  We investigate the dynamical effects of electron-phonon coupling (EPC) on the
superconducting properties of two-dimensional (2D) systems, calculating the
Eliashberg function in terms of dynamically renormalized phonons. By studying
different approximations for the phonon self-energy, we identify the important
role of charge fluctuations in shaping the superconductivity properties, not
only through the renormalization of phonon frequencies and damping rates but
also through structural changes in the phonon spectral function. With the
dynamical effects treated consistently, we argue that a part of the phonon
spectral weight necessarily shifts to low frequencies due to the coupling to
the 2D gapless plasmon. Furthermore, we find that the EPC leads to excess
phonon spectral weight as well - i.e., phonon production - which generally
tends to enhance the transition temperature. Our calculations point out that
the influence of phonon production becomes greater as the density and the
effective mass of electrons increase.
