  The electronic structure and thermodynamic stability of tetragonal
$\rm{BiFeO_3}$(001) surfaces have been investigated using density functional
theory. In this work, four different structures having different lattice
constants with two possible surface terminations have been studied. We have
found that the surface electronic structure and the thermodynamic stability is
quite sensitive with respect to the nature of the surface termination. The
$\rm{FeO_2}$ terminated surface is found to be energetically more stable
compared to $\rm{BiO}$ terminated surface in all the cases. Interestingly, we
have found evidences of half-metallicity and spin-polarized two-dimensional
hole gas (2DHG) at the one mono-layer thick surface in all the structures. The
effect of the surface thickness have been systematically studied. It has been
demonstrated that the half-metallic 2DHG survives only in one of the structures
for all the thicknesses, incidentally, which is the most thermodynamically
stable structure.
