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

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