Phonon anomaly in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$*

Electron-phonon coupling (EPC) plays in many exotic phenomena displayed by strongly-correlated electron materials, such as the pairing mechanism in high-$T_{C}$ superconductors and metal-insulator transition in ruthenates. Many studies show a strong renormalization of quasi-particle band structure near Fermi energy associated with the coupling to boson modes $\sim $40-50 meV. However, there is no clear picture of the origin of these modes. With angle-resolved electron energy loss spectroscopy, we have studied the lattice dynamics of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ surface. The data indicate a phonon anomaly of modes $\sim $50 meV, identified as the in-plane Cu-O stretching modes, where the energy shifts (softens) while the spectral linewidth and weight vary along (0,0)-($\pi $,0) direction. The other feature $\sim $80 meV, attributed to apical oxygen vibrations, exhibits distinct dispersion toward ($\pi $,$\pi )$. Such behaviors are observed both above and below the superconducting T$_{C}$. These detailed measurements provide new insights into the nature of EPC in such materials. * Supported by China NSF-10704084, NSF DMR-0346826, NSF and DOE (NSF-DMR-0451163 and DMS{\&}E).