  Bose-Fermi systems such as mixtures of electrons with excitons or
exciton-polaritons are extensively discussed as candidates to host a variety of
intriguing phenomena, including polaron formation, drag effects, supersolidity,
and superconductivity. In this work, assuming the strong-coupling regime
between the semiconductor excitons and cavity photons, we develop the many-body
theory approach addressing the interplay of different types of interaction
among various species in such a mixture, wherein we take into account dynamical
density responses of both the Bose-condensed exciton-polaritons and the
two-dimensional electron gas inside an optical microcavity. As was anticipated
previously, at high enough polariton densities the lower hybrid mode of the
system's excitation spectrum acquires a roton minimum, making the system prone
to superconducting pairing in the vicinity of the roton instability. We analyze
the possibility of polariton-BEC-mediated superconductivity in the electron gas
taking into account full momentum and frequency dependence of the gap, as well
as in the Eliashberg approach where the momentum dependence is neglected, and
in the Bardeen-Cooper-Schrieffer approach that discards the frequency
dependence and dynamical effects. Considering the interaction screening in
Thomas-Fermi and in random-phase approximations, we estimate the critical
temperatures of superconductivity to be not larger than 0.1 K in the vicinity
of instability. As possible realizations of the coupled polariton-electron
system, semiconductor quantum wells and two-dimensional transition metal
dichalcogenides are considered.
