  Scanning tunnelling spectroscopy (STS) measurements find that the surface of
Bi-2212 is characterized by nanoscale sized regions, "gap patches," which have
different magnitudes for the d-wave energy gap. Recent studies have shown that
the tunnelling conductance can be fit using a BCS-type density of states for a
d-wave superconductor with a local quasiparticle scattering rate. The fit is
made with a scattering rate which varies linearly with energy and has a slope
that is positively correlated with the local value of the gap. We revisit a
model of quasiparticle scattering by impurities and spin fluctuations which was
previously used to describe the lifetimes of nodal quasiparticles measured by
angle-resolved photoemission (ARPES). We argue that the broadening of the local
density of states is in general determined by the imaginary part of the
self-energy of the system averaged over a small region. The size of this region
is set by a mean free path which depends upon the energy. At low energies, this
region is found to be significantly larger than a gap "patch", so that the
density of states measured by STS is homogeneous in this energy range. At
higher energies where the mean free path is comparable with the patch size, the
density of states is inhomogeneous. We show that a local self-energy in the
impurity-plus-spin fluctuation model, while not strictly linear, yields a local
density of states (LDOS) nearly identical to the full theory, and argue that it
is consistent with the STS data as well as the phenomenological linear
scattering rate extracted from experiment. We also explore the qualitative
consequences of this phenomenology for the spectral widths observed in ARPES
and predict the existence of Fermi arcs in the superconducting state.
