Sharp optical-phonon softening near optimal doping in ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_{4+\delta }$ observed via inelastic x-ray scattering

We report a direct observation of a sharp Kohn-type anomaly in the doubly degenerate copper-oxygen bond-stretching phonon mode occurring at $\mathbf{q}=\left(0.3,0,0\right)$ in ${\text{La}}_{2-x}{\text{Ba}}_x{\text{CuO}}_{4+\delta }$, with $x=0.14\pm 0.01$, thanks to the high $\mathbf{Q}$ resolution of inelastic x-ray scattering. This anomaly is clearly seen when the inelastic signal is analyzed using a single mode but is also consistent with a two-mode hypothesis, possibly due to a splitting of the degenerate modes due to symmetry-breaking stripes. Our observation shows that the effect persists at the stripe-propagation vector in a superconducting system close to optimal doping.

Figure 1

Inelastic x-ray scattering example spectrum of ${\text{La}}_{1.86}{\text{Ba}}_{0.14}{\text{CuO}}_{4+\delta }$ at $\mathbf{Q}=\left(3.11,0.04,0\right)$. At zero energy one observes the elastic line, while at positive energy, several phonon modes are observed corresponding to Stokes processes. In the inset we zoom on to the high-energy optical Cu-O bond-stretching mode. Anti-Stokes modes are not visible because of the unfavorable detailed balance at low temperature.

Figure 2

(Color online) Inelastic x-ray scattering example spectra from ${\text{La}}_{1.86}{\text{Ba}}_{0.14}{\text{CuO}}_{4+\delta }$ at different wave vectors $\mathbf{Q}=\left(3+q_x,q_y,0\right)$ (see text). In the left column the data are fitted using the degenerate Cu-O bond-stretching mode hypothesis. In the right column the data are fitted supposing two Cu-O bond-stretching mode.

Figure 3

(Color online) Left panel: ${\text{La}}_{1.86}{\text{Ba}}_{0.14}{\text{CuO}}_{4+\delta }$ Cu-O BS dispersion. Blue squares represent frequencies from the model fit with one mode, while red hollow circles represent a fit with two modes. The line corresponds to a shell-model lattice dynamics calculation. Right panel: ${\text{La}}_{1.86}{\text{Ba}}_{0.14}{\text{CuO}}_{4+\delta }$ Cu-O BS width, symbols as above.