Different temperature-dependent local displacements in the underdoped and overdoped

$\chem{Cu}$ K-edge extended X-ray absorption fine-structure (EXAFS) measurements have been used to study the temperature-dependent local structure of the underdoped and overdoped superconducting $\chem{La}$$_{2-x}$$\chem{Sr}$x$\chem{CuO_4}$ system. The Debye-Waller factor of the in-plane $\chem{Cu}$-$\chem{O}$ bonds has been used as an order parameter to measure the local displacements, revealing an anomalous behavior for the underdoped case while the overdoped system shows a negligible change as a function of temperature. The results provide an evidence for a close relationship between local $\chem{Cu}$-$\chem{O}$ displacements, electron-lattice interaction and charge inhomogeneity in the copper oxide superconductors