Local quasiparticle lifetimes in a d-wave superconductor*

Recently, scanning tunnelling spectroscopy (STS) measurements have exhibited good fits to conductance spectra at the surface of Bi-2212 using a BCS-type model for a d-wave superconductor and assuming a local quasiparticle scattering rate varying linearly with energy. Employing a model of quasiparticle scattering by impurities and spin fluctuations we argue that the broadening of the local density of states is in general given by the self-energy of the system averaged over a small region. The size of this region at low energies is shown to be significantly larger than a gap ``patch'', a region over which the gap is roughly constant in this system; states measured by STS are therefore very homogeneous in this energy range. At energies above a scale determined by disorder, STS averages over states localized within a gap ``patch'', and lifetimes are correspondingly inhomogeneous. We show that the local self-energy in the impurity-plus-spin fluctuation model can explain the data as well as the phenomenological linear scattering rate extracted from experiment.