  Similar to what has recently been achieved with Bose-Bose mixtures [Lamporesi
et al., Phys. Rev. Lett. 104, 153202 (2010)], mixed-dimensional Fermi-Fermi
mixtures can be created by applying a species-selective one-dimensional optical
lattice to a two-species Fermi gas ($\sigma \equiv \{\uparrow, \downarrow \}$),
such a way that only one of the species feel the lattice potential and is
confined to a quasi-two-dimensional geometry, while having negligible effect on
the other, that is leaving it three dimensional. We investigate the ground
state phase diagram of superfluidity for such mixtures in the BCS-BEC
evolution, and find normal, gapped superfluid, gapless superfluid, and phase
separated regions. In particular, we find a stable gapless superfluid phase
where the unpaired $\uparrow$ and $\downarrow$ fermions coexist with the paired
(or superfluid) ones in different momentum space regions. This phase is in some
ways similar to the Sarma state found in mixtures with unequal densities, but
in our case, the gapless superfluid phase is unpolarized and most importantly
it is stable against phase separation.
