In-plane copper-oxygen bond-stretching mode anomaly in underdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_{4+\delta }$ measured with high-resolution inelastic x-ray scattering

We have measured the lattice dynamics of underdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_{4+\delta }$ $(x=0.08)$ using high resolution inelastic x-ray scattering spectroscopy. The very good intrinsic $\mathbf{Q}$ resolution of this experiment allows us to differentiate between the two proposed scenarios for the Cu-O bond stretching phonon broadening reported at $\mathbf{q}=(0.25,0,0)$. The results suggest that this phonon broadening is due to intrinsic damping rather than to a steep dispersion, having important implications on the interpretation of phonon anomalies in cuprates.

Figure 1

Representative energy scans below $20\mathrm{K}$. Error bars represent $\pm 1$ s.d. (a), (b) Inelastic x-ray scattering (IXS) energy scans taken on ${\mathrm{La}}_{1.92}{\mathrm{Sr}}_{0.08}\mathrm{Cu}{\mathrm{O}}_{4+\delta }$ (LSCO) at $T=16\mathrm{K}$, with $\Delta E=5.5\pm 0.2$ (a) and $\Delta E=3.0\pm 0.2\mathrm{meV}$ (b). (c) Inelastic neutron scattering (INS) energy scans taken on ${\mathrm{La}}_{1.875}{\mathrm{Ba}}_{0.125}\mathrm{Cu}{\mathrm{O}}_4$ (LBCO) at $T=10\mathrm{K}$ from Ref. 7. (d) Comparison of the BS mode fit of panel (a) and (b). (e) Comparison of the BS mode fit in LSCO $(\Delta E=3.0\mathrm{meV})$ and the BS mode fit in LBCO from Ref. 7.

Figure 2

Left: dispersion of the phonon modes with polarization vector $\widehat{\mathbf{e}}\parallel [1,0,0]$. Dispersion along the $(1-\xi 0,0)$ line has ${\Delta }_1$ longitudinal symmetry and along the $(0,1-\nu 0)$ line has ${\Delta }_3$ transverse symmetry. Calculations (lines) are compared to the experimental results, as extracted from IXS spectra fit (circles). Gray and dashed filled correspond to two different experimental runs. Dashed bars with risers correspond to the fitted FWHM. Top $x$ scale for ${\Delta }_1$ symmetry corresponds to $\mathbf{Q}=(4-\xi ,0,0)$. Right: calculated IXS cross section (lines) compared to the intensities extracted from IXS spectra fit for the longitudinal acoustic (LA), and all optic modes but the 4th one, which is extinguished in this Brillouin zone. All fitted intensities are corrected by the Bose factor.