# Comment on "Superconductivity at low density near a ferroelectric quantum critical point: Doped SrTiO3"

## TL;DR

This paper critically evaluates a proposed pairing mechanism in doped SrTiO3, showing that the coupling to soft TO phonons is negligible and challenging previous theoretical claims about superconductivity in this low-density material.

## Contribution

The authors identify errors in prior calculations of phonon-electron coupling and provide evidence that this coupling is too weak to explain superconductivity in doped SrTiO3.

## Key findings

- Corrected eigenvector calculations show negligible coupling to TO phonons.
- Transport data set upper bounds on the coupling constant much lower than previous estimates.
- The use of Eliashberg theory is inappropriate when phonon frequency exceeds Fermi energy.

## Abstract

W\"olfle and Balatsky Phys. Rev. B 98, 104505 (2018) have proposed a microscopic pairing mechanism for doped SrTiO$_3$ (STO) based on the ${\textit gradient}$ coupling of electronic density to the soft TO phonon mode. Since this coupling to TO phonons is usually weak, this conclusion is surprising, especially for a low density superconductor such as STO, where the density of states is small. A crucial step in the argument made by W\"olfle and Balatsky is that the displacement vector of the TO mode is not strictly perpendicular to the momentum vector, making a deformation coupling possible. We show that they have made a mistake in computing the eigenvector and have grossly overestimated this lack of orthogonality. When corrected, the coupling is negligible. We also use transport data to put upper bounds on the coupling constant which are much smaller than the estimate by W\"olfle and Balatsky. Finally, we also object to their use of the Eliashberg equation when the phonon frequency is larger than the Fermi energy.