We investigate and compare different approaches for handling the dynamical effects of electron-phonon coupling (EPC) on the superconducting properties of two-dimensional (2D) systems in the presence of plasmons. By calculating the Eliashberg function in terms of dynamically renormalized phonons, we discuss the approximations frequently used in the literature and compare them to a fully dynamical treatment of the screening effects. We emphasize the importance of consistent calculations of charge fluctuations that shape the superconducting properties, not only through the renormalization of phonon frequencies and damping rates but also through structural changes in the phonon spectral function. We find that part of the phonon spectral weight necessarily shifts to low frequencies owing to the coupling with the 2D gapless plasmon. Additionally, we find that the EPC leads to excess phonon spectral weight—i.e., phonon production—which generally tends to enhance the transition temperature.

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• Phonons
• Plasmons
• Superconductivity
• Two-dimensional electron system
• Random phase approximation