Odd-parity superconductivity from phonon-mediated pairing: Application to Cu$_x$Bi$_2$Se$_3$

Motivated by the proposed topological state in Cu$_x$Bi$_2$Se$_3$, we study the possibility of phonon-mediated odd-parity superconductivity in spin-orbit coupled systems with time-reversal and inversion symmetry. For such systems, we show that, in general, pure electron-phonon coupling can never lead to a triplet state with a higher critical temperature than the leading singlet state. The Coulomb pseudopotential, which is the repulsive part of the electron-electron interaction and is typically small in weakly correlated systems, is therefore critical to stabilizing the triplet state. We introduce a chirality quantum number, which identifies the electron-phonon vertex interactions that are most favorable to the triplet channel as those that conserve chirality. Applying these results to Cu$_x$Bi$_2$Se$_3$, we find that a phonon-mediated odd-parity state may be realized in the presence of weak electronic correlations if the chirality-preserving electron-phonon vertices are much stronger than the chirality-flipping vertices.