Oxygen phonon branches in overdoped ${\mathrm{La}}_{1.7}{\mathrm{Sr}}_{0.3}{\mathrm{Cu}}_3{\mathrm{O}}_4$

The dispersion of $\mathrm{Cu}–\mathrm{O}$ bond-stretching vibrations in overdoped ${\mathrm{La}}_{1.7}{\mathrm{Sr}}_{0.3}{\mathrm{CuO}}_4$ (not superconducting) has been studied by high-resolution inelastic neutron scattering. It was found that the doping-induced renormalization of the so-called breathing and half-breathing modes is larger than in optimally doped ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{CuO}}_4$. On the other hand, the phonon linewidths are generally smaller in the overdoped sample. Features observed in optimally doped ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{CuO}}_4$ which suggest a tendency towards charge stripe formation are absent in overdoped ${\mathrm{La}}_{1.7}{\mathrm{Sr}}_{0.3}{\mathrm{CuO}}_4$.

Figure 1

(Color online) Energy scans taken along the line $Q=(5-q,0,0)$ at $T=10\mathrm{K}$. Successive scans were offset by 50 counts for the sake of clarity. Lines depict fit curves. The peaks are associated with longitudinal $\mathrm{Cu}\text{−}\mathrm{O}$ bond-stretching vibrations, except for the double peak at $\mathbf{q}=0.5$ below $60\mathrm{meV}$ which is related to $\mathrm{Cu}\text{−}\mathrm{O}$ bond-bending vibrations.

Figure 2

(Color online) Upper panels: dispersion of the longitudinal high-energy phonon branches in the (100) and (110) directions, respectively, for various doping levels in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. Lines are a guide to the eye. Data for $x=0$ and 0.1 were taken from Ref. 1. Data for $x=0.15$ were taken from Ref. 6. Lower panels: the data points show the linewidths (full width at half maximum) of the phonon peaks and the lines show the experimental resolution. The wavyness of the lines is due to focusing effects. Recent data (Ref. 7) for $x=0.15$ measured with an improved $q$ resolution in the transverse direction are included as open symbols.

Figure 3

(Color online) Dispersion of the transverse high-energy phonons in the (110) direction for various doping levels in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. Lines are a guide to the eye. Data for $x<0.3$ were taken from Ref. 1. The displacement pattern of the zone boundary mode is shown in Fig. 4 (middle).

Figure 4

Displacement patterns of zone-boundary bond-stretching modes. Top: longitudinal mode in the (110) direction (breathing mode). Middle: transverse mode in the (110) direction quadrupolar mode). Bottom: longitudinal mode in the (100) direction (half-breathing mode). Circles and solid points denote oxygen atoms and copper atoms, respectively. Only the displacements in the $\mathrm{Cu}\text{−}\mathrm{O}$ planes are shown. All other displacements are small for these modes.

Figure 5

(Color online) Energy scans taken at the zone boundary in the (110) direction at $T=10\mathrm{K}$. The peak in (a) corresponds to the longitudinal and in (b) to the transverse $\mathrm{Cu}\text{−}\mathrm{O}$ bond-stretching mode. The two modes are of breathing or quadrupolar character, respectively. The width of the quadrupolar mode is resolution limited.

Figure 6

(Color online) Energy scan taken at $Q=(0,0,15)$ at $T=10\mathrm{K}$. The intensity distribution was fitted with two Gaussians corresponding to the contributions of the $O_z^z$ mode (around $48\mathrm{meV}$) and another mode of $A_g$ type (around $56\mathrm{meV}$). The energy resolution was about $4\mathrm{meV}$.

Figure 7

Top: experimentally observed frequency renormalization in LSCO of the half-breathing mode $(\pi ,0)$ and of the breathing mode $(\pi ,\pi )$, respectively (Ref. 21). Data for $x<0.30$ were taken from Refs. 1, 5, 6. Bottom: predicted doping dependence of the frequencies of the half-breathing mode and of the breathing mode, respectively, after Horsch and Khaliullin (Ref. 20). The triangles denote experimental data known at the time of the report.

Figure 8

(Color online) Comparison of energy scans taken at $Q=(5-0.3,0,0)$ on optimally doped (Ref. 7) and on overdoped LSCO at $T=10\mathrm{K}$. The data were corrected for background scattering and normalized to yield the same integrated intensity. Both scans were taken with the same experimental set up. Note that the width observed on the optimally doped sample is somewhat larger than that reported in Ref. 6 because of a better momentum resolution in the later experiment.