Introduction

The structure of the high temperature superconducting cuprates has proven to be a complex subject, but nonetheless rewarding because of the wealth of structural and physical phenomena which can be observed within these systems. From the basic components of covalent CuO$_2$ and ionic metal-oxygen planes, a rich and varied family has been constructed since the first discovery in 1986 of superconductivity in the La$_{2-x}$Ba$_x$CuO$_4$ system [4]. The challenge to achieve a detailed understanding of the nature and properties of these cuprate structures is motivated by a fundamental desire to understand their, as yet unexplained, superconducting mechanism, and novel normal state properties. It is further motivated by the technological need to optimise their performance for applications, and the continuing possibility of designing new materials with still higher values of T$_c$.

Although, with their extraordinary properties, they are usually treated as a class of materials unique to themselves, the high-T$_c$ cuprates can be most effectively considered as being members, or at least close relations, of a number of pre-existing groups of materials which are well understood. Crystallographically, the ideal structures of all the cuprate systems have much in common with the perovskite ABO$_3$ structure type. In this system, a metal atom B sits at the centre of an oxygen octahedra, and a large metal atom A occupies the spaces formed when these octahedra link into a corner-sharing network. A huge array of perovskite-based systems are known, and have been of special interest for their magnetic and ferroelectric properties long before the advent of high-T$_c$ superconductivity [5]. The systems can all be developed from the ABO$_3$ structure by considering the various possibilities of stacking the two constituent layers, AO and BO$_2$. The system A$_2$BO$_4$ is, for example, such a combination of layers with a stacking sequence AO-BO$_2$-AO; with A=La and B=Cu this is essentially the structure of the cuprate La$_2$CuO$_4$. Although the basic perovskite structure is cubic and truly three-dimensional, it is through a combination of layer stacking, intergrowths, and ordered vacancies, that the strongly two-dimensional nature of the cuprates derive. A complete scheme of classification for the cuprates based upon the perovskite description has been advanced [6].

Listed in Table 1.1 are some of the most commonly studied cuprate structural families, along with their respective stacking sequences, and transition temperatures. In the text that follows each cuprate system will be identified by its principal cation, and a particular phase by its layer sequence (so that YBa$_2$Cu$_3$O$_7$ becomes Y-123 for example). There are many more systems than can be listed here, and comprehensive structural surveys are available [7,8]. An important factor for superconductivity is the number, $n$, of adjacent CuO$_2$ layers, the value of T$_c$ increases with $n$ up to a maximum for $n=3$. In single layer structures the Cu atoms are octahedrally coordinated, but for $n>1$ the layers are formed of Cu-O tetrahedra, with each adjacent layer separated by an oxygen-deficient layer of an A atom. It is within these planes of copper oxide that superconductivity takes place, and it is the mixed valence state of the Cu which appears to be the crucial factor governing superconductivity in this. Each of the perovskite A$_{n-1}$Cu$_n$O$_{2n}$ blocks is separated by further layers, which can be of a variety of compositions and structures, but which fulfill the common role of acting as a charge reservoir for the doping of carriers to the CuO planes. It is the oxidation of the average Cu valence above 2+ by this mechanism of charge transfer from the AO reservoir layers which is responsible for the metallic, and ultimately the superconducting behaviour, of the CuO$_2$ layers.

Table 1.1: The basic structures of just some of the most commonly studied high-T$_c$ cuprate systems.

System	Typical layer sequence	n	T$_c$
La$_{2}$CuO$_{4+x}$	LaO-CuO$_2$-LaO	n=1	38K
La$_{2-x}$(Sr,Ba)$_x$CuO$_4$	La$_{2-x}$(Sr,Ba)$_x$O-CuO$_2$-La$_{2-x}$(Sr,Ba)$_x$	n=1	38K
YBa$_2$Cu$_3$O$_7$	CuO-BaO-CuO$_2$-Y-CuO$_2$-BaO-CuO	n=2	95K
YBa$_2$Cu$_4$O$_8$	CuO-CuO-BaO-CuO$_2$-Y-CuO$_2$-BaO-CuO-CuO	n=2	90K
Bi$_2$Sr$_2$Ca$_{n-1}$Cu$_n$O$_{4+2n}$	BiO-SrO-CuO$_2$-Ca-CuO$_2$-SrO-BiO	n=1	20K
		n=2	90K
		n=3	110K
Tl$_2$Ba$_2$Ca$_{n-1}$Cu$_n$O$_{2+2n}$	TlO-BaO-CuO$_2$-Ca-CuO$_2$-BaO-TlO	n=1	80K
		n=2	110K
		n=3	125K
Pb$_2$Sr$_2$YCu$_3$O$_8$	CuO-PbO-SrO-CuO$_2$-Y-CuO$_2$-SrO-PbO-CuO	n=2	70K
HgBaCa$_{n-1}$Cu$_n$O$_{2+2n}$	HgO-BaO-CuO$_2$-Ca-CuO$_2$-BaO-HgO	n=2	114K
		n=3	135K

The orderly picture of the cuprates presented by Table 1.1 is only a starting point, for their real structures are far from those of the crystallographic ideal. In his review [7], Sleight states the pervasive themes of the oxide superconductors to be defects, disorder, inhomogeneity and thermodynamic instability: `It is highly doubtful that a good understanding of the high-T$_c$ superconductors can be obtained unless we learn to model the inhomogeneities that abound.'. The inhomogeneities include point defects due to mixed cation substitutions, oxygen vacancies and interstitials, as well as extended defects, dislocations, distortions due to the strain between layers, and twinning as a response to the relief of strain. In addition, the compositions given in Table 1.1 are only to be considered as nominal values; the true stoichiometries are always different from these, and may vary over a considerable range. These intrinsic structural complexities are a characteristic of metastable multi-atom structures, and they are, it would seem, intimately linked to the exciting novel properties of these systems. Indeed, if the ideal structures were to be actualised (and some of the ideal compositions are too unstable to even prepare) then in the main, they would result in anti-ferromagnetic insulators. The metallic states of the real systems, which are essential for superconductivity, are the result of oxidation or cation doping, and the deliberate introduction of such defects can be employed to control the doping of carriers to the CuO$_2$ planes, and hence control the superconducting properties. The intentional application of defects to increasing the magnetic flux pinning properties is also a vital means of increasing values of critical current, a factor so important for technological applications.

Inevitably, the non-ideal crystalline nature presents very difficult problems for reliable synthesis and characterisation of samples, and differentiating the effectiveness of specific defects is likewise problematic. X-ray scattering is now well established as an essential tool in the determination of crystal structures, and in the case of diffuse scattering as a means of observing deviations from the ideal structure due to both structural fluctuations and defect structures [9]. The advantages of x-rays for studying relatively small samples, and the ability to investigate bulk properties, means the technique can provide an invaluable complement to the other probes of atomic structure such as high-resolution electron microscopy (HREM), electron diffraction, neutron diffraction, EXAFS, positron annihilation, etc. It is certainly an important tool for characterising the high-T$_c$ cuprates, and it is the aim of this thesis to apply x-ray scattering techniques to the study of one of the structurally most involved, and consequently least well understood, of the cuprates, the system $\rm {Bi_2Sr_2CaCu_2O_{8+\delta}}$ (or Bi-2212).

The bismuth-based family $\rm {Bi_2Sr_2Ca_{n-1}Cu_nO_{4+2n}}$ is closely related to both thallium $\rm {Tl_2Ba_2Ca_{n-1}Cu_nO_{4+2n}}$ and mercury $\rm {HgBa_2Ca_{n-1}Cu_nO_{2+2n}}$ families, and together the n=3 phases of these rank amongst the highest T$_c$ values yet achieved; 110K, 128K, and 135K respectively. The charge reservoir AO layers have a rocksalt-type structure, but in all three systems they are strongly distorted. In the case of the Bi-systems, the distortions are so strong that the whole structure is modulated, and this modulation has long-range order with a well-defined periodicity which is incommensurate with that of the ideal structure. Such incommensurate structures are also observed with only short-range ordering in the Tl-systems. The incommensurate nature of the structure has compounded greatly the difficulties of determining the structural characteristics of an already complex system, and many uncertainties remain. The potential implications of the incommensurate modulation for the superconducting properties, perhaps through control of charge transfer has, for instance, led to much inconclusive debate.

The incommensurate modulation of the Bi-systems are the most dramatic response to a structural frustration which is common to all of the cuprates: the mismatch in lattice dimensions between the covalently bonded perovskite CuO$_2$ planes and the rocksalt-type structure of the charge reservoir layers. Frustration is a characteristic of all layered structures to some degree and as such the modulation of the Bi-systems forms part of a wider field of interest which is associated with the general properties of incommensurate structures. In this field, the interest is focused on the mechanisms which drive the incommensurate phenomena, and the factors which determine amplitudes, wavevectors, and other characteristics of the modulation.

The aim of the experiments which make up this thesis attempt to study the structure of Bi-2212 with reference to its dual properties as both a high-T$_c$ cuprate and an incommensurate structure. As such, Chapter 2 first sets out to place the cuprate systems in this wider context. It starts by introducing the general concepts of incommensurate structures with particular reference to the class of two dimensional layered misfit structures with which the cuprates are most closely associated, and examples of systems which share features of relevance are discussed. The formalism of basic x-ray scattering theory and the extension necessary for its application to incommensurate systems will also be set out in this chapter.

In the field of high-T$_c$ a tremendous amount of literature has been published since 1986, and the portions relating to the structure and properties of Bi-2212 have occupied a considerable share. The relevant parts of the literature will therefore be reviewed, where necessary, at the most appropriate points throughout the thesis. Chapter 3 starts with an extensive review of studies concerned with determining the structure of Bi-2212, and emphasises the many still unresolved differences between proposed models, and the contradictions of some experimental observations. An attempt to resolve some of these questions, is then described, by a comparative study of x-ray scattering measurements obtained from a variety of Bi-2212 single crystals grown by four different research groups. The results are successful in clarifying precisely which features are fundamental to the structure, and which are sample-dependent. Extra features not accounted for by current models are also identified, and high-resolution measurements of these show them to be characteristic of defects in the incommensurate modulation.

In Chapter 4 the properties of the incommensurate modulation are investigated by studying its temperature response in situ up to 450$^o$C. The results reveal the invariance of the incommensurability due to pinning, and the fundamental control exerted by oxygen diffusion above 300$^o$C is also characterised. A possible new oxygen-deficient high-temperature phase is tentatively identified.

The important role of oxygen in the high-T$_c$ cuprates is now beyond doubt, and the experiments in Chapter 5 attempt to distinguish changes in diffuse scattering associated with changes in oxygen content or ordering. The specific case of Bi-2212 is reviewed at the start of this chapter, for which there have been many investigations of the transport properties as a function of oxygen content, but remarkably few into the possible associated changes of microstructure.

The final question relating to the Bi-2212 structure regards the possible existence of low temperature structural changes which could be associated with the superconducting transition. There have been a variety of reports from other structural probes for evidence of changes in the buckling of the CuO$_2$ layers, and an anomaly in the temperature dependence of the O(apical) Debye-Waller factor. Chapter 6, therefore, presents the most detailed x-ray measurements to date made over the temperature range T=200-10K of the variation in intensity of fundamental and satellite reflections. It is found that the results follow well the expected Debye-Waller behaviour, and show no evidence for any changes in either the period or amplitude of the modulation. The results thus place an upper limit upon the magnitude, or the lengthscale, of any possible structural involvement.

The thesis is completed by Chapter 7 which describes the development of a method to simulate the diffuse scattering from a computer model of the Bi-2212 structure, with the aim of aiding the interpretation of the experimental observations from previous chapters. The approach involves large scale numerical calculations of the scattered intensity which are designed to exploit the parallel architecture of a Connection Machine 200. The preliminary results which are presented show the method to be well capable of accommodating the incommensurate features of Bi-2212, and to be suitable for more detailed studies in the future.