Temperature dependence of the bond-stretching phonon anomaly in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6.95}$

We report the results of a detailed inelastic neutron-scattering study of renormalization of the bond-stretching phonons in optimally doped ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ (YBCO). In agreement with previous work, this renormalization is strongest halfway to the zone boundary in the [010] direction where it manifests itself as a transfer of phonon spectral weight from $\sim 60$ to $\sim 50\text{meV}$. Phonon renormalization begins on cooling from well above the superconducting transition temperature, $T_c$, but strongly accelerates at $T_c$. In contrast with the ${\text{La}}_{2-x}{\text{Sr}}_x{\text{CuO}}_4$ system, where a similar longitudinal phonon anomaly occurs, the anomaly in YBCO appears to extend in the transverse direction along the branch that connects the longitudinal mode at the in-plane wave vector ${\mathbf{q}}_{\text{in}}=\left(0,0.25\right)$ and the transverse bond-stretching mode at ${\mathbf{q}}_{\text{in}}=\left(0.25,0.25\right)$. Therefore, phonon renormalization in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6.95}$ seems to be consistent with a quasi-one-dimensional behavior.

Figure 1

(Color online) (a) Schematic of the bond-stretching (and bending) vibrations along $a$ and $b$ directions. Small green/Large red circles represent Cu/O, respectively. Eigenvectors of the zone center (b) ${\Delta }_4$ and (c) ${\Delta }_1$ bond-stretching phonons.

Figure 2

(Color online) Dispersion relation of the out-of-phase Cu–O bond-stretching vibrations in optimally doped YBCO as seen on a twinned sample. The energies are plotted against the nominal wave vector calculated from an average lattice constant $a_{\text{ave}}=\left(a+b\right)/2$. The actual wave vectors for the scattering from each of the two twin domains differ by $2\delta =2\pi /a-2\pi /b$. The arrows show the zone center for phonons propagating along the $a^{\ast }$ or the $b^{\ast }$ direction. The ellipsoid depicts the instrumental resolution.

Figure 3

(Color online) Comparison of phonon spectra measured at $\mathbf{q}=\left(0,0.25,0\right)$ in the defocused condition and $T=10$ and 300 K for (a) ${\Delta }_1$ and (b) ${\Delta }_4$. Intensities have been divided by the Bose factor.

Figure 4

(Color online) Energy scans taken at different temperatures. [(a)–(c)] Lines correspond to fits with 1 or 2 G. The 100 K data points have been omitted for clarity from (b). The double peak in (a) and (c) corresponds to the $a^{\ast }$ mode (the higher energy peak) and the $b^{\ast }$ mode (the lower energy peak). These two peaks merge to a single one in (b). (d) Data were taken with the final energy $E_f=13.4\text{meV}$ for $\hslash \omega \ge 46\text{meV}$ and with $E_f=12.5\text{meV}$ $43\le \hslash \omega \le 48\text{meV}$. The 12.5 meV data were corrected for the different resolution volume by multiplying by ${\left(13.4/12.5\right)}^2$. The resulting intensities were averaged in the overlapping energy range (46.5–48 meV). The 200 K data were divided by the Bose factor and 23 counts were subtracted to correct for the temperature dependence of the background. Blue/red arrows (on the right/left side) indicate intensity gain/loss. (e) Integrated intensity loss from 200 to 10 K integrated around the 60 meV peak in the data shown in (d).

Figure 5

(Color online) Temperature dependence of phonon-scattering intensity. (a) Difference between the intensity at 57 meV and the average of intensities at 53 and 49 meV at $\mathbf{Q}=\left(0,3.25,0\right)$. (b) Backround-subtracted intensity at $\mathbf{Q}=\left(0,2.75,-2\right)$ at 60 meV. Open and solid circles represent different datasets (see text).