  Critical velocities of superfluid Fermi gases in optical lattices are
theoretically investigated across the BCS-BEC crossover. We calculate the
excitation spectra in the presence of a superfluid flow in one- and
two-dimensional optical lattices. It is found that the spectrum of low-lying
Anderson-Bogoliubov (AB) mode exhibits a roton-like structure in the
short-wavelength region due to the strong charge density wave fluctuations, and
with increasing the superfluid velocity one of the roton-like minima reaches
zero before the single-particle spectrum does. This means that superfluid Fermi
gases in optical lattices are destabilized due to spontaneous emission of the
roton-like AB mode instead of due to Cooper pair breaking.
