Superconductivity at very low density: The case of strontium titanate

Doped strontium titanate becomes superconducting at a density as low as n=510^170. 16em{0ex}cm^-3, where the Fermi energy is orders of magnitude smaller than the longitudinal-optical-phonon frequencies. In this limit, the only optical mode with a frequency which is smaller than the Fermi energy is the plasmon. In contrast to metals, the interaction strength is weak due to screening by the crystal, which allows the construction of a controllable theory of plasmon superconductivity. We show that plasma mediated pairing alone can account for the observed transition temperatures only if the dielectric screening by the crystal is reduced in the slightly doped samples compared with the insulating ones. We also discuss unique features of the plasmon mechanism, which appear in the tunneling density of states above the gap.