  We investigate the quantum and thermal phase diagram of fermionic polar
molecules loaded in a bilayer trapping potential with perpendicular dipole
moment. We use both a BCS theory approach that is most realiable at
weak-coupling and a strong-coupling approach that considers the two-body bound
dimer states with one molecules in each layer as the relevant degree of
freedom. The system ground state is a Bose-Einstein condensate (BEC) of dimer
bound states in the low density limit and a paired superfluid (BCS) state in
the high density limit. At zero temperature, the intralayer repulsion is found
to broaden the regime of BCS-BEC crossover, and can potentially induce system
collapse through the softening of roton excitations. The BCS theory and the
strongly-coupled dimer picture yield similar predictions for the parameters of
the crossover regime. The BKT transition temperature of the dimer superfluid is
also calculated. The crossover can be driven by many-body effects and is
strongly affected by the intralayer interaction which was ignored in previous
studies.
