Microwave conductivity of d-wave superconductors

We calculate the microwave conductivity of a ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ superconductor in the presence of elastic impurity scattering and inelastic scattering due to antiferromagnetic spin fluctuations. The low-temperature conductivity does not simply reflect the linear temperature dependence of the number of quasiparticles in a d-wave system, as often supposed. We compare with recent data on high-quality ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6.95}$ single crystals.