# Phonon mediated superconductivity in low carrier-density systems

## TL;DR

This paper investigates how phonon-mediated superconductivity behaves in low carrier-density systems, revealing that the critical temperature increases as the chemical potential decreases, especially when it becomes comparable to phonon frequencies.

## Contribution

It provides a detailed analysis of superconductivity at low carrier densities, challenging the validity of the Migdal-Eliashberg approximation in this regime, and derives the dependence of $T_c$ on chemical potential and phonon frequency.

## Key findings

- $T_c$ increases as $rac{ ext{chemical potential}}{ ext{phonon frequency}}$ decreases.
- In the dilute limit, $T_c$ scales with $ ext{phonon frequency}$ and Rydberg energy as $T_c o 0$.
- Migdal-Eliashberg approximation becomes invalid when $rac{ ext{chemical potential}}{ ext{phonon frequency}} o 0$.

## Abstract

Motivated by the observation of superconductivity in SrTiO$_3$ and Bi, we analyze phonon-mediated superconductivity in three-dimensional systems at low carrier density, when the chemical potential $\mu$ (equal to Fermi energy at $T=0$) is comparable to or even smaller than the characteristic phonon frequency $\omega_L$. We consider the attractive part of the Bardeen-Pines pairing interaction, in which the frequency-dependent electron-phonon interaction is dressed by the Coulomb potential. This dressing endows the pairing interaction with momentum dependence. We argue that the conventional Migdal-Eliashberg (ME) approximation becomes invalid when $\mu \leq \omega_L$ chiefly because the dominant contribution to pairing comes from electronic states away from the Fermi surface. We obtain the pairing onset temperature, which is equal to $T_c$ in the absence of phase fluctuations, as a function of $\mu/\omega_L$. We find both analytically and numerically that $T_c$ increases as the ratio $\mu/\omega_L$ becomes smaller. In particular, in the dilute regime, $\mu \rightarrow 0$, it holds that $T_c\propto\omega_L\left(\frac{Ry}{\omega_L}\right)^\eta$, where $\text{Ry}$ is the Rydberg constant and $\eta \sim 0.2$.