Novel pairing mechanism for superconductivity at a vanishing level of doping driven by critical ferroelectric modes

Superconductivity (SC) occurring at low densities of mobile electrons is still a mystery since the standard theories do not apply in this regime. We address this problem by using a microscopic model for ferroelectric (FE) modes, which mediate an effective attraction between electrons. When the dispersion of modes, around zero momentum, is steep, forward scattering is the main pairing process and the self-consistent equation for the gap function can be solved analytically. The solutions exhibit unique features: Different momentum components of the gap function are decoupled, and at the critical regime of the FE modes, different frequency components are also decoupled. This leads to effects that can be observed experimentally: The gap function can be non-monotonic in temperature and the critical temperature can be independent of the chemical potential. The model is applicable to lightly doped polar semiconductors, in particular, strontium titanate.