Introduction

The crystallographic structures of the high-T$_c$ superconducting cuprates are now well established. There remains, however, much interest in the existence and nature of possible structural changes associated with the electronic transition to the superconducting state. Although no major structural transition takes place, there has been ample evidence from a range of techniques demonstrating the major role played by the crystalline lattice in these materials (for an early review see Ranninger [157]). A strong electron lattice coupling is believed to be either actively participating in the high-T$_c$ superconducting mechanism, or at least sensitively mirroring the interactions in some way [158,159,160]. Local structural distortions and instabilities have been reported in cuprates at temperatures above the superconducting transition, as T$_c$ is approached, and at T$_c$ itself. The most dramatic results being for $\rm {Tl_2Ba_2CaCu_2O_8}$ where neutron diffraction and pair distribution analysis [161] revealed correlated displacements of O and Cu perpendicular to the CuO$_2$ plane coincident with the onset of superconductivity.

Studies enquiring into the structure in the vicinity of the superconducting transition have tended to fall into one of two camps. (i) Probes of the local structure such as EXAFS, and PDF, have offered strong indications of structural changes across T$_c$; and (ii) measurements of the average structure using diffraction techniques which have on the whole, and to the contrary, failed to confirm the existence of any specific changes. The analysis of EXAFS results from Y-123 by Conradson and Mustre de Leon [162,163] for instance, produced evidence for a double-well potential along the c axis at the apical oxygen site which they observed to soften in the vicinity of T$_c$. Ion channelling experiments also revealed anomalous changes across T$_c$, specifically in the oxygen sub-lattice of Y-123 [164,165]. Despite the most exhaustive of neutron diffraction studies into various R-123 systems Schweiss [166], however, failed to find any evidence for the existence of a double-well potential. The general trend would seem therefore to indicate that local lattice distortions over a length scale of only a few unit cells are of importance to the superconducting mechanism.

In the case of the Bi-systems, which are structurally closely related to the ${\rm Tl_2Ba_2CaCu_2O_8}$ system, the ever present incommensurate modulation severely hampers the reliable observation of such subtle results. As has already been stated in previous chapters, the modulation originates with the mismatch in bond lengths between the BiO and CuO$_2$ layers. The widely studied high-temperature tetragonal to low-temperature orthorhombic transition in La$_{2-x}$Sr$_x$CuO$_4$ [23], which is known to be influential to superconductivity, is similarly related to the relief of stress due to mismatch between LaO and CuO$_2$ sheets. The related tilting of the CuO$_6$ octahedra has been observed to directly change at T$_c$ by Braden [167] suggesting that a coupling exists between the orthorhombic distortion and superconductivity. Unlike the lanthanum compounds, however, no phase transitions are known to occur in the Bi-systems, with no change in the incommensurate wavevector observed down to at least 10K [93,100]. The change in mismatch stress which must accompany a change in temperature may nonetheless produce a more subtle response within the structure.

Investigations of the temperature dependence of the $\rm {Bi_2Sr_2Ca_Cu_2O_{8+\delta}}$ lattice have therefore concentrated upon phonon behaviour and upon the electronic structure. Neutron scattering experiments have indicated large anharmonic effects for oxygen modes below T$_c$ [168]. Whilst positron annihilation studies, which are strongly influenced by the incommensurate modulation, have observed discontinuous changes across T$_c$ indicating a change in the electronic state of the Bi$_2$O$_2$ layer at the onset of superconductivity [169]. Similar to Y-123, ion channelling has also measured an anomalous change near T$_c$ [170]. Also observed (in the n=3 phase) is evidence of structural instabilities and lattice softening in the normal state at temperatures between T=160K and 120K [171]. A discontinuity in Young's modulus and a peak in the internal friction have also been observed at 145K [172]. Determination of the elastic constants from sound velocity data has shown a small softening in a longitudinal mode at T$_c$ [173], whilst quite strong softening of in-plane shear modes has been observed at a variety of temperatures; results which could be caused, according to Wu [174], by the mismatched structure becoming thermodynamically unstable at lower temperatures. A number of other acoustic experiments have consistently reported internal friction peaks (that is rapid changes with temperature in the damping of the ultrasonic signal), which are suggestive of structural phase changes at similar temperatures of 250K and 160K (see Anderson [175] and references therein).

Most recently, EXAFS studies by Bianconi [176,177] of the Cu site configurations have investigated the involvement of the modulated structure with electron-lattice interactions. The evidence reported shows a non-homogeneous change in the Cu-O(apical) Debye-Waller factor close to T$_c$, with hardening at some Cu sites and softening at others. The different sites are distinguished by their position in the corrugation of the CuO$_2$ plane, i.e. the phase of the incommensurate modulation at each site. It has further been proposed that the non-homogeneous Cu sites could be correlated with the ordering of polarons in the CuO$_2$ layers; the implication is that confinement of polarons along one dimensional stripes in the corrugated CuO$_2$ plane is at the origin of the superconducting pairing mechanism [178,179,180]. This is a similar interpretation to that of the local structural distortions observed by PDF in other systems but is given here, it is claimed, a long range order by the incommensurate modulation. It would therefore produce a change detectable in a diffraction experiment caused by the weakening of the modulation at the onset of polaron formation at a temperature slightly above T$_c$.

A complete knowledge of the behaviour of the modulation at low temperature is therefore necessary for two important reasons. To allow the elimination of modulation related changes from the results of other experiments which are seeking exclusively superconductivity related changes. And to establish whether the proposed structural links between the modulation and the superconducting mechanism itself in the bismuth compounds has any basis. The only previous dedicated study of the incommensurate modulation at low temperature, carried out by Takenaka [93], was limited to a low resolution x-ray diffraction measurement of the positions of the incommensurate satellite reflections down to 10K; no change was observed. Whilst the satellite positions are linked to the period of the modulation, it is the intensity of the satellite reflections which relate information about the amplitude of the modulation, i.e. the magnitude of the atomic displacements. The temperature dependence of the intensity is in turn related to the Debye-Waller factor. Presented in this chapter are the most precise measurements to date on the temperature response of the incommensurate modulation in Bi-2212; the study utilises single crystal x-ray scattering methods over the temperature range T=200K to 18K.