Misfit Layer Structures

One category of incommensurate phases exhibited by low-dimensional structures can be simply described as being composed of the stacking of two, or more, alternating layer types, the in-plane periodicities of the different layer types being mutually incommensurate along at least one direction. It is to this category of so-called misfit layer structures to which the high-T$_c$ cuprates, well known for their highly two-dimensional nature, can be most closely identified.

The basis of a classification scheme for misfit structures has been systematised by Makovicky and Hyde [14]. Their extensive review is rooted in the mineralogical origins of the first naturally occurring examples to be found but goes on to include many recently synthesised misfit layer compounds, such as the sulphides to be discussed later, which demonstrate just how quickly the field has expanded from what were initially thought to be quite rare geological oddities to instead be a wide-ranging and important structural phenomenon. The degrees of incommensurability, the types of layers and their chemical composition, and in particular the type of match between the interlayer surfaces are all applied in the classification. The interest of the work is a qualitative understanding of the structural modulations which result from the interactions between layers.

A mineralogical example is that of the tochilinites, described by Organova [15], in which tetragonal sulphide layers of edge-sharing FeS$_4$ tetrahedra alternate with a octahedral hydroxide layer of (Mg, Al, Fe)(OH)$_2$. The nature of the incommensurability will be dependent upon the match, or mismatch, which exists between the intralayer lattice vectors of the tetragonal layer and those of the hydroxide layer. In general terms, the coincidence of two layer types can be such that each of the two pairs of layer periodicities may independently be either commensurate (C), semicommensurate (S), or incommensurate (I), and thus the overall match can potentially be a combination of any two of these. A concise notation is to represent the possible combination as SC, IC, II and so on. So, for example, in one variety of the tochilinites both layer types are commensurate along one direction, while along the other there is a coincidence match every 3 cells of the tetragonal layer with every 5 cells of the octahedral layer, and the structure is thus described as SC.

The geometry of the interlayer interface (i.e. the bonding surfaces of the layers) in the tochilinites is of the most common type. There are only a limited number of layer types known amongst misfit layer compounds, and the result is that the interface geometries are mostly either of the hexagonal/hexagonal type or of the hexagonal/tetragonal type. The two types are abbreviated by Makovicky [14] to H/H and H/Q respectively. The tochilinites are an example of the latter kind. The H/H category, on the other hand, is based upon the combination of chemically distinct octahedral layers, and also includes the graphite intercalation compounds [16], almost all of which form II structures with no significant signs of modulation. In contrast, the H/Q category shows a wide variety of structures and both component layers are frequently modulated along one direction.

An important trait of the misfit compounds which presents itself again in the tochilinites is that of vacancy ordering. The presence in tochilinite of Fe vacancies in the tetragonal sulphide layers allows a variation in the stoichiometry, 2Fe$_{1-x}$S with 0.08$\leq$x$\leq$0.28, to maintain an agreement with the interlayer match. In the SC structure, the fractional occupancy of selected Fe sites is ordered with the period of the 3 cell coincidence match of the tetragonal layer. The interaction between layers is such that this Fe vacancy pattern is duplicated in the adjacent hydroxide layers, which have full occupancy, as a preference for Fe$^{3+}$ at certain octahedra, matching with the necessary 5 octahedral cells. A whole series of tochilinite varieties are known, from different geological origins, which change the interlayer match through IC, SS, to II by means of different chessboard arrangements of Fe vacancies.

In the high-T$_c$ cuprates vacancy ordering has found its most remarkable expression in the abundance of oxygen vacancy superstructures to be found in the Cu-O(reservoir) layers of the Y-systems[17]. In YBa$_2$Cu$_3$O$_{7-\delta}$ the oxygen sites in the reservoir layers form Cu-O-Cu chains. The two bookends of the phase correspond to $\delta$=1.0, in which all the oxygen chain sites are vacant (T$_c$=0K) and $\delta$=0.0 in which the chains are fully occupied (T$_c$=90K). A complete phase diagram has been experimentally established by scattering experiments [18,19,20] which includes a series of superstructures, with long- or short-ranged order, ranging over the intermediate values of $\delta$ and with which T$_c$ also varies between 0 and 90K. The midpoint $\delta$=0.5, for example, corresponds to a 2x$a$ superstructure formed of alternating full and empty chains, and is responsible for a plateau at 60K in T$_c$ versus $\delta$. At higher oxygen contents successively longer period superstructures, 3x$a$, 5x$a$, form as $\delta$=0 is approached, and at low concentrations partially filled chains produce more complicated arrangements. The relationship between the ordering and T$_c$ has also been established, and the understanding of the mechanism for it draws heavily upon the model of charge transfer to the conducting CuO$_2$ planes. This is an example of how interlayer interactions not only play an important role in determining a system's structure but may also be inextricably linked to the material's physical properties as well.

A strong association may be made between the influence that layer matching has upon a structure, and the magnitude of the interlayer interactions which exist within it, both in terms of structural modulations and material properties. The bonds which cross layer interfaces will be strained along any lattice direction which departs from a commensurate match, causing adjacent layers to alternately suffer compression and tension. Such lattice strain will be contained by the fluctuation of local charge balance. It is to alleviate such strain that substitutional and configurational modifications are favoured by a lattice. When such mechanisms are insufficient or unable to accommodate excessively strained layer matches then more dramatic structural transformations are to be expected. The result then is a broad range of behaviour found extending from cases with such weak interactions that the layers are essentially independent with barely any modulation to semicommensurate `lock-in' cases with strong interlayer interactions, and to an extreme where the interactions are so strong that the layers are actually broken up into strips by the excessive strain.

The sulphide known as cannizarite, of approximate stoichiometry (Pb,Bi)$_2$S$_2$, is an example of an IC structure for which sufficient studies have been made to allow analysis of its modulated structure [21]. Cannizarite is the basis for a very large series of compounds which ultimately transpose to an overall CC structure by means of the stress-induced break-up of the layers into alternating strips of commensurate and incommensurate units of differing compositions. It is again of the H/Q type with a two atom thick tetragonal (Pb,Bi)S layer alternating with a five atom thick layer of (Pb,Bi)$_2$S$_3$ octahedra. The positions of Pb and Bi are believed to be mixed in both layers. The structure is notable because of the 45 degree rotation of the tetragonal layer with respect to the more usual orientation relative to the octahedral layer. Both layer types are sinusoidally modulated in unison, and a complex variety of modulation waves of differing direction and wavelength, dependent to a fine degree upon stoichiometry, are found.

The remarkable range of misfit structures to be found amongst layered sulphides highlights the extent to which subtle compositional changes are critical. At a fundamental level, the effect of chemical composition can be understood in terms of the difference in cation sizes and the extent to which layers must adapt to accommodate ionic radii different in size from that of the lattice ideal. The atomic arrangements which connect across a layer interface of the H/Q variety are illustrated in Figure 2.3; it is clear that the two layers can only be stacked in this form, undistorted, if the bond lengths meet the condition $AO/BO = \sqrt{2}$. If this condition is not met then distortions will ensue. The mismatch in the bond lengths between the two layers can be expressed by

$$t = \frac {AO} {\sqrt{2} BO}$$ (4.2)

where t is known as the tolerance factor, and was first introduced by Goldshmidt [22] to consider distortions in perovskite systems. A particularly well studied high-T$_c$ cuprate system which has been understood in this way is $\rm {La_{2-x}Sr_xCuO_4}$, reviewed by Goodenough [23]. Starting with the unsubstituted structure with x=0, it can be calculated from the values of the ionic radii [24] that at room temperature $t < 1$ and the CuO$_2$ (BO) layers will be under compression and the LaO (AO) layers under tension. This is responsible for an orthorhombic distortion of the tetragonal structure. Further, because of the different thermal expansions of the bonds, La-O being the larger, then $t$ also decreases with decreasing temperature. At high temperatures ($\approx$ 540K) there is a transition to the tetragonal structure when $t >1$. In the substituted structure La$^{3+}$ is replaced by the larger ion of Sr$^{2+}$ and this too has an effect, increasing $t$ as $x$ increases. With the result that the transition to the orthorhombic phase is shifted to lower temperatures as substitution is increased, eventually being completely suppressed by $x=0.22$.

Figure 2.3: The H/Q layer match represented in (a) by an hexagonal AO layer, and in (b) by a tetragonal BO$_2$ layer.