Role of electron-phonon interaction in a magnetically driven mechanism for superconductivity

We use the renormalization group method to examine the effect of phonon mediated interaction on d-wave superconductivity, as driven by spin fluctuations in a quasi-one-dimensional electron system. The influence of a tight-binding electron-phonon interaction on the spin-density-wave and d-wave superconducting instability lines is calculated for arbitrary temperature, phonon frequency and antinesting of the Fermi surface.The domain of electron-phonon coupling strength where spin-density-wave order becomes unstable against the formation of a bond-order-wave or Peierls state is determined at weak antinesting. We show the existence of a positive isotope effect for spin-density-wave and d-wave superconducting critical temperatures which scales with the antinesting distance from quantum critical point where the two instabilities merge. We single out a low phonon frequency zone where the bond-oder-wave ordering gives rise to triplet f-wave superconductivity under nesting alteration, with both orderings displaying a negative isotope effect. We also study the electron-phonon strengthening of spin fluctuations at the origin of extended quantum criticality in the metallic phase above superconductivity. The impact of our results on quasi-one-dimensional organic conductors like the Bechgaard salts where a Peierls distortion is absent and superconductivity emerges near a spin-density-wave state under pressure is emphasized.