Charge density wave (CDW) correlations are prevalent in all copper-oxide superconductors. While CDWs in conventional metals are driven by coupling between lattice vibrations and electrons, the role of the electron-phonon coupling (EPC) in cuprate CDWs is strongly debated. Using Cu $L_3$ edge resonant inelastic x-ray scattering, we study the CDW and Cu-O bond-stretching phonons in the stripe-ordered cuprate ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_x{\mathrm{CuO}}_{4+\delta }$. We investigate the interplay between charge order and EPC as a function of doping and temperature and find that the EPC is enhanced in a narrow momentum region around the CDW ordering vector. By detuning the incident photon energy from the absorption resonance, we extract an EPC matrix element at the CDW ordering vector of $M\simeq 0.36\mathrm{eV}$, which decreases to $M\simeq 0.30\mathrm{eV}$ at high temperature in the absence of the CDW. Our results suggest a feedback mechanism in which the CDW enhances the EPC which, in turn, further stabilizes the CDW.

• Received 11 October 2019
• Revised 4 March 2020
• Accepted 30 July 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.097002