CHAPTER 1
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In these circumstances, the dilute regular solution theory implied by the basic point defect approach is no longer applicable.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 at high temperatures. In such cases, the structures of the intermediate phases show how ordering occurs, and it is a reasonable inference that analogous ordering processes operate over a short range within the non-stoicheiometric phase. On that basis, the structure of the non-stoicheiometric crystal could be regarded as not completely random and homogeneous in composition, but involving small regions or microdomains of differing superstructure order (and hence differing composition). The close relationship between the structures concerned implies that they can inter-grow coherently within a single crystal or particle; at the high temperatures for which non-stoicheiometric behaviour represents a stable sate, self-diffusion processes are important, so that each microdomain is a dynamically fluctuating region, smaller in extent than the coherent scattering length for X-ray diffraction effects. On this view, put forward particularly by Ariya and Wadsley, a non-stoicheiometric crystal is inherently a hybrid crystal in Ubbelohde's sense; the concept must at present be regarded as non-proven, but has found increasing acceptance, although attempts to formulate more precisely its thermodynamic implications raise a number of problems. Some residual ordering of defect centres in non-stoicheiometric phases can be inferred from the results of careful thermodynamic studies of the ferrous oxide phase Fe1-xO and the praseodymium oxide phase PrO2-x and from thorough X-ray diffraction studies of the La2O3 — CeO2 solid solution system.
If, in place of simple vacancies and interstitials, non-stoicheiometric systems must be viewed structurally as based upon defect complexes and a considerable measure of short-range order, an inescapable conclusion is that attempts to interpret their properties in terms of classical point defect theory, without reference to their real structure, lead to quite fallacious results. This has not yet been fully appreciated.
1 Crystallographic Shear
The immediate concern is with those structures in which defects are virtually eliminated bf planar singularities and, in particular, with the so-called 'shear structures'.
The term 'crystallographic shear' (which is better not abbreviated to 'shear') was first applied by Wadsley to the structural principle underlying the homologous series of intermediate phases in the oxides of molybdenum, tungsten, vanadium, and titanium, discovered by Magneli and his co-workers. It has proved to be of wider significance, describing the relationship between crystal structures that are based on the same packing of anions, and extending to the defect structure of simple oxides.
If ionic crystals are considered as built up by the linkage of co-ordination polyhedra, e.g. the [MO6] octahedra in the oxides of the six-co-ordinate cations, crystallographic shear provides a means of altering the anion: cation ratio in some simple parent structure, such as the ReO3 or rutile type structures, without the introduction of point defects and without change in cation co-ordination, but with a change in the co-ordination number of certain anion positions. Along some crystallographic plane, the linkage between coordination polyhedra is changed, a closer linkage (e.g. replacement of apex-sharing by edge-sharing, as in ReO3-type structures, Figure 1A, or edge-sharing by face-sharing, as in rutile-based structures, Figure 1B) eliminating a set of anion sites and conversely.
The consequences of such planar structural features can be exemplified by the tungsten oxide WO2.90 or W20O58 (Figure 2). WO3 itself has a distorted variant of the ReO3 structure, with [WO6] octahedra linked through all the apical oxygen atoms. In W20O58, the octahedra are nearly regular and linked as in ReO3 except along regularly recurring planes, the (130) planes of the ReO3 structure, along which there are groups of octahedra (infinite columns in the direction normal to the diagram) which share edges. The slices of structure so formed have a lower oxygen : tungsten ratio (in this instance 8 : 3) than the slabs of unperturbed parent structure enclosed between successive slices of edge-sharing octahedra. Inspection shows that in this, and all, shear structures, the crystallographic planes parallel to the slice of altered composition are of two kinds: A, containing metal atoms (or metal and oxygen atoms) and B, containing only oxygen atoms. In the parent structure, there is regular alternation ABABAB..; at the discontinuity, one oxygen-only sheet is omitted to give the stacking sequence ABABAABAB..... In W20O58, type A planes have the composition WO, type B planes the composition O2. The resulting structure is that which would result if a type B sheet were abstracted from the parent structure, followed by a mutual displacement of the two slabs of structure so as to restore six-fold coordination to the cations and continuity to the oxygen sublattice.
In this particular structure, the two type A sheets collapse to form a single sheet (denoted as A2) with twice the normal density of tungsten atoms). Across the plane concerned, the crystallographic shear plane (CS plane), 'normal' and 'interstitial' cation sites switch meanings as between slabs of parent structure separated by the CS plane. This displacement is equivalent to a shear, the direction and magnitude of which is defined by a shear vector which approximates to a rational lattice vector of the parent structure, but is not necessarily an exact lattice vector because the new polyhedron linkage in the CS plane may involve significant distortion.
The hypothetical shear process can be represented by the operation of some displacement vector < a b c/u v w> on the {hkl} planes of the parent structure, the CS operation being symbolized as <a b c/u v w> {hkl}. In order that a set of lattice points may be eliminated, the shear vector must have a component normal to the shear plane; a true mechanical shear, with the displacement lying in the shear plane can produce only an antiphase boundary, without change in stoicheiometry (Figure 3). The crystallographic shear plane and the shear vector together determine the structure, and therefore the composition, of the collapsed slice; the total composition is determined by the slabs of more or less undistorted parent structure. An isolated CS plane constitutes an extended (two dimensional) defect; regular recurrence of the CS operation on some particular set of (hkl) planes generates a new ordered structure. The new repeating unit of the crystal is therefore defined by an enlarged unit cell, of lowered symmetry, with at least one long interplanar spacing, the distance between successive CS planes. Since the structure at the CS planes depends only on their orientation and shear vector, they could in principle be separated by any arbitrary width of parent structure.
If the spacing of CS planes is completely regular, the result is to generate from a parent structure MX[??] a homologous series of stoicheiometrically defined intermediate, mixed valence compounds with the general formula MXam-m; m measures the number of anion sites eliminated by crystallographic shear, n the breadth of the slabs of parent structure, as measured by the number of co-ordination octahedra in unbroken rows between CS planes. Attainment of perfect regularity of CS plane spacing clearly depends upon some long-range interactions, and this problem is considered below. If, on the other hand, the interaction forces are not sufficient to impose regularity of spacing, then a new type of non-stoicheiometry must result: the composition of a crystal could be a continuous function of the chemical potential of its components, changes in composition being taken up as alterations in the distribution of CS plane spacings. Such a crystal could be regarded from an alternative standpoint: as made up of lamellae of different compositions, in coherent intergrowth — microdomains, effectively infinite in two dimensions but only a few, or even one, unit cells in thickness. With the advent of electron microscopic techniques it has become possible to investigate the microstructure of CS phases at this level, and some results are discussed later, in reference to particular systems; they raise questions as to the stage at which a defined structure is recognized as a distinct compound, and the distinction between such a structurally defined entity and a phase in the thermodynamic sense.
Considerable interest clearly attaches to the thermodynamic definition of the CS phases: whether they display stoicheiometric variability and, in particular, whether at high temperatures thermal fluctuations of composition produce variations in CS plane spacing and a consequential broadened composition range. As will be seen, the evidence from known series of compounds is that CS planes interact strongly enough to impose a high degree of order, so that where the slabs of parent structure are up to 6 — 8 octahedra wide the compounds are strictly stoicheiometric. In some systems indeed, they exist only as very high temperature phases. On the other hand, the kinetics of CS plane ordering and readjustment may be very sluggish, so that disorder, at the microdomain level, cannot easily be eliminated.
In a topological sense, crystallographic shear describes the relationship between crystal structures that are based on a common mode of packing of anions and a constant co-ordination of cations, notably octahedral coordination. In continuous three-dimensional structures, the progenitor structures are the compounds MX3, involving co-ordination octahedra linked through all their apices. Of this type, the ReO3 structure is based upon cubic close-packing and the PdF3 type upon hexagonal close-packing of anions. The operation of recurrent crystallographic shear on the former involving one or two CS processes, is illustrated in Figure 3; the relations between the structures derived or derivable from the PdF3 type have been discussed by Andersson (Figure 4). These considerations have a predictive value, in foreseeing the possible existence of polymorphic forms (e.g. high-pressure structural transformations), of new series of intermediate mixed valence and ternary phases, and of defect structures in the compounds concerned.
2 The Mechanism and Kinetics of Crystallographic Shear
Crystallographic shear has been considered so far in purely geometric terms but since CS compounds are formed (albeit in an often disordered state) by solid-state reactions and in the reduction of the parent oxides, the transformations involved pose questions as to the mechanism by which reaction and ordering take place. Two stages are involved; how one whole sheet of anion sites can be virtually eliminated to define the locus of collapse of the crystal structure, to form a CS plane, and how CS planes attain a regular spacing. Both steps seem to require extensive co-operative shifts of atomic positions.
