Doping dependence of softening in the bond-stretching phonon mode of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4\left(0⩽x⩽0.29\right)$

The doping dependence of the softening in $\mathrm{Cu}\text{−}\mathrm{O}$ bond-stretching optical phonons along [100] direction has been systematically studied on ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ using inelastic x-ray scattering. The softening increases with $x$ in the underdoped region and starts to saturate or makes local maximum around the optimum doping $\left(x\sim 0.15\right)$. This behavior is discussed in terms of the coupling between the phonons and the low energy charge fluctuations on the doped ${\mathrm{CuO}}_2$ planes.

Figure 1

Inelastic x-ray scattering spectrum of ${\mathrm{La}}_2{\mathrm{CuO}}_4$ at $Q=\left(3.10,0,0\right)\left[\zeta =0.90\right]$ and $T\simeq 20\mathrm{K}$ (filled circles). Solid lines are fits. The elastic response (resolution function) is also plotted with a dashed line. Inset: Polarization patterns of the highest energy LO bond-stretching phonon modes $\left(\simeq 80\mathrm{meV}\right)$ within a ${\mathrm{CuO}}_2$ plane for $\zeta =1.0$ and 0.5. The filled and open circles indicate the $\mathrm{Cu}$ and O atomic positions, and the arrows show the displacements of O atoms.

Figure 2

Inelastic x-ray scattering spectra of the bond-stretching phonon branches in (a) ${\mathrm{La}}_{1.96}{\mathrm{Sr}}_{0.04}{\mathrm{CuO}}_4$ at $T\simeq 20\mathrm{K}$, (b) ${\mathrm{La}}_{1.88}{\mathrm{Sr}}_{0.12}{\mathrm{CuO}}_4$ at $T\simeq 13\mathrm{K}$, and (c) ${\mathrm{La}}_{1.71}{\mathrm{Sr}}_{0.29}{\mathrm{CuO}}_4$ at $T\simeq 26\mathrm{K}$ near the Brillouin zone boundary $\left[Q=(3,0,0)\right]$ along [100] crystal axis direction. Scans having different $Q$ are shifted vertically.

Figure 3

(a) The dispersion relation of the highest energy longitudinal optic branches along [100] direction for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. Closed triangles are neutron data of the $x=0.15$ (optimally doped) sample reported by Pintschovius et al. (Ref. 5). Lines are fitted cosine curves. (b) The linewidth (FWHM) as a function of $q$ for some selected $x$ samples. Lines are a guide to the eye. The different $x$ data are shifted vertically by the values in the parentheses for clarity in both (a) and (b).

Figure 4

$\mathrm{Sr}$-doping $\left(x\right)$ dependence of the amplitude of softening in the $\mathrm{Cu}\text{−}\mathrm{O}$ bond-stretching vibration along [100] direction and the superconducting transition temperature in the ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ system. Closed and open circles are the present x-ray data and the neutron data reported by Pintschovius et al. (Refs. 3, 5, 18) respectively. Open squares are the calculation using a $t\text{−}J$ model by Rösch et al. (Ref. 19) Solid line is a guide to the eye, and the dashed line shows an expected linear dependence on $x$ when there is no enhancement by the charge collective mode from the calculations by Khaliullin et al. (Ref. 22) (see text).