The physical origin of the electron-phonon vertex correction

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.

Quantitative analysis of nonadiabatic effects in dense H3S and PH3 superconductors

“…The application of the above Eliashberg equations to describe the electron-phonon superconductivity is justified for systems in which the value of the phonon energy scale (Debye frequency, ω D ) to the electron energy scale (Fermi energy, ε F ) ratio is negligible. Otherwise the Eliashberg equations should be generalized by taking into account the lowest-order vertex correction [33][34][35][36][37]: and…”

High-temperature superconductivity in the Ti-H system at high pressures

“…Starting from BCS theory, several first-principles Green's function methods had been proposed to calculate the superconducting properties. Migdal-Eliashberg formalism is one of these, and can accurately describe conventional superconductors 44 . Within the Migdal approximation 45 , the adiabatic ratio λω D / F ( m * /M ) is small, since the vertex correction O( m * /M ) can compare to the bare vertex and then be neglected.…”

Non-Adiabatic Effects in Superconducting Intermetallic Borocarbides

“…Moreover, the ratio ω D /ε F is of the order of 0.01 and therefore it would seem that the vertex corrections due to the breakdown of Migdal's theorem for the electron-phonon interactions should also have important consequences on the thermodynamic properties of superconducting state. The generalization Eliashberg equations can be written in the following form [18,19,[21][22][23]:…”