  The finite temperature phase diagram for the 2D attractive fermion Hubbard
model with spin-dependent disorder is considered within Bogoliubov-de Gennes
mean field theory. Three types of disorder are studied. In the first, only one
species is coupled to a random site energy; in the second, the two species both
move in random site energy landscapes which are of the same amplitude, but
different realizations; and finally, in the third, the disorder is in the
hopping rather than the site energy. For all three cases we find that, unlike
the case of spin-symmetric randomness, where the energy gap and average order
parameter do not vanish as the disorder strength increases, a critical disorder
strength exists separating distinct phases. In fact, the energy gap and the
average order parameter vanish at distinct transitions, $V_{c}^{\rm gap}$ and
$V_{c}^{\rm op}$, allowing for a gapless superconducting (gSC) phase. The gSC
phase becomes smaller with increasing temperature, until it vanishes at a
temperature $T^{\ast}$.
