Impact of Electron-phonon Coupling on Superconductivity of Strontium Titanate

Recent experimental work has led to new effort to understand the mechanism behind this very dilute superconductor, and highlights the potential role of ferroelectricity. As an incipient ferroelectric, STO shows a softening of a zone center optical phonon accompanied by a huge dielectric constant. This transverse optical phonon does not couple to the electrons, but the associated longitudinal optical phonon can provide an effective attraction. However, the energy of the longitudinal phonon considerably exceeds the Fermi energy, challenging the screened pseudopotential mechanism by which the electron-phonon mediated attraction can overcome Coulomb repulsion. In this work we consider an effective electron-electron interaction that includes the bare electronic Coulomb repulsion and an optical phonon mediated interaction, renormalized by electronic screening in the random phase approximation. We solve numerically the superconducting gap equation and explore how the dynamics of the optical phonons and the Fermi energy size of the conduction electrons affect the instability in the superconducting channel.