The experimental observation of superconductivity in doped semimetals and
semiconductors, where the Fermi energy is comparable to or smaller than the
characteristic phonon frequencies, is not captured by the conventional theory.
In this paper, we propose a mechanism for superconductivity in ultralow-density
three-dimensional Dirac materials based on the proximity to a ferroelectric
quantum critical point. We derive a low-energy theory that takes into account
both the strong Coulomb interaction and the direct coupling between the
electrons and the soft phonon modes. We show that the Coulomb repulsion is
strongly screened by the lattice polarization near the critical point even in
the case of vanishing carrier density. Using a renormalization group analysis,
we demonstrate that the effective electron-electron interaction is dominantly
mediated by the transverse phonon mode. We find that the system generically
flows towards strong electron-phonon coupling. Hence, we propose a new
mechanism to simultaneously produce an attractive interaction and suppress
strong Coulomb repulsion, which does not require retardation. For comparison,
we perform same analysis for covalent crystals, where lattice polarization is
negligible. We obtain qualitatively similar results, though the screening of
the Coulomb repulsion is much weaker. We then apply our results to study
superconductivity in the low-density limit. We find strong enhancement of the
transition temperature upon approaching the quantum critical point. Finally, we
also discuss scenarios to realize a topological p-wave superconducting state
in covalent crystals close to the critical point
