BCS superconductivity near the band edge: Exact results for one and several bands

TL;DR

This paper provides exact analytical results for BCS superconductivity near the band edge in low-density regimes, revealing nonanalytic behaviors of the critical temperature in both two and three dimensions, with implications for materials like SrTiO3.

Contribution

It offers the first exact solutions in the low-density limit for BCS superconductors, including multiband systems, highlighting nonanalytic behaviors of $T_c$ and emphasizing the importance of self-consistent chemical potential treatment.

Findings

In 2D, $T_c$ rises faster than any power of density at low n.

In 3D, $T_c$ has zero slope at weak coupling and infinite slope at strong coupling.

Exact results are confirmed by numerical simulations.

Abstract

We revisit the problem of a BCS superconductor in the regime where the Fermi energy is smaller than the Debye energy. This regime is relevant for low-density superconductors such as SrTiO$_3$ that are not in the BEC limit, as well as in the problem of "shape resonances" associated with the confinement of a three-dimensional superconductor. While the problem is not new, exact results were lacking in the low-density limit. In two dimensions, we find that the initial rise of the pairing temperature $T_c$ at low density $n$ is nonanalytic and faster than any power of $n$. In three dimensions, we also find that $T_c$ is nonanalytic, but starts with zero slope at weak coupling and infinite slope at strong coupling. Self-consistent treatment of the chemical potential and energy dependence of the density of states are crucial ingredients to obtain these results. We also present exact results for multiband systems and confirm our analytical expressions by numerical simulations.