Discussion

The results of Section 4.3.2 demonstrate uncontrovertibly the inert relationship between incommensurability and temperature in Bi-2212. While the b axis has a steady linear response, and the layer mismatch must therefore vary continuously with temperature, the incommensurate value of the modulation remains fixed. This demonstrates that the incommensurate state cannot be considered in the context of a 'floating phase'. The result is a very precise confirmation of that already suggested by electron diffraction experiments, while the additional intensity information provides a new insight. The interplay in behaviour between the satellite and fundamental reflections indicates the amplitude of the atomic modulation function to steadily decrease with temperature with a concordant gain in the order of the average structure. Although the reverse of what might naively be expected to be the effect of increasing temperature, the results make for a straightforward model of the modulation's behaviour which is illustrated in Figure 4.14. With the incommensurate value pinned at $\approx$0.21${\bf b}^*$, the real space period of the modulation becomes solely dependent upon the value of the b lattice parameter. The effect of increasing temperature is to increase b and thereby to stretch the modulation along that direction. The effect, as would be the case for the stretching of any corrugated plane, is a flattening of the corrugations, which is observed in the decreasing modulation amplitude. It is interesting to note that the modulation does not respond to the c axis, which is increasing at the same time and might therefore be expected to stretch the modulation amplitude along with it. This shows that the dominant parameter controlling the behaviour of the modulation is the in-plane Cu-O bond length which is also responsible for determining the b axis.

Figure 4.14: A simple model illustrating the response of the modulation to temperature. The incommensurate value is fixed but the modulation is stretched by the extension of the b axis with increasing temperature, and the result is a decrease in the amplitude of the modulation.

Despite the common picture of incommensurate modulations, with wavevectors which vary continuously or discreetly to some degree with temperature, the phenomenon of systems with temperature independent wavevectors is well documented [116]. Such phases exist in the 2H-TaSe$_2$ system, and also in NbSe$_3$ for example. In the Landau free-energy description of incommensurate phases, any change of the modulation amplitude as a function of temperature should also lead to a change of wavevector. Explanations for such temperature independent behaviour have been established, however, by considering the pinning of the wavevector to the underlying lattice. The mechanism for pinning can arise by two specific means. Pinning by impurities is one important mechanism, such as in 1T-VSe$_2$ where poor stoichiometry plays an important role. In cases where the coherence length of the modulation is large, indicating impurity pinning to be of minor consequence, then a further mechanism comes about through the properties of strongly non-sinusoidal functions. The contribution to the free energy of higher harmonics of a non-sinusoidal modulation become highly significant in a way in which those of a purely sinusoidal one do not, and the sum of such terms is able to effectively pin higher-order commensurate modulations. Or, to use the language of Chapter 2, these are described as semi-commensurate modulations. Such situations commonly arise in systems with competing periodicities, called discommensurations, the phenomena is described by the Devils Staircase approach [12].

In the case in question, both approaches could potentially account for the temperature independent wavevector. The wavevector value of 0.21${\bf b}^*$ sits close enough to the commensurate value of $\frac{1}{5}$ (or $\frac{4}{19}$) to potentially be semicommensurate, and a theoretical model of the modulation with discommensurate features has even been developed by Walker [79]. The strength of the higher order satellites, observable to third order in these experiments, is evidence of a strong higher harmonic component. The sawtooth modulation function proposed to describe the oxygen occupancy of the BiO layer by Petricek [86] is strongly deviated from the sinusoidal modulation which describes the CuO$_2$ planes; this is reflected to an extent in all of the other structural refinements. However, the coherence of recognised discommensurate modulations in the chalcogenides is typically in excess of 4000${\rm\AA}$. By comparison, pinning due to a heavily defected structure, possibly by poor stoichiometry or dislocations, is one which might seem more readily identifiable in Bi-2212.

The presence of dislocation networks in Bi-2212 has been confirmed in electron microscopy studies by Lee [117] and discussed in detail by Shang [118]. They were first identified by Fung [119] who described them as anti-phase boundaries. They involve extended boundaries in the ${\bf a-b}$ plane across which sites in the BiO layer are displaced by a 1/2$<$1 1 0$>$. They effectively correspond to a dislocation in the BiO ribbons and could therefore be a potential source of pinning. The disappearance of the dislocations upon heating between 400 and 460$^o$C was suggested to be evidence of a structural transition in this range by Yang [120], and it is indeed in this temperature window that the changes in the modulation wavevector have been observed. Another effect claimed to occur in this region by Bidkin [92] is the disappearance of the diffuse streaks. Unfortunately, the data here is not conclusive due to the much stronger mosaic scattering which obscures their measurement. The possibility cannot be eliminated that the streaks merely reduce in intensity in accord with the other reflections but to a level too low to be detectable. If the disappearance of the streaks is authentic, it would suggest that their origin relates in some way to the dislocations, and that both changes are a consequence of the increased mobility of the oxygen in the Bi$_2$O$_2$ layers at the initiation of the second oxygen diffusion process. Their disappearance was described as being permanent by Yang, however, while the streaks are observed to be at least partially restored upon cooling in these measurements (Bdikin [92] also described the transition as fully reversible).

A further pinning mechanism, potentially the dominant one, which could be realisable in Bi-2212 derives from the structural model of Le Page. The model is one which has strongly non-sinusoidal characteristics, involving a random sequence of 5$b$ or 4.5$b$ commensurate blocks which duplicate the incommensurate period. Instead of a semicommensurate phase pinned by discommensuration though, it is more reminiscent of a chaotic phase where the interaction with the underlying CuO$_2$ layers is so strong as to cause the modulation to lock-on to the periodicity of these layers at random (the situation was shown in Figure 2.1(c) of Chapter 2). A chaotic phase model would account for the dominance of the Cu-O bond length, the temperature independence of the wavevector, and the random sequencing of the proposed commensurate blocks. If this picture is correct, then it would indicate that a peculiarly strong inter-layer interaction is at work in the Bi-systems. The absence of long range order in the modulations of other compounds such as the Tl-systems, might then be explained by a weaker inter-layer interaction in these systems.

Above 300$^o$C oxygen diffusion processes enact the dominant influence upon the structure. The combined results of the two experimental sections presented show there to be two quite distinct oxygen diffusion processes at work, each exerting its own particular authority. This has also been evidenced in many experiments which have found the effects of annealing upon other properties, such as T$_c$, to be very different above and below about 450$^o$C [121,122,113]. The first of the processes commences close to 300$^o$C and is observed in this experiment as a growing inhomogeneity in the sample. The loss of oxygen is also responsible for a permanent extension of approximately 0.4$\%$ in both b and c lattice parameters. Such an extension in the c axis is the most commonly observed feature to be associated with oxygen loss, and likewise a comparable contraction upon oxygen absorption (see the review of this in Section 5.2). The observations made by electron diffraction methods of vacuum annealing in the vicinity of 300$^o$C, supports the assumption that the changes observed in this work are also due to oxygen diffusion. The results here confirm that such diffusion mechanisms have little or no influence upon the period of the incommensurate modulation.

The second phase of oxygen diffusion commences above 400$^o$C, and in this temperature range at least, appears to be much slower than the first diffusion process. Its different nature is also seen in the reversal of the c axis extension which was effected in the first. The measurements over the prolonged period of time at 450$^o$C appear to be monitoring the controlled decomposition of the sample, due to this slow but continuous escape of oxygen from the structure. It is during this period that changes in the modulation period are finally induced, close to the amorphisation as has been reported in all previous studies (although it is usually observed only fleetingly as a rapid transition at much higher temperatures). The change involves the gradual appearance of diffuse scattering, and eventually new satellites, with ${\bf b}^*$ wavevector components $\beta \leq 0.20$. Although a diffuse distribution of scattering is found from 0.20${\bf b}^*$ to 0.10${\bf b}^*$, very strong, well defined reflections were found at $\approx 0.14{\bf b}^*$. This corresponds to a new modulation wavelength of $\approx 7.1{\it b}$, or $\frac{3}{2}$x4.7 the old wavelength. Assuming this is the result of the reducing oxygen content, then it is strong evidence to suggest a phase transition to a new oxygen deficient state, intermediate on the route to decomposition. It is noted, that the value observed here for this new period is close to that also observed to appear, by many authors, in electron diffraction studies of rare-earth doped Bi-systems. For instance, Chen [123] has measured the appearance of satellites at $\approx 0.125{\bf b}^*$ in ${\rm Bi_2Sr_2Ca_{1-x}Pr_xCu_2O_{8+\delta}}$ when $x$ exceeds 0.6, and Inoue [124] observed a similar long period modulation in ${\rm Bi_2Sr_2Ca_{1-x}Y_xCu_2O_{8+\delta}}$. The appearance of these new modulations are observed, as is also the case here, as a phase separation, the new modulation coexisting with the original modulation in the same sample. Changes in the lattice parameters very like those presented here in Figure 4.13 (i.e. contraction of the c axis and expansion of the basal plane) are also commonly reported to accompany the appearance of this new modulation period.

The important question which must be answered is to which oxygen sites in the structure do these two separate diffusion processes relate. The same conclusions of distinct diffusion mechanisms at these temperatures have also been reached by other means [111,125,112] (reviewed in Section 5.2), but without any knowledge as to the structural changes involved. The sequence of diffusion as a function of temperature and time is determined by the activation energy of the different oxygen sites; precise data does not exist for this but a good indication [126] of how tightly bound a particular oxygen will be is its bond length. Data on oxygen bond lengths is available from the structural refinements discussed in Chapter 3. In ascending order of expected activation energy the oxygen sites would be: inter-ribbon and bridging, rocksalt, apical, and the most tightly bound of all, Cu planar. It can be surmised from this that the first diffusion process must almost certainly be associated with any interchain oxygens or with the bridging sites, and this is a frequent assertion of many oxygen studies. The persistence of the modulation well beyond this initial oxygen diffusion process is strong evidence to conclude that the modulation remains stable and well ordered even when the excess oxygen of the Bi$_2$O$_2$ layers has been removed. This is just as would be expected in the picture of the BiO layer being strongly pinned to the potential of the CuO$_2$ layer, and would account for the very similar descriptions of the modulation given by all the structural refinements despite very different values being determined for oxygen content; even one, for example, in which the bridging positions were observed to be entirely vacated [83].

Figure 4.15: Three examples of how the modulated structure may change to give new values of $\beta$. For simplicity the modulation period of the starting structure (lower strip) has been approximated to 5 unit cells (or $\beta$=1/5). To produce the experimentally observed value of 0.14$\approx$2/15 it can be seen that an alteration of the structure between bridging positions is necessary.

However, the much slower second process did produce a very definite effect upon the modulation and it might be argued that this could therefore correspond to the change in the bridging oxygens. The change, though, involved the transition of the incommensurate period to a preferred value of $\beta\approx0.14$, and this value does not seem consistent with a change due to the absence of the bridging positions alone. To illustrate this, in Figure 4.15 the modulation periods have been approximated to their nearest semicommensurate counterparts, and the position of the bridging sites, repeating once for every five unit cells, is indicated. If the modulation is considered to still be pinned then it cannot adjust continuously to the changing oxygen content, and the first discontinuous change in period which is possible is then when every second bridging position has been vacated and the modulation relaxes to a value of $\beta=\frac{1}{10}$ or 0.10, which could after further oxygen removal be followed by $\beta=\frac{1}{15}$. But, in order to achieve a change to the value which is observed in this experiment, of $\beta=\frac{2}{15}$, a change in the structure of the central rocksalt sites of the Bi$_2$O$_2$ layers is also required. A fact which then suggests that the bridging positions have already been vacated by the first diffusion process and that the second diffusion corresponds to the rocksalt oxygen sites. This is a more dramatic reorganisation of the structure and hence the transition is coincident with an extension of the b axis and the collapse of the c axis. This behaviour can also be accounted for in this picture of oxygen removal by considering the bonding between adjacent BiO layers. In the fully oxygenated structure there is little bonding between adjacent layers, with the Bi lone pair believed to be directed along the c axis forcing the layers apart. However, under substantially reduced oxygen conditions, the Bi could only maintain the favoured oxygen coordination by forming bonds with oxygens in the adjacent layer and directing the lone pair instead within the basal plane and thus expanding the rocksalt region.

The above interpretation does run contrary to the assumptions made in the high-temperature electron diffraction studies discussed at the start of this chapter. In these it was asserted that the deference of the satellites to the low temperature oxygen diffusion indicated that the oxygen was not removed from the BiO layers but rather from the perovskite block. From the wider discussion of the modulations properties though, there appears no compelling reason to assume that the structure does not remain impervious to the removal of the bridging oxygens. This then overcomes the apparently anomalous requirement that the most tightly bound oxygen sites should be those which become mobile at the lowest temperature. The ordering responsible for the 2$a$x2$b$ superstructure observed in electron diffraction experiments must therefore be either the result of some additional interaction between the electron beam and the oxygen of the CuO$_2$ planes, although this is thought not to be the case, or it is located elsewhere in the structure. The same argument must also be applied to the twinning reported by Yang [110] after vacuum annealing. Indeed, they note themselves that the proposed CuO$_2$ twinning model is of a previously unknown character for a perovskite structure. It should also be pointed out that oxygen vacancies, of the density suggested by the twinning and the 2$a$x2$b$ models within the CuO$_2$ planes would surely be highly detrimental to the superconducting properties. Whilst Gao [109] has observed these features to be present in samples with close to optimum values of T$_c$.

In conclusion, the fundamental properties of the incommensurate modulation of Bi-2212 have been investigated using x-ray scattering to observe in situ the response of the structure to high temperature (over the range room temperature to 450$^o$C). The results have conclusively demonstrated the temperature independent nature of the incommensurability, and several pinning mechanisms which would offer explanations for this have been discussed. The strong influence exerted by oxygen diffusion over the structure has also been highlighted, and the results clearly indicate the existence of two distinct oxygen diffusion processes, the first being dominant over the temperature range 300-400$^o$C, and the second commencing above 400$^o$C. However, it has also been found that the wavevector of the modulation is in no way a continuous function of oxygen content; a result contrary to the assumptions which have previously been widespread in the literature. Only after very prolonged annealing at 450$^o$C under vacuum was evidence for a structural transition observed, which involved the appearance of satellite reflections with a new wavevector of ${\bf q}\approx0.14{\bf b}^*+{\bf c}^*$. It is tentatively suggested that this transition may relate to the formation of a new oxygen-deficient structure, due to the removal of oxygen from both the bridging and rocksalt sites of the Bi$_2$O$_2$ layer.