Doping dependent intrinsic line width of the Cu-O bond-stretching phonon with q=(0.25 0 0) in $La_{2-x}Sr_xCuO_4$

We have recently found that the charge inhomogeneities provide significant broadening in the Cu-O bond stretching phonon of $La_{2-x}Sr_xCuO_4$, and the line shape of the phonon at zone boundary is well reproduced by the simple model which takes charge inhomogeneous effect into account [1]. The question is, now, how large intrinsic line width of the phonon at q=(0.25 0 0), where the giant phonon softening and broadening exist [2], is apart from the charge inhomogeneous effect on the line width. In this talk, we will show the doping dependence of the intrinsic line width of the phonon from x=0.05 to x=0.30. Interestingly, the intrinsic line width as a function of doping follows the superconducting transition temperature. We will discuss relationship between the phonon and the superconductivity in $La_{2-x}Sr_xCuO_4$. \\[4pt] [1] S. R. Park at al., accepted for publication in PRB (2011).\\[0pt] [2] D. Reznik et al., Nature {\bf 440}, 1170 (2006).