Ultrasonic attenuation in d-wave superconductors

We consider a BCS-like model of interacting electrons on a square lattice. Within mean field theory, our model has an instability towards the formation of a d-wave superconducting state. We calculate the longitudinal ultrasonic attenuation rate in the superconducting phase in the clean limit for parameters appropriate to the cuprates. We find that the temperature dependence of the attenuation rate is similar to that observed in conventional superconductors for phonon wavevectors along the direction of the antinodes of the gap. However, the attenuation rate for phonon wavevectors along the nodal direction is found to be strikingly different in its temperature and doping dependence. The attenuation rate as(T) increases in magnitude and shows a sign change in its curvature (when plotted as a function of temperature) with increasing value of dopant concentration.