  The superconducting state of SRO is widely believed to have chiral p-wave
order that breaks time reversal symmetry. Such a state is expected to have a
spontaneous magnetization, both at sample edges and at domain walls between
regions of different chirality. Indeed, muon spin resonance experiments are
interpreted as evidence of spontaneous magnetization due to domain walls or
defects in the bulk. However, recent magnetic microscopy experiments place
upper limits on the magentic fields at the sample edge and surface which are as
much as two orders of magnitude smaller than the fields predicted theoretically
for a somewhat idealized chiral p-wave superconductor. We investigate the
effects on the spontaneous supercurrents and magnetization of rough and pair
breaking surfaces for a range of parameters within a Ginzburg-Landau formalism.
The effects of competing orders nucleated at the surface are also considered.
We find the conditions under which the edge currents are significantly reduced
while leaving the bulk domain wall currents intact, are quite limited. The
implications for interpreting the existing body of experimental results on
superconducting SRO within a chiral p-wave model are discussed.
