Incommensurate Crystal Phases

In the ninety-year history of crystallography the concept of lattice periodicity has been paramount to the development of the subject, and to much of solid state science, for the major part of that time. Translational lattice periodicity was seen as the distinguishing characteristic between ideal ordered crystals and disordered, or amorphous phases, which have no long-range order in their atomic positions. Over the past two decades, however, the notion of what can truly be considered as crystalline has had to be dramatically revised with the advancing discoveries of what are termed incommensurate, or more broadly, quasiperiodic crystal phases; materials which possess perfect long range order but which lack translational periodicity in one or more of their lattice directions. Although doubted by many physicists upon their conception, the existence of such phases should not have been so surprising considering there was no theoretical basis, beyond the desire for beautiful simplicity, to the assumption that the thermodynamic ground state of a solid be three-dimensionally periodic.

The first incommensurate phases were discovered more than twenty years ago in magnetic systems where the magnetisation has a helical structure with a pitch that does not have a rational relation to the underlying lattice. Many materials have since been found (some as common and everyday as quartz or the mineral calverite AuTe$_2$) which display two or more periodicities which are incommensurate with each other. Simply stated, the two periodicities $q_1$ and $q_2$ are incommensurate if they cannot be expressed as a ratio of two integers:

$$\frac{q_1}{q_2}\neq\frac{M}{N}\hspace{0.5in}M,N=1,2,3....$$ (4.1)

In general, however, the term incommensurate is applied quite loosely (and is frequently interchangeable with the term modulated) to any phase which cannot be simply described by the ratio of small integers. Indeed, in practice a truly incommensurate phase may well be indistinguishable from one in which the ratio is of two very large integers. The simple case of equation 2.1 being a rational fraction with small arguments would describe the more familiar situation of a commensurate superstructure (e.g. a unit cell doubling due to a vacancy ordering at every other site). The intermediate stage of semicommensurate has also been used to describe cases comprised of two moderately small integers (e.g. 7/4, 12/7) which are often suggestive of a `lock-in' value.

Incommensurate phases are understood to be the result of a conflict between various competing forces within a system. A simple model commonly used to illustrate this is shown in Figure 2.1. A chain of atoms connected by harmonic springs are imposed upon a periodic potential with its own fixed period b. The equilibrium separation of the chain atoms, $a_o$, which would in general be non-commensurate with b, is then perturbed by this periodic potential. On average, the atoms will favour the minima of the potential and the resulting atomic spacings will be dependent upon the strength of the potential relative to the harmonic interaction. A commensurate structure is shown favoured in Figure 2.1(a) for a strong potential, whilst Figure 2.1(b) shows the case for a weak potential where an incommensurate phase results. For a very strong potential a chaotic phase is possible where the atoms are pinned at random in the potential's minima. This simple semi-microscopic model which is able to exhibit many of the phenomena associated with incommensurate phases was originally introduced by Frenkel and Kontorova (later developed by Frank and Van der Merwe [10] and references therein).

Figure 2.1: A simple model which is able to display many of the properties of incommensurate phases. The system is in a commensurate phase in (a), an incommensurate phase in (b), and in (c) the system is pinned in a chaotic phase.

The behaviour and properties of an incommensurate system may show many unique features but precisely which may differ widely from system to system, and will be dependent upon the microscopic origin of the incommensurability. Classifying different systems along these lines allows two general types of incommensurate structure to be distinguished. (i) The modulated structure, which has a basic crystalline lattice with a modulation wave imposed upon it, the wavelength of which is incommensurate with the basic lattice. The modulation could be of any local variable such as atomic position, charge density, electric polarisation, magnetisation, occupation, etc. The origin of the incommensurability in this case is then the result of competition between interactions of different ranges. (ii) The composite structure consists of two or more interpenetrating sublattices of different chemical compositions with mutually incommensurate lattice periodicities. In this case the origin is the presence of two incompatible length scales. The two categories are not necessarily mutually exclusive. It is often found in the composite structures, which are the more complicated of the two, that interaction between the sublattices will also lead to a modulation of the structure.

The periodic potential of Figure 2.1 is realised in a material system by an underlying crystal lattice, and would fall into the category of the composite structures. For example, when the model is extended to two dimensions, the model resembles the situation of gas atoms adsorbed on to a crystal surface. At low gas pressures and temperatures commensurate phases are formed, but for high enough pressures the high density of adsorbed atoms pack into an incommensurate phase. An example of a modulated structure would be the Peierls mechanism in conductors. Here the interaction of conduction electrons with the atomic lattice leads to a modulation of the electron density and an accompanying distortion of the lattice. Known as a Charge-Density Wave, this well understood electron-phonon phenomenon is observed in the superconducting perovskite $\rm {BaPb_{1-x}Bi_xO_3}$ system for example, and in the layered metal chalcogenides such as $\rm {TaSe_2}$.

The phase diagram of an incommensurate system will also be dependent upon the specific microscopic mechanism at its origin, however, some general comments can be made which apply to both composite and modulated categories. Whenever some parameter is varied, such as $a_o$ in the model of Figure 2.1, or the temperature or pressure in the case of adsorbed gases, a system may exhibit two fundamentally different natures. The `floating' phase is one, Figure 2.2(a), in which the change in the modulation wave vector is continuous, passing through commensurate values without any `lock-in'; this behaviour may occur in two dimensions. The other is the `staircase' phase, Figure 2.2(b), where the wavevector does lock-in at commensurate values. When an infinity of these lock-in values exist, it is termed a `devil's staircase', Figure 2.2(c). Hysteresis is often a property of such staircase phases. The phase diagram properties have been treated in greater theoretical than experimental depth as yet. In particular, a phenomenological Landau approach has been reviewed by Levanyuk [11], and a microscopic approach reviewed by Bak [12] and by Janssen [13].

Figure 2.2: The three possible phase diagrams of an incommensurate system, each with a fundamentally different nature: (a) the floating phase; (b) the staircase phase; and (c) the devil's staircase.

The high-T$_c$ cuprates can perhaps be most closely identified with the two-dimensional misfit layer structures. These belong essentially to the composite structure type but with displacive modulations often playing a strong role. The properties of the misfit layer structures have been most recently reviewed by Makovicky and Hyde [14]. The next section of this chapter will introduce the system of classification established to aid understanding of such structures, and use an example of the well understood ABS$_3$ sulphides to illustrate the ideas. The following section will discuss the basis for a reciprocal space approach to incommensurate phases, this being the experimentally observed picture in x-ray scattering. The remainder of the chapter will briefly establish the mathematical nomenclature for x-ray diffraction, and show how this can be extended to include deviations from an ideal translationally periodic crystal in a general way. The results of which will offer specific descriptions for the incommensurate reciprocal space features, as well as for the temperature response due to thermal vibrations, and for the effect of crystal defects.