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The experimental observation of superconductivity in doped semimetals andsemiconductors, where the Fermi energy is comparable to or smaller than thecharacteristic phonon frequencies, is not captured by the conventional theory.In this paper, we propose a mechanism for superconductivity in ultralow-densitythree-dimensional Dirac materials based on the proximity to a ferroelectricquantum critical point. We derive a low-energy theory that takes into accountboth the strong Coulomb interaction and the direct coupling between theelectrons and the soft phonon modes. We show that the Coulomb repulsion isstrongly screened by the lattice polarization near the critical point even inthe case of vanishing carrier density. Using a renormalization group analysis,we demonstrate that the effective electron-electron interaction is dominantlymediated by the transverse phonon mode. We find that the system genericallyflows towards strong electron-phonon coupling. Hence, we propose a newmechanism to simultaneously produce an attractive interaction and suppressstrong Coulomb repulsion, which does not require retardation. For comparison,we perform same analysis for covalent crystals, where lattice polarization isnegligible. We obtain qualitatively similar results, though the screening ofthe Coulomb repulsion is much weaker. We then apply our results to studysuperconductivity in the low-density limit. We find strong enhancement of thetransition temperature upon approaching the quantum critical point. Finally, wealso discuss scenarios to realize a topological $p$-wave superconducting statein covalent crystals close to the critical point.

Superconductivity near a ferroelectric quantum critical point in ultralow-density Dirac materials