Microwave Conductivity of a High Purity d-wave Superconductor

The cusp-like behavior of the microwave conductivity observed in clean ortho-II YB$_{2}$Cu$_{3}$O$_{6.50}$ at low temperature and low frequency is shown to be related directly to a linear in frequency dependence of the impurity scattering rate with a negligibly small value at zero frequency. In the weak scattering limit, the conductivity decreases linearly with the frequency. In the vortex state, assuming a random (Gaussian) distribution of vortices, we show that the magnetic field profoundly alters the impurity scattering rate, which now acquires a finite zero frequency value. As a consequence, we predict a Drude-like line shape in the microwave conductivity at low frequency.