Anomalous Dispersion of Longitudinal Optical Phonons in ${\mathrm{Nd}}_{1.86}{\mathrm{Ce}}_{0.14}{\mathrm{CuO}}_{4+\delta }$ Determined by Inelastic X-Ray Scattering

The phonon dispersions of ${\mathrm{Nd}}_{1.86}{\mathrm{Ce}}_{0.14}{\mathrm{CuO}}_{4+\delta }$ along the $[\xi ,0,0\mathrm{}]$ 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 at about $\mathbf{q}=(0.2,0,0)$. These signatures provide evidence for strong electron-phonon coupling in this electron-doped high-temperature superconductor.