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We report a persistent low-energy phonon broadening around $q_{B} \sim 0.28$r.l.u. along the Cu-O bond direction in the high-$T_c$ cuprateBi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212). We show that such broadeningexists both inside and outside the conventional charge density wave (CDW)phase, via temperature dependent measurements in both underdoped and heavilyoverdoped samples. Combining inelastic hard x-ray scattering, diffusescattering, angle-resolved photoemission spectroscopy, and resonant soft x-rayscattering at the Cu $L_3$-edge, we exclude the presence of a CDW in theheavily overdoped Bi-2212 similar to that observed in the underdoped systems.Finally, we discuss the origin of such anisotropic low-energy phononbroadening, and its potential precursory role to the CDW phase in theunderdoped region.

Persistent low-energy phonon broadening near the charge order $q$-vector in bilayer cuprate Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$