Effects of phonon interaction on the pairing in the high-T$c$ superconductors

TL;DR

This paper investigates how phonon interactions influence d-wave superconducting pairing in high-Tc materials, revealing that phonons can enhance T_c and induce mixed gap symmetries without increasing T_c, with implications for isotope effects and experimental properties.

Contribution

It introduces a detailed analysis of phonon effects on d-wave pairing, including mixed gap symmetries and their impact on T_c and isotope coefficients, using coupled BCS gap equations.

Findings

Phonons can induce s-wave components in the d-wave gap.

Anisotropic phonon interactions can significantly enhance d-wave pairing and T_c.

The combined pairing state shows reduced isotope coefficient despite increased T_c.

Abstract

We study the effects of phonon interaction on the superconducting pairing in the background of a d-wave gap, mediated by antiferromagnetic (AFM) spin fluctuations, using coupled BCS gap equations. We found that phonon interaction can induce a s-wave component to the d-wave gap in the (D+S) form with an interaction anisotropy and in the (D+$i$S) form without anisotropy, respectively. In either case, however, T$_c$ is not enhanced compared to the pure d-wave pairing without phonon interaction. On the other hand, anisotropic phonon interaction can dramatically enhance the d-wave pairing itself and therefore T$_c$, together with the AFM spin fluctuation interaction. This (D$_{AFM}$ + D$_{ph}$) type pairing exhibits strongly reduced isotope coefficient despite the large enhancement of T$_c$ by phonon interaction. Finally, we study the combined type of (D$_{AFM}$ + D$_{ph}$ +$i$S)) gap and calculate the penetration depth and specific heat to be compared with the experiments.