  We study a quantum phase transition between fermion superfluid (SF) and band
insulator (BI) of fermions in optical lattices. The destruction of the band
insulator is driven by the energy gain in promoting fermions from valance band
to various conducting bands to form Cooper pairs. We show that the transition
must take place in rather shallow lattice height, $V_{o}/E_{R}$ between 2.23
and 4.14. The latter is the prediction of mean field theory while the former is
the value for opening a band gap. As one moves across resonance to the molecule
side, the SF-BI transition evolves into the SF-Mott insulator transition of
bosonic molecules. We shall also present the global phase diagram for
SF-Insulator transition for the BCS-BEC family.
