Can supersymmetry naturally explain the positron excess?

It has often been suggested that the cosmic positron excess observed by the High-Energy Antimatter Telescope (HEAT) experiment could be the consequence of supersymmetric dark matter annihilating in the galactic halo. Although it is well known that evenly distributed dark matter cannot account for the observed excess, if substantial amounts of local dark matter substructure are present, the positron flux would be enhanced, perhaps to the observed magnitude. In this paper, we attempt to identify the nature of the substructure required to match the HEAT data, including the location, size and density of any local dark matter clump(s). Additionally, we attempt to assess the probability of such substructure being present. We find that if the current density of neutralino dark matter is the result of thermal production, very unlikely $(\sim {10}^{-4}$ or less) conditions must be present in local substructure to account for the observed excess.