Phonon anomalies and lattice dynamics in the superconducting oxychlorides Ca${}_{2-x}$CuO${}_2$Cl${}_2$

We present a comprehensive study of the phonon dispersion in an underdoped, superconducting Ca${}_{2-x}$CuO${}_2$Cl${}_2$ crystal. We interpret the results using lattice dynamical calculations based on a shell model, and we compare the results to other hole-doped cuprates, in particular, to the ones isomorphic to La${}_{2-x}$Sr${}_x$CuO${}_4$ (LSCO). We found that an anomalous dip in the Cu-O bond stretching dispersion develops in oxychlorides with a simultaneous marked broadening of the mode. The broadening is maximum at $\approx$$[\pi /(2a\left)00\right]$, which corresponds to the charge-modulation propagation vector. Our analysis also suggests that screening effects in calculations may cause an apparent cosine-shaped bending of the Cu-O bond-stretching dispersion along both the ($q$ 0 0) and ($q$ $q$ 0) directions, which is not observed on the data close to optimal doping. This observation suggests that the discrepancy between experimental data and ab initio calculations on this mode originates from an overestimation of the doping effects on the mode.

Figure 1

Drawing[21] of the unit cell of Ca${}_2$CuO${}_2$Cl${}_2$.[22] The square coordination of copper with its four nearest-neighbor oxygen ions in the CuO${}_2$ planes is shown. The chlorine ions are located in the apical site above the copper.

Figure 2

Chart[25] of the magnetization vs temperature across the superconducting transition at 100 Oe for the sample 2b measured by IXS. (Inset) Front and side views of the crystal, as measured in diffraction and IXS, in a grease film, with the glass fiber holder.

Figure 3

Plot[31] of the measured IXS intensities as a function of energy on Ca${}_{2-x}$CuO${}_2$Cl${}_2$ for the propagation vector corresponding to the middle of the Brillouin zone along three different symmetry in-plane lines. Left column shows the overall data, while the right column shows the respective zoom at the high energy of the Cu-O bond stretching mode. (Top) Longitudinal at (0.25 0.25 0), with ${\Sigma }_1$ symmetry. (Middle) Longitudinal at (0.25 0 0), with ${\Delta }_1$ symmetry. (Bottom) Transverse at (0 0.25 0), with ${\Delta }_3$ symmetry. Red lines are fit to the data described in the text. The contribution from the Cu-O bond stretching mode is singled out as a black line.

Figure 4

Drawing[21] of the atomic displacements of the in-plane modes that we have measured at the zone center $\Gamma$, as calculated by our lattice dynamical model described in the text. For each mode, we give its character, and the calculated frequency at $\Gamma$. We highlight the octahedron formed by the anions, with the two apical chlorines and the four square coordinated oxygens, around the copper, in order to single out the differences and similarities with analogue displacements in hole-doped cuprates with T structure as, e.g., shown in Ref. 24. The calcium cations lies outside the octahedron, on the vertical of the apical chlorine one.

Figure 5

Chart[31] of the Ca${}_{2-x}$CuO${}_2$Cl${}_2$ phonon dispersion (blue circles) measured by IXS and calculated frequencies using the shell model of lattice dynamics described in the text (red line). We also indicate, on the left panel, the frequencies at the zone center $\Gamma$, extrapolated from their positions in the infrared spectra reported in Ref. 42 (empty triangle-right).

Figure 6

Calculated (lines) and experimental (symbols) inelastic scattering intensities for phonon modes in Ca${}_{2-x}$CuO${}_2$Cl${}_2$ CuO, with character ${\Delta }_1$ (left), ${\Delta }_3$ (center), and ${\Sigma }_1$ (right).[31] The intensity units are arbitrary for calculations. The experimentally reported areas of peak units are in $I/I_0\times$ meV and are multiplied by a common factor for all modes for a given line in the Brillouin zone. Mode assignment is based on the closest experimental frequency to a given calculated mode. The only exception is the acoustic and(or) Eu(4) mode at $q=(3,0.5,0)$, for which both assignments are proposed (and match) the calculated one. Labels correspond to the modes character at the zone center as explained in the text.

Figure 7

(Top) Ca${}_{2-x}$CuO${}_2$Cl${}_2$ Cu-O bond-stretching dispersions (blue circles) measured by IXS and calculated frequencies using a shell model of lattice dynamics; (top, left) along ($q$ 0 0) (${\Delta }_1$) and (top, right) along ($q$ $q$ 0) (${\Sigma }_1$). Black lines corresponds to a model with lower doping and a screening wave number ${\kappa }_s=\frac{\pi }{2a}$, while red lines corresponds to a model with higher doping, and a screening wave number ${\kappa }_s=\frac{\pi }{a}$. The blue dashed line is a cosine function compared to the data of ${\Delta }_1$ symmetry in order to show the marked anomaly at $q\sim$ 0.75. Bottom panel: Blue full circles are the FWHM of the IXS data corresponding to the Cu-O bond-stretching mode along ($q$ 0 0), showing a maximum around $q\sim$ 0.75.[31]