  We study the superfluid behavior of ultracold atomic Fermi gases with a short
range attractive interaction in a two-dimensional optical lattice (2DOL) using
a pairing fluctuation theory, within the context of BCS-BEC crossover. We find
that the mixing of lattice and continuum dimensions leads to exotic phenomena.
For relatively large lattice constant $d$ and small hopping integral $t$, the
superfluid transition temperature $T_c$ exhibits a remarkable reentrant
behavior as a function of the interaction strength, and leads to a pair density
wave ground state, where $T_c$ vanishes, for a range of intermediate coupling
strength. In the unitary and BCS regimes, the nature of the in-plane and
overall pairing changes from particle-like to hole-like, with an unexpected
nonmonotonic dependence of the chemical potential on the pairing strength. The
BEC asymptotic behaviors exhibit distinct power law dependencies on the
interaction strength compared to cases of pure 3D lattice, 3D continuum, and
1DOL. These predictions can be tested in future experiments.
