Possible phononic mechanism for d_x²-y² superconductivity in the presence of short-range antiferromagnetic correlations

We discuss the high-temperature superconductors in a regime where the antiferromagnetic (AF) correlation length is only a couple of lattice spacings. In the model proposed here, these short-range AF fluctuations play an essential role in the dressing of the carriers, but the attraction needed for superconductivity arises from a transverse phonon oxygen mode with a finite buckling angle as it appears in YBa₂Cu₃O_7-δ. A simple fermion-phonon model analog to the Holstein model is introduced to account for this effect. We argue that the model has a d_x²-y²-wave superconducting ground state. The critical temperature (T_c) and the O-isotope effect coefficient (α_O) versus hole density (x) are in qualitative agreement with experiments for the cuprates. The minimum (maximum) of α_O(T_c) at optimal doping is caused by a large peak in the density of states of holes dressed by AF fluctuations, as discussed in previous van Hove scenarios.