Theory of superconductivity mediated by topological phonons

Topological phononic insulators are the counterpart of three-dimensional quantum spin Hall insulators in phononic systems and, as such, their topological surfaces are characterized by Dirac cone-shaped gapless edge states arising as a consequence of a bulk-boundary correspondence. We propose a theoretical framework for the possible superconducting phase in these materials, where the attractive interaction between electrons is mediated by topological phonons in nontrivial boundary modes. Within the BCS limit, we develop a self -consistent two-band gap equation, whose solutions show that the superconducting critical temperature has a nonmonotonic behavior with respect to the phononic frequency in the Kramers-like point. This remarkable behavior is produced by a resonance that occurs when electrons and phonons on the topological surfaces have the same energy: this effectively increases the electron-phonon interaction and hence the Cooper pair binding energy, thus establishing an optimal condition for the superconducting phase. With this mechanism, the Tc can be increased by well over a factor 2, and the maximum enhancement occurs in the degenerate phononic flat-band limit.