X-ray diffuse scattering experiments from bismuth-based high-$T_c$ superconductors

A detailed study, by x-ray diffuse scattering, of the recently found two-dimensional (2D) displacive short-range-order (2DSRO) superstructure, with doubled periodicity along the orthorhombic ${\mathbf{a}}_o$ direction and perpendicular to the known long-range structural modulation, from the high-$T_c$ superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ (Bi-2212) is reported. The investigation has been extended to high and low temperatures for optimally doped crystals, to crystals with different doping levels, and to the one layer compound ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{6+\delta }$ (Bi-2201). The results show that the 2DSRO is present at room temperature, for all studied crystals and with the same commensurate $2{\mathbf{a}}_o$ periodicity; significant differences in intensity and in the extent of the 2DSRO are however observed. The most striking feature is that both, the intensity of the diffuse scattering and the extent of the 2DSRO goes through a maximum for the optimal doped crystals and decreases for overdoped and underdoped samples, they are also smaller for the one layer Bi-2201 which has a lower $T_c$. The reversible temperature dependence reveals that the diffuse scattering is unchanged between $35\mathrm{K}$ and $300\mathrm{K}$, but starts washing out for higher temperatures and vanishes around $450\mathrm{K}$, temperature above which another scattering, one dimensional in character, is found. This one-dimensional (1D) short-range order (1DSRO) corresponds to linear correlated displacements along the pseudotetragonal directions of the Cu-O-Cu chains. These findings tend to show that these short-range ordering features may be of importance for a better understanding of high-$T_c$ materials, at least those from the bismuth-based family.

Figure 1

X-ray diffuse scattering pattern of two Bi-2201 crystals measured with the fixed film-fixed crystal technique. The incident x-ray beam $(\lambda =1.45\mathrm{\AA })$ and the ${\mathbf{c}}_o^{*}$ axis are parallel to each other. Traces of reciprocal planes with $h$ index constant are indicated as dotted lines. The two types of Bi-2201 samples (a) and (b) differ by the relative orientation of the different domains. In panel (c) a typical diffractogram for the optimally doped Bi-2212 compound is displayed for comparison.

Figure 2

Precession diffractogram for a Bi-2201 crystal with slightly misoriented domains. Solid lines represent the reciprocal planes with $h$ index constant.

Figure 3

Profiles along the ${\mathbf{b}}_o^{*}$ direction taken from the precession diffractogram of the Bi-2201 sample. Data after background substraction are displayed as open circles and diamonds for the horizontal and most intense tilted domains, respectively. The best obtained fits are plotted as solid lines.

Figure 4

Experimental x-ray diffractograms measured in the low temperature regime: (a) at $T=35\mathrm{K}$ and (b) at room temperature. In both cases we clearly see the diffuse structures characteristic of the 2DSRO with the same intensity (see text for details).

Figure 5

Experimental x-ray diffractograms of the optimally doped Bi-2212 samples measured in the two different temperature regimes: (a) at room temperature, showing the presence of the $2{\mathbf{a}}_o$ diffuse structure, and (b) at $T=651\mathrm{K}$ where only the 1D structure can be appreciated (indicated as dashed black lines). The solid white lines in both figures correspond to the profiles taken for the quantitative analysis.

Figure 6

(a) Profiles of the diffuse structure taken along the $(1∕2klm)$ line measured in the temperature range between 297 and $463\mathrm{K}$ together with the solid line obtained for the best fit (solid lines). (b) Example of 1D diffuse structure at $651\mathrm{K}$: raw data and extracted profile. (c) Raw profiles as a function of the temperature for the 1D diffuse structure taken along the white line in Fig. 5b.

Figure 7

(a) Domain size and chain length obtained from the linewidth of the diffuse 2D and 1D diffuse structures. (b) Integrated intensity as a function of temperature. The lines have been drawn as a guide to the eye.

Figure 8

Detail of the precession diffractogram measured for three of the Bi-2212 samples: (a) underdoped $80\mathrm{K}$ (U80K). (b) Optimal doping (OD) and (c) overdoped $83\mathrm{K}$ (O83K). We observe the presence of diffuse spots at the $h=\pm 1∕2$ positions.

Figure 9

(a) Profiles of the diffuse structure taken along the $\left(h201\right)$ line (${\mathbf{a}}_o$ direction) and along the $(1∕2k0m)$ (${\mathbf{b}}_0$ direction) as a function of doping. Circles represent the data after background substraction and the solid lines in (b) the results of the best fit obtained with a four Voigt function.

Figure 10

Domain size from the analysis of the linewidth of the diffuse spots displayed in Fig. 2b. The dashed line represent the best fit obtained for the data and the solid one the empirical parabola of the $T_c$ vs doping (see text for details).

Figure 11

(Color online) Allowed displacive modes obtained from the symmetry analysis.