Anomalous Dispersion of Longitudinal Optical Phonons in $\mathbf{N}{\mathbf{d}}_{\mathbf{1.86}}\mathbf{C}{\mathbf{e}}_{\mathbf{0.14}}\mathbf{C}\mathbf{u}{\mathbf{O}}_{\mathbf{4}+\mathit{\delta }}$ Determined by Inelastic X-ray Scattering

The phonon dispersions of $\mathrm{N}{\mathrm{d}}_{1.86}\mathrm{C}{\mathrm{e}}_{0.14}\mathrm{C}\mathrm{u}{\mathrm{O}}_{4+\delta }$ along the $[\xi ,0,0]$ direction have been determined by inelastic x-ray scattering. Compared to the undoped parent compound, the two highest longitudinal phonon branches, associated with the Cu-O bond-stretching and out-of-plane oxygen vibration, are shifted to lower energies. Moreover, an anomalous softening of the bond-stretching band is observed around $\mathbf{q}=(0.2,0,0)$. These signatures provide evidence for strong electron-phonon coupling in this electron-doped high-temperature superconductor.