defective crystal (its configurational entropy and the changes in vibrational entropy due to the defects) and Ed is the energetic cost of creating a complementary pair of point defects. Ed depends upon a number of factors, including the band gap, and is lower for quasi-covalent crystals (e.g. the transitionmetal chalcogenides) than for essentially ionic crystals (e.g. most of the oxides), but is typically of the order of several electron volts. It follows that the calculated equilibrium concentration of point defects, particularly in the metallic oxides but also in the sulphides etc. is extremely small, even at high relative temperatures θ = T/Tm (Tm = the melting point of the crystal).

By contrast, the apparent defect concentration in non-stoicheiometric compounds, with a chemically significant range of existence, is very high, affecting the occupancy of 0.1 — 20% of the lattice points of at least one sublattice of the crystal.

In these circumstances, the dilute regular solution theory implied by the basic point defect approach is no longer applicable. Interactions between defects, and between the defects and the 'solvent' crystal lattice, mediated by coulombic forces, electron delocalization, and repulsions, become important. Moreover, the view that non-stoicheiometric compounds involve only the extension of point defect theory — displacement of the intrinsic defect equilibrium in response to changes in the chemical potential of the components — runs into the difficulty that, to account for defect concentrations of the magnitude indicated, the energy of creation of point defects would have to be unacceptably small. Efforts to include interaction effects into a generalized statistical thermodynamic treatment have had only limited success, because the actual structure of non-stoicheiometric compounds and solid solutions shows that some highly specific factors are involved. These depend both on the structure of the parent phase (e.g. the types of defect complex found in hyper- and hypo-stoicheiometric fluorite structures, the ordering effects in transition-metal chalcogenides related to the B8 and C6 structure types) and on the electronic structure of the elements and compounds concerned (e.g. the variety of defect structures within the NaCl structure type, as shown by FeO, TiO, and NbC.

Although no simple general description is any longer tenable, the results produced by increasingly strong interaction effects can be summarized schematically, as in Table 1.

There are differences between essentially ionic and essentially metallic structures as regards their non-stoicheiometric behaviour. In the latter, point defects are perturbations of the inner potential; changes in composition change the electron population at the top of the conduction band, i.e. usually in the anti-bonding or non-bonding part of the band. Because the local perturbations can be screened by the conduction electrons, long-range interactions are small and it may still be appropriate to speak of isolated point defects. Nevertheless, evidence is accumulating that interactions are strong enough to lead to marked site-preference energy effects, so that there is a considerable degree of correlation between the positions of vacancies (e.g. the preferential location of vacant carbon sites in NbC1-x on third neighbour positions), even though this is not strong enough to lead to long-range order. l n ionic compounds, coulombic forces and crystal-field interactions exert more profound effects, so that the parent crystal structure around nominal point defects is modified. At significant defect concentrations most 'point' defects are transformed into defect complexes, which may aggregate into defect clusters. Whether this invariably happens cannot be affirmed; the statement is correct for each one of the few defect structures that have been examined in detail.

A second feature shown in Table 1 is that interaction between defect centres, whatever their nature, tends to the formation of ordered structures. It can be maintained that, for attainment of true inner equilibrium at low temperatures, the configurational entropy of any random defect system must be reduced to zero by ordering. In principle, any assemblage of atoms, in any proportions, could order in a sufficiently large unit cell ; in practice, non-stoicheiometric crystals either unmix into phases of simple composition and structure or form a succession of intermediate compounds — a homologous series of intermediate mixed valence phases, derived according to some common structural pattern from the structure of the parent non-stoicheiometri compound. In an increasing number of instances, indeed, it has been found that such ordered intermediate phases are invariably formed, and that the apparent existence of non-stoicheiometric phases with wide ranges of composition is illusory. In a number of systems, however, (e.g. the transition-metal chalcogenides, CeO2-xetc.) there is a transition from univariant thermodynamic behaviour, with successions of sharply defined phases, at low temperatures, to the bivariant behaviour of a non-stoicheiometric phase...