Two- and three-dimensional incommensurate modulation in optimally-doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$

X-ray scattering measurements on optimally doped single crystal samples of the high-temperature superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ reveal the presence of three distinct incommensurate charge modulations, each involving a roughly fivefold increase in the unit cell dimension along the $\mathbf{b}$ direction. The strongest scattering comes from the well known $(H,K\pm 0.21,L)$ modulation and its harmonics. However, we also observe broad diffraction which peak up at the $L$ values complementary to those which characterize the known modulated structure. These diffraction features correspond to correlation lengths of roughly a unit cell dimension, ${\xi }_c\sim 20\mathrm{\AA }$ in the $\mathbf{c}$ direction, and of ${\xi }_b\sim 185\mathrm{\AA }$ parallel to the incommensurate wave vector. We interpret these features as arising from three-dimensional incommensurate domains and the interfaces between them, respectively. In addition we investigate the recently discovered incommensurate modulations which peak up at $(1∕2,K\pm 0.21,L)$ and related wave vectors. Here we explicitly study the $L$ dependence of this scattering and see that these charge modulations are two dimensional in nature with weak correlations on the scale of a bilayer thickness, and that they correspond to short-range, isotropic correlation lengths within the basal plane. We relate these new incommensurate modulations to the electronic nanostructure observed in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ using STM topography.

Figure 1

(Color online) A schematic diagram depicting one and a half unit cells of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ is shown. The light (orange) pyramids represent ${\mathrm{CuO}}_2$ plus apical oxygen complexes, while the dark (blue) octahedra represent ${\mathrm{BiO}}_2$ plus apical oxygen atom complexes. The Cu ions reside at the center of the base of the light pyramids, while the Bi ions reside at the center of the octahedra. The ${\mathrm{CuO}}_2$ and ${\mathrm{BiO}}_2$ bilayer thicknesses are indicated as 3.3 and $3.2\mathrm{\AA }$, respectively, while the bilayer-bilayer separation of $c∕2\sim 15.4$ is also shown.

Figure 2

Synchrotron x-ray scattering scans on a logarithmic intensity scale along $[0,K,0]$ (top panel) and $[0,K,1]$ (lower panel) in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. A total of four harmonics of the modulated structure with the form $(0,4-n\times \tau ,L)$ are observed at both $L=0$ and $L=1$, and they clearly alternate between resolution-limited and broad along both $K$ and $L$ (see Fig. 3). The broad component to the scattering is well described by a Lorentzian-squared line shape (see Fig. 4).

Figure 3

Rotating anode x-ray scattering measurements on a logarithmic intensity scale show the $L$ dependence of the principal Bragg peaks, such as $[0,4,L]$ (top panel); the first order harmonic of the modulated structure $[0,3.79,L]$ (middle panel); and the second order harmonic of the primary modulated structure $[0,3.58,L]$ (bottom panel). The harmonics alternate with $L$ between resolution-limited Bragg scattering and broad scattering which is well described by a Lorentzian-squared line shape (see Fig. 4).

Figure 4

X-ray scattering scans along $K$ (top) and $L$ (bottom) taken from Figs. 2, 3 and plotted on a linear intensity scale. Both plots show the second order harmonics of the primary modulation [0, 3.58, 1] and [0, 3.58, 5]. These positions correspond to wave vectors with broad (as opposed to resolution-limited) line shapes. Also shown is the fit to each line shape using a Lorentzian-squared form, Eq. (1), which is clearly excellent. As described in the text, this line shape is typically used to describe the three-dimensional random field domain state in disordered magnets.

Figure 5

X-ray scattering scans of the form $[H,3.58,5]$ which show the new modulated structure corresponding to a doubling of the unit cell along $\mathbf{a}$ is shown. The dashed line in the upper panel shows where the scans reside in reciprocal space, passing through the second order harmonic of the primary modulation. The middle and lower panels show these scans on logarithmic (middle) and linear (bottom) intensity scales, with Lorentzian line shapes [Eq. (2)] fit to the data. The bottom panel also demonstrates the lack of temperature dependence to this scattering below room temperature.

Figure 6

X-ray scattering scans of the form $[1∕2,K,5]$ which display the new modulated structure corresponding to a doubling of the unit cell along $\mathbf{a}$ is shown. The upper panel shows where the scans reside in reciprocal space, passing through the first four harmonics of this new modulated structure. The data in the bottom panel is fit to Lorentzian line shapes [Eq. (2)] centered at each of the four harmonic positions along $\mathbf{K}$. Also shown for comparison in the same panel are $\mathbf{K}$ scans of the first and second harmonics of the primary modulated structure, whose linewidth defines the instrumental resolution.

Figure 7

X-ray scattering scans of the form $[1∕2,K,L]$ which observe the new modulated structure corresponding to a doubling of the unit cell along $\mathbf{a}$ are shown. The top panel shows $L$ scans of the second order $(K=3.58)$ harmonic of this scattering, at different temperatures below $250\mathrm{K}$, as well as a background scan of the form $[0.3,3.45,L]$. The lower panel shows the $L$ dependence of each of the first four harmonics of this structure at $10\mathrm{K}$. Remarkably over the 10 Brillouin zones from $L=10$ to $L=-10$, a near complete rod of scattering is observed for the first order harmonic $(K=3.79)$, while higher order harmonics peak up weakly at $L=\pm 5$.

Figure 8

(Color online) The correlation areas of the new modulated structure corresponding to a doubling of the unit cell along $\mathbf{a}$ extracted from our x-ray scattering analysis are superposed on STM topographs (Ref. 9) of the surface of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. Our measurements indicate isotropic short range correlations with a correlation length of $\sim 27\mathrm{\AA }$, and these correspond very well to the extent of the “S” microstructure which appears to be imprinted on the primary modulated structure of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$, as observed in real space with the STM measurements.