Charge-screening role of $c$-axis atomic displacements in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ and related superconductors

The importance of charge reservoir layers for supplying holes to the ${\mathrm{CuO}}_2$ planes of cuprate superconductors has long been recognized. Less attention has been paid to the screening of the charge transfer by the intervening ionic layers. We address this issue in the case of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, where CuO chains supply the holes for the planes. We present a simple dielectric-screening model that gives a linear correlation between the relative displacements of ions along the $c$ axis, determined by neutron powder diffraction, and the hole density of the planes. Applying this model to the temperature-dependent shifts of ions along the $c$ axis, we infer a charge transfer of 5–10% of the hole density from the planes to the chains on warming from the superconducting transition to room temperature. Given the significant coupling of $c$-axis displacements to the average charge density, we point out the relevance of local displacements for screening charge modulations and note recent evidence for dynamic screening of in-plane quasiparticles. This line of argument leads us to a simple model for atomic displacements and charge modulation that is consistent with images from scanning-tunneling microscopy for underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$.

Figure 1

Normalized bulk susceptibility measured on warming in a 1 Oe field after zero-field cooling. Vertical bars at the top indicate the superconducting transitions.

Figure 2

Typical Rietveld refinement of the $Pmmm$ structural model (line) to the $x=0.56,T_c=58$ K sample data (circles) at 10 K. Vertical bars indicate calculated peak positions; the line at the bottom indicates the difference between data and model. Goodness of fit parameters are ${\chi }^2=3.59,R_{\mathrm{wp}}=4.99\%$, and Bragg factor $R=6.86\%$.

Figure 3

Superconducting transition temperature vs oxygen content in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$: data from Liang et al. [19] (crosses), Cava et al. [14] (squares), and present work (circles).

Figure 4

Lattice parameters in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$: (a) $c$ and (b) $a$ and $b$. Data from Cava et al. [14] at $T=5$ K (squares) and present work at 10 K (circles).

Figure 5

(a) Diagram of the orthorhombic unit cell of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, with the distinct sites labeled. Note that there are mirror planes at the Y and the chain layers. We choose to place the origin of $z$, the relative coordinate along $c$, at the chain layer. (b) Schematic of the unit cell projected along [100], labeled by nominal ionic valences. Long arrows indicate the transfer of doped holes from the Cu-O chains to the ${\mathrm{CuO}}_2$ planes; short arrows indicate the compensating atomic displacements. The dashed line outlines the unit cell.

Figure 6

Plot of $p$ vs $x$ in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ from Liang et al. [19] (crosses) and calculated, as discussed in the text, from structural parameters determined by Cava et al. [14] (squares) and in the present work (circles).

Figure 7

Relative temperature dependence of the $z$ parameters for the ${\mathrm{Cu}}_{\mathrm{pl}}$ (squares), Ba (diamonds), ${\mathrm{O}}_{\mathrm{pl}}$ (triangles), and ${\mathrm{O}}_{\mathrm{ap}}$ (circles) sites determined by Rietveld refinement for the $x=0.56$ sample. Uncertainties are comparable to the fluctuations.

Figure 8

Temperature dependence of the normalized change in the lattice parameters $a,b$, and $c$. Lines through the data points are fits with Eqs. (5) and (6), where the respective values of the parameter $T_0$ are indicated by the vertical bars.

Figure 9

(a) Orthorhombic strain vs temperature for the $x=0.56,T_c=58$ K sample. The line is calculated from the fits to $a\left(T\right)$ and $b\left(T\right)$, shown in Fig. 8. (b) Temperature dependence of the strain relative to the strain at 10 K for all four samples. The inset shows the low-temperature strain values as a function of $x$, providing an implicit symbol legend.

Figure 10

(a) Upper ${\mathrm{CuO}}_2$ plane of a bilayer in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$, with O sites (brown) dimpled towards the Ca layer below, and Cu atoms (gray) above. (b) Same layer after application of $B_{1g}$ symmetry displacements of the O sites, with O atoms shaded according to height along the $c$ axis, with dark being furthest from the Cu layer. (c) Modulation of the $B_{1g}$ displacements at $\mathbf{Q}=(0.25,0)$. (d) Resulting pattern if we associate hole density with the O sites with the shortest bond lengths in (c). Here, Cu sites $(+)$ are also set to the same color as the hole-rich O sites, for comparison with STM images in [32].