A Review of Structure Studies

The first report of superconductivity in the Bi-Sr-Cu-O system was by Michel [35] in 1987. There soon followed other reports, with substantial improvements in T$_c$, and efforts to identify the structure responsible for superconductivity [38,36,39]. Michel's attempt to synthesise such a bismuth based high-T$_c$ cuprate, in the wake of discoveries in the lanthanum and yittrium systems, was suggested by the already known tendency of bismuth to form layered oxides in the systems known as Aurivillius phases [40]. Identification of the n=2 structure came with the first single crystal x-ray study of Subramanian [37] who found the average crystallographic structure to be orthorhombic with a unit cell of ${\it a \approx b}$=5.4${\rm\AA}$, c=30.9${\rm\AA}$, and containing four formula units. The structure was similarly identified by Tarascon [41] but was assigned a unit cell based on the perovskite sublattice, the parameters related by $a= \sqrt{2} a_p$ where $a_p=3.85{\rm\AA}$ is the Cu-O-Cu cube edge, and caused because the Bi-O direction is rotated through $\frac{\pi}{4}$ with respect to the perovskite sublattice. The structure within the Bi layer was however mistakenly classified by both studies as being of this classical Aurivillius type, a statement which was subsequently taken up in other studies [42,43,44].

Figure 3.1: The idealised crystal structures of the three members of the bismuth-based family Bi$_2$Sr$_2$Ca$_{n-1}$Cu$_n$O$_{4+2n}$ with n=1,2 and 3 (relating to the number of CuO$_2$ layers in each unit cell).

Recognition that the $\rm { Bi_2O_2}$ layers were different to those of the Aurivillius phase came in an x-ray refinement by Sunshine [45] and in a neutron refinement by Bordet [46]. Its was noted by Sunshine that the bismuth bonding and geometry was highly unusual with double layers of edge-sharing octahedra, with a coordination consistent with Bi$^{3+}$, rather than the eight-fold coordination of the Aurivillius structure. The same conclusion was drawn by Torardi [47] from Bi-2201 and Tl-2201 systems. An important observation of this comparison was that the c axis of the Bi phase was 1.4${\rm\AA}$ longer than in the Tl phase, despite essentially equivalent average structures, and the average ionic radii of Bi and Tl being almost identical. Torardi correctly concluded that this was a reflection of the very weak Bi-O interlayer bonding. In all of these early refinements the oxygen sites within the $\rm { Bi_2O_2}$ layers were poorly defined with excessively large thermal parameters. So large, they could only be indicative of large static displacements away from the refined sites. An attempt to refine the positions from neutron powder data by Sequeira [48], however, led to anomalous values of Bi-O bond lengths. Evidence that the displacements were at least locally ordered, rather than uncorrelated local displacements, came from atomic pair distribution function (PDF) analysis of Tl-2212 by Dmowski [49]. The presence of significant local distortions were more clearly confirmed in a later neutron study by Bordet [50] who proposed a model where the oxygens are shifted off the centre of the rocksalt cell to instead occupy one of four possible sites arranged symmetrically around the rocksalt position: this allowed for an ordering of the Bi-O bonds into chains running along the ${\bf b}$ axis, and accounted for some of the disorder.

None of these early studies made any attempt to incorporate information from the incommensurate satellites. The many dramatic structure images obtained by high resolution electron microscopy (HREM) [51,52,53,54,55,56] clearly demonstrated this would be necessary for the screen of apparent disorder to be lifted from the $\rm { Bi_2O_2}$ layers. Matsui [53] observed alternating dark and light bands corresponding to zones of contraction and expansion in the layers with displacements too large to be ignored in refinements of the average structure. The bands repeated along the b axis with a periodicity of $\approx$ 4.7b, and were stacked orderly along the c axis so as to form a body-centred supercell. The images were in good accord with the positions of the satellites, whose symmetry had been explored in detail by electron diffraction [52,57,58,59], and were found to be only observable in ${\bf b}^*$-${\bf c}^*$ and ${\bf a}^*$-${\bf b}^*$ reciprocal planes with a wavevector (for Bi-2212) of ${\bf q} = \frac{1}{4.7}{\bf b}^* + {\bf c}^*$. That oxygen was a significant participant in the modulation was suggested by the large differences measured by Fischer [60] between x-ray and neutron intensities of the satellites. The commensurate $\gamma{\bf c}^*$ component results from the body-centred stacking observed by Matsui [53]. In the case of the n=1 system this does not apply with the ${\bf c}^*$ component varying between 0.3${\bf c}^*$ to 0.6${\bf c}^*$, while the $\beta{\bf b}^*$ component remains identical. It was pointed out by Chen [58] that the difference between the incommensurate value, in real space, of 4.76b and a potential commensurate value of 5b is $\approx 1.3{\rm\AA}$, the presence of the ${\bf c}^*$ component is effective in reducing this difference to only $\approx 0.7{\rm\AA}$ along the wavevector direction. Also, Imai [61] observed that the separation between modulation wave fronts in real space, when calculated from the relevant lattice parameters and q values, is rendered similar, close to 20${\rm\AA}$, in both Bi-2201 and Bi-2212 systems by this.

It should be noted that some confusion in notation has arisen between authors in ascribing the modulation to either a or b lattice directions; this is essentially arbitrary as both a and b lattice parameters are similar. The sole consequence is the choice of space group, which may be Amaa related for an a axis modulation or Bbmb related for a b axis one. As both descriptions are otherwise equivalent, the choice of the b axis is made in this work and will be used consistently throughout, and the differences between authors will be neglected.

With so few experimental facts, several alternate mechanisms were initially proposed to be at the origin of the modulation. Their respective arguments have been discussed previously by Zandbergen [56] and Toledano [62]. The two-dimensional nature of the Bi$_2$O$_2$ layers along with the propensity for bismuth to disproportionate into Bi$^{3+}$ and Bi$^{5+}$ suggested a charge density wave, similar to that in the Ba(Bi,Pb)O$_3$, to be responsible. The CDW also requires the nesting of the Fermi surface, and evidence for this came from early band structure calculations based on the average structure which did show pockets below the Fermi level [63,64]. However, the coordination is typical of Bi$^{3+}$, and no evidence for the presence of Bi$^{^5+}$ has come from any of the many spectroscopic studies [65,66]. That the displacements could be the result of an occupational modulation was suggested by Kirk [67] based upon scanning tunneling microscopy (STM) images which appeared to show rows of Bi vacancies every nine or ten atomic sites. Certainly the occurrence of vacancies and other compositional variations hold an influence over the period of the modulation [68,69], but evidence for such ordered arrays of vacancies have not been observed in any of the similar studies subsequently undertaken [70].

The suggestion that the modulation could be the result of lattice mismatch between the component layers was advanced early on by Zandbergen [56], and later by several others. The mismatch is between the crystallographic units of the rocksalt type charge reservoir layer and the more rigid perovskite CuO$_2$ layers. This can be quantified using equation 2.2 for the tolerance factor discussed in Chapter 2. Because of the inflexibility of the strong intralayer covalent bonding of the CuO$_2$ sheets, it is the intralayer Cu-O bond length of $\approx 1.9{\rm\AA}$, which ultimately determines the lattice parameters a and b. Using this value in equation 2.2 an ideal length for the rocksalt intralayer bond of $\approx 2.7{\rm\AA}$ is obtained, which is the value required for the stable undistorted stacking of the two layer types to be possible. When compared to the standard Bi-O bond length of between 2.3${\rm\AA}$ and 2.4${\rm\AA}$ [24] favoured in other bismuth oxides, the magnitude of the structural frustration becomes clear. This mismatch places the BiO layers under tension and the CuO$_2$ layers under compression. The difference is comparable to the magnitude of the Bi atom displacements along the b axis of $\approx 0.4{\rm\AA}$ observed by HREM.

A complete structural determination, which must incorporate the satellite reflections, is a considerable challenge. Traditional Rietveld refinement methods analyse only fundamental reflections and so refine only an average unmodulated structure. These can be adapted in two ways to incorporate the satellites, but the number of refinable parameters will increase dramatically in both cases. The least satisfactory of the two alternates is to approximate the incommensurate structure to a commensurate supercell; the earliest attempt analysed Bi-2201 by approximating to a 5b supercell [71]. The results were successful in illustrating that large displacements of Cu and O along the c axis accompanied the Bi displacements along b. This strong disruption of the CuO$_2$ planes in Bi-2201 which is present, though less dramatic in the other systems, was suggested as explanation for its comparatively much lower T$_c$ value (T$_c$=40K for Bi-2201 and T$_c$=80K for Tl-2201). The more satisfactory solution is the extension of the refinement method using the superspace group description of modulated crystals described in Chapter 2. Such methods had been only recently developed prior to the discovery of high-T$_c$ by Petricek and Coppens [72], and the solution of the Bi-2212 structure in a single crystal x-ray study by Gao and Coppens [73] was to be its first large scale application . They determined in full amplitudes and phase relations of cation displacements in each layer, and showed the sinusoidal modulation of the Cu sites involved only displacements along the c axis, perpendicular to the plane, in agreement with the strong covalent bonding within the plane. Oxygen site refinements could still not be made, however, due to the oxygen, which is a weak scatterer of x-rays, being in the presence of the very strong scattering contribution of bismuth.

A breakthrough in understanding came with the discovery by Tarascon [74] that the satellites were locked on to commensurate values in compounds where Cu is replaced by either Fe, Co, or Mn; a refinement using traditional methods, without need for approximation, was then possible. The compounds are insulating but in all senses isostructural to Bi-2212. The x-ray refinement by Le Page [75] of ${\rm Bi_{10}Sr_{15}Fe_{10}O_{46}}$, found it to be modulated with a commensurate period of 5b, the modulation was sinusoidal but unlike Bi-2212, there was no second harmonic contribution (perhaps as a result of the commensurate state). The results supplied a model for the oxygen distribution in the $\rm { Bi_2O_2}$ layers consistent with the need to establish physically sensible intralayer bond lengths. The oxygen was found to occupy two distinct types of sites, which they called the rocksalt-type accounting for roughly 70$\%$ of the oxygen, and the bridging-type accounting for the remainder; these are illustrated in Figure 3.2. The rocksalt regions coincide with the zones of compression in the modulation, and the bridging-type correspond to the zones of expansion which link these regions. This works to reduce the Bi-O bond length, which in the undistorted rocksalt structure would be $2.7{\rm\AA}$ and too long compared to the favoured value. An important feature of the bridging region is that it allows the accommodation of an additional oxygen atom, filling the space created at the point of greatest expansion between two Bi atoms. The result is the incorporation of an additional oxygen atom every 5 unit cells.

Figure 3.2: An idealised representation of the two possible arrangements in the Bi$_2$O$_2$ layers identified by Le Page with oxygens in (a) the ideal rocksalt, and (b) the bridging positions.

In Bi-2212 Le Page [75] suggested the structure could be similarly locked onto the underlying Cu-O lattice, and the incommensurability would then be the result of a pseudo-periodic insertion of oxygen at intervals of either 4.5 or 5 unit cells (i.e. between every 9 or 10 Bi atoms). The overall average of the random sequence of such commensurate blocks would reproduce the incommensurate period observed in diffraction experiments. Interpretation of HREM images of the BiO planes have drawn Onozuka [76,77] and Goodman [78] to propose similar models. A theoretical model based on the concept has also been explored by Walker [79] in which commensurate domains are separated by domain walls formed at the extra oxygen position.

The presence of oxygen in excess of the ideal stoichiometric value of 8.0 per formula unit (p.f.u.). had already been indicated by compositional analysis and efforts to determine oxygen content using thermogravimetric methods (a full description is given in chapter 5). This being a question of significance to the superconducting properties, it had led to much debate about the possible location of the extra oxygen. An early discussion by Goodenough [80], for example, put the case for oxygen incorporation into either the oxygen deficient Ca layers or into the Bi$_2$O$_2$ layers. Evidence that the Ca layers remain oxygen deficient, however, comes from the $\rm {La_{2-x}Sr_xCaCu_2O_6}$ system where oxygen is believed to be readily incorporated into the Ca layers but as a consequence, superconductivity is difficult to attain due to the unfavourable Cu coordination which results [7]. The case for extra oxygen located in the Bi$_2$O$_2$ layers had already been made prior to the results of Le Page [75], and models which implicated the modulation in this had been crafted by Zandbergen [56], Torardi [81], and a model very close to that of Le Page suggested by Hewat [69]. Although the extra oxygen position is said by Le Page [75] to 'cause' the modulation, this is not in fact the case. The underlying cause is rather the lattice mismatch which has already been discussed. It is to accommodate this mismatch that the Bi atoms adjust their coordination and thereby create the extra space in which oxygen atoms may be accommodated. A more correct statement may perhaps be that, the extra oxygen is responsible for stabilising the long range coherence of the modulation with a particular period. Whether the period of the modulation might then be expected to vary with oxygen stoichiometry is a question which will be addressed in Chapter 4.

The relationship with oxygen stoichiometry established a definite connection between the modulation and superconductivity, through its potential to mediate the supply of carriers to the CuO$_2$ layers. This property plays a role equivalent to that of oxygen intercalation in the CuO reservoir layers in the Y-systems, and to Sr substitution in the La-systems. The importance of understanding the details of the mechanism has stimulated a substantial and continued effort to better describe the microstructure of the Bi$_2$O$_2$ layers, and to model the properties of the modulation.

Table 3.1: A summary of the structural refinements which have been undertaken, and their results for unit cell and compositional parameters. Where superspace methods have been applied the superspace group is given, otherwise the size of the commensurate supercell is given (along with the spacegroup for the average structure).

		$a$	$b$	$c$	q	super	Composition	oxygen
						cell	Bi:Sr:Ca:Cu	$8+\delta$
	Levin [82]	5.407	5.412	30.771	0.210$b^*$	5$b$	1.94:1.78:0.72:1.86	0.40
	single crystal x-ray					(Bbmb)
	Calestani [83]	5.416	5.416	30.965	0.20$a^*$	5$a$	(data not given)	0.0
	single crystal x-ray					(Amaa)
	Beskrovnyi [84]	5.397	5.401	30.716	0.211$a^*$	19$a$	2.00:2.14:0.72:2.00	0.21
	single crystal neutron					(Amaa)
	Yamamoto [85]	5.396	5.397	30.649	0.211$b^*$	N Bbmb	2.15:1.92:0.75:2.00	1.00
	powder x-ray and neutron					/11$^-$1
	Petricek [86]	5.408	5.413	30.871	0.210$a^*$	M A2aa		0.10
	single crystal x-ray					/1$^-$11
	Gao [87]	5.415	5.415	30.861	0.2095$a^*$	M A2aa	2.02:1.75:0.96:2.00	0.14
	combined x-ray & neutron					/1$^-$11

A number of detailed structural refinements have now been undertaken, and these are outlined along with their main differences in Table 3.1. Good agreement exists amongst the studies upon the nature of the cation displacements, which are illustrated in Figure 3.3, and can be described in two components: (i) a longitudinal modulation wave with displacements along the ${\bf b}$ axis, which originates in the Bi$_2$O$_2$ layers and decays rapidly to almost zero amplitude within the CuO$_2$ layers; and (ii) a modulation with transverse displacements along the ${\bf c}$ axis which is at its most pronounced in the CuO$_2$ layers. Harmonic analysis of the displacements has been presented in each case.

Figure 3.3: A projection of the structure in the b-c plane to illustrate the displacement of cations (solid circles) and oxygens (open circles), over roughly one period of the modulation i.e. five unit cells (a single unit cell has been outlined). The diagram is taken from the results of Yamamoto [85] but is representative of all the studies. Only Cu-O bonds shorter than 2.5Å, and Bi-O bonds shorter than 2.7Å are plotted.

However, important points of contention exist amongst the studies upon the organisation of oxygen atoms within the Bi$_2$O$_2$ layers. In agreement with Le Page [75], each offers a model which includes the ordering of short (2 to 2.5${\rm\AA}$) Bi-O bonds into parallel chains running along the modulation direction. These chains illustrated in Figure 3.4(a), are varyingly described as BiO double strings [83], or zig zag chains [82], and are separated from each other perpendicular to the modulation by $\approx 3{\rm\AA}$. Disagreement centres upon the oxygen arrangements within the expanded regions of these chains, and the resulting inclusion, or not, of additional oxygen sites. The extra oxygen atom in the bridging position, as described by Le Page [75], is indicated by an arrow in Figure 3.4(a). This same oxygen position is refined, but with differing occupancy probabilities, by Beskrovnyi [84] and Levin [82] using supercell approximations, and by both Petricek [86] and Gao [87] using superspace methods. In contrast Calestani [83] failed to find any such evidence for an extra oxygen in this bridging position, and in its place presented a model with a Bi-Bi distance of 3.1${\rm\AA}$, and which was suggested to be a covalent Bi-Bi bond similar to that in Bi metal.

Figure 3.4: Two possible atomic arrangements of a single BiO layer represented in the ${\bf a}-{\bf b}$ plane. In (a) well defined chains of Bi-O bonds exist with an extra oxygen atom inserted pseudo-periodically between every 9 or 10 Bi atoms in the 'bridging' position (taken from Le Page [75]). In (b) the more complicated arrangement refined by Yamamoto [85], in a truly incommensurate model, involves additional extra oxygen located in the expanded bridging region, and a split site (arrowed) with an occupational probability of $\frac{1}{2}$ .

Extra oxygen in additional sites to the bridging position have also been identified. Levin [82] found oxygen in a position which effectively links adjacent chains. Evidence for this interchain position was also found by Calestani [83], though it was not included in the final refinement. The interchain position has also been refined, in studies of related compounds; by Torardi [81] for Y doped Bi-2201, by Miehe [88] for Bi-2223, and Jirak [89] for the isostructural Bi$_2$Sr$_2$MnO$_{6.5}$. Some evidence was also found by Calestani [83] of extra oxygen located out of the BiO plane between adjacent rocksalt areas, a position coincident with the points of maximum layer separation; this possibility was rejected because of the short O-O distances ($<1.6{\rm\AA}$) involved. The variation in the number and occupancy of these additional sites in each refinement is made evident in the final column of Table 3.1, where the value for the excess oxygen content, $\delta$, varies from 0.0 to 1.0. The exceptionally high value of $\delta=1.0$, a value which has not been substantiated by any other experiments which have determined $\delta$, comes from a study by Yamamoto [85]. This study is one of the two most elaborate to date (the other being by Gao [87]), involving the combined refinement of both x-ray and neutron datasets using superspace methods. The representation of the Bi$_2$O$_2$ layer in Figure 3.4(b) is modeled after the results of this study. In this picture there are now four additional oxygens located within the expanded region, of two distinct types which are similar, though not identical to, the bridging and interchain positions found in the other studies. A more realistic picture is perhaps obtained with lower occupancy values for these sites than obtained by Yamamoto [85], but the potential for variations in the oxygen arrangements is made clear by this model, and is most likely closer to the real structure than the single bridging position settled for in other refinements.

It cannot be ignored that substantial differences also exist for values of cation composition determined by the studies. The implications for mixed occupancies and vacancies are discussed by the respective studies, though little discussion is offered as to the implications they may have for the modulation or the oxygen arrangements. Partial occupation of the Ca site by Sr, varying between 4% to 28%, was observed by Beskrovnyi [84], Gao [87], and Yamamoto [85], and it is noted that the distribution must be random to comply with symmetry. The Sr site was observed to be partially occupied by both Ca and Bi by Yamamoto [85], whilst vacancies in the Sr layer (of order 10%), were reported by Beskrovyni [84], Levin [82], and Gao [87]. The location of the Sr substitutions, or vacancies, would be most favoured at locations adjacent to the extra oxygen insertion in the BiO layer. An anomalous x-ray scattering study by Lee [90] of the Bi distribution across the cation sites produced similar results.

Despite the detailed pictures of the structure which have emerged, achieving at least a certain level of agreement, it is surprising there should still remain controversy over the positions and nature of the satellites which are observed arranged around the fundamental reflections. Previous single crystal diffraction studies using both electron and x-ray methods have consistently observed additional features in incommensurate positions which are not accounted for by any of the current refinement models. To illustrate the situation Figure 3.5 is a schematic diagram showing the position of the first and second order satellites which are observed around the fundamental reflections, and are described by the wave vector

$${\bf q}^* = 0.21{\bf b}^* + {\bf c}^*$$ (5.1)

and arise from the modulation already described. However, additional reflections, much weaker than the primary satellites, have been observed at the odd (0 0 l) positions where $l=2n+1$ [52,91,92,93,94,95]. Such reflections violate the established centrosymmetric space group (Amaa or Bbmb), and were proposed by Eibl [91] on the basis of electron diffraction studies to indicate a primitive Bravais lattice. While Zaretskii [96] and Zhigaldo [95] observing the same features using x-rays, proposed instead a second modulation parallel to the c axis to account for them. In addition, asymmetry about the ${\bf c}^*$ axis has been observed in the main satellites by electron and x-ray diffraction [42,92,94,95,97] which confers a small incommensurate ${\bf c}^*$ component upon the modulation wavevector ${\bf q}$, similar to that found in the Bi-2201 compound, and implies a lower monoclinic symmetry.

Figure 3.5: A schematic diagram of an area of reciprocal space in the ${\bf b}^*-{\bf c}^*$ plane showing fundamental reflections (large circles) along with their companion incommensurate satellites, 1st order (large squares) and 2nd order (small squares) described by the wavevector ${\bf q}$.

Supposed satellite reflections have also been observed lying either side of fundamental reflections with the characteristic 0.21${\bf b}^*$ value but with no ${\bf c^*}$ component [52,92,98,99,100]. These reflections can commonly be seen to be present in the [1 0 0] zone electron diffraction patterns of many published studies, but are frequently passed by without comment [101], or have led authors to mistakenly assume, in contradiction to the body-centred symmetry of the modulation, a wavevector ${\bf q}=0.21{\bf b}^*$ to describe all satellites [91]. It was suggested very early on by Shaw [52] that they could be the result of some further ordering in the structure. More systematic x-ray studies by Patterson [100] showed them to be broadened along the ${\bf c^*}$ direction and suggested a second, longer incommensurate wavevector, or like Novomlinsky [99], that the additional reflections could still be described by ${\bf q}^*=0.21{\bf b}^*+{\bf c}^*$, but were satellites of the forbidden $l=2n+1$ fundamental positions. Similarly, Bdikin [92] linked their presence to that of the reflections on the forbidden positions, and concluded that they derived from the sensitivity of super-reflections to symmetry violations and hence indicated the structure to have a primitive monoclinic unit cell.

All of these additional errant features, then, are frequently interpreted as suggestive of a second weaker modulation being present in the structure, or of there being some lowering of symmetry in the modulation. Certainly, all suggest further structural subtleties which have yet to be incorporated into the models derived by the refinements. The inconsistencies between studies is strong evidence to suppose that they may be strongly dependent upon either cation composition or oxygen content. The wide compositional range of the Bi-2212 phase, as evidenced by the values in Table 3.1, would certainly support this. The motivation, then, for the experimental work of this chapter was to study in detail the diffuse scattering that is associated with each fundamental reflection, and by looking at a variety of crystals to ascertain which features are fundamental to the structure of the Bi-2212 phase, and which can be attributed to sample dependent factors.