# d-wave superconductivity from electron-phonon interactions

## TL;DR

This paper investigates how electron-phonon interactions can lead to d-wave superconductivity in a quasi-2D model, emphasizing the importance of vertex corrections and spatial fluctuations in predicting the order parameter's symmetry.

## Contribution

It introduces an extended Migdal-Eliashberg theory that includes vertex corrections and spatial fluctuations, revealing the emergence of d-wave superconductivity near half-filling.

## Key findings

- D-wave superconductivity appears close to half-filling.
- Increasing filling transitions the order parameter from d-wave to s-wave.
- Large Coulomb pseudopotential suppresses s-wave but not d-wave superconductivity.

## Abstract

I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-2D Holstein model. I use an extended Migdal-Eliashberg theory which includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb pseudopotential on the superconductivity (such as is found in contemporary superconducting materials like the cuprates), which results in the destruction of the s-wave states, while leaving the d-wave states unmodified.