The physical origin of the electron-phonon vertex correction

TL;DR

This paper explains the complex structure of the electron-phonon vertex correction by decomposing it into two physical terms, showing how reduced exchange interaction enhances effective electron-phonon coupling, relevant for high-temperature superconductivity.

Contribution

It provides a physical interpretation of the vertex correction's structure and demonstrates how weakening exchange interactions can enhance electron-phonon coupling in materials.

Findings

Vertex correction decomposed into two physical terms

Lowering electron density enhances effective coupling

Long-range electron-phonon interactions increase Tc

Abstract

The electron-phonon vertex correction has a complex structure both in momentum and frequency. We explain this structure on the basis of physical considerations and we show how the vertex correction can be decomposed into two terms with different physical origins. In particular, the first term describes the lattice polarization induced by the electrons and it is essentially a single-electron process whereas the second term is governed by the particle-hole excitations due to the exchange part of the phonon-mediated electron-electron interaction. We show that by weakening the influence of the exchange interaction the vertex takes mostly positive values giving rise to an enhanced effective coupling in the scattering with phonons. This weakening of the exchange interaction can be obtained by lowering the density of the electrons, or by considering only long-ranged (small q) electron-phonon couplings. These findings permit to understand why in the High-Tc materials the small carrier density and the long ranged electron-phonon interaction may play a positive role in enhancing Tc.