Effects of phonon interaction on pairing in high-Tcsuperconductors

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 $(D+iS)$ 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 $({D}_{\text{AFM}}+{D}_{\text{ph}})$ type pairing exhibits strongly reduced isotope coefficient despite the large enhancement of ${T}_{c}$ by phonon interaction. Finally, we study the combined type of $({D}_{\text{AFM}}+{D}_{\text{ph}}+iS)$ gap and calculate the penetration depth and specific heat to be compared with the experiments.