  We study the stability of superfluid Fermi gases in deep optical lattices in
the BCS--Bose-Einstein condensation (BEC) crossover at zero temperature. Within
the tight-binding attractive Hubbard model, we calculate the spectrum of the
low-energy Anderson-Bogoliubov (AB) mode as well as the single-particle
excitations in the presence of superfluid flow in order to determine the
critical velocities. To obtain the spectrum of the AB mode, we calculate the
density response function in the generalized random-phase approximation
applying the Green's function formalism developed by C\^ot\'e and Griffin to
the Hubbard model. We find that the spectrum of the AB mode is separated from
the particle-hole continuum having the characteristic rotonlike minimum at
short wavelength due to the strong charge-density-wave fluctuations. The energy
of the rotonlike minimum decreases with increasing the lattice velocity and it
reaches zero at the critical velocity which is smaller than the pair breaking
velocity. This indicates that the superfluid state is energetically unstable
due to the spontaneous emission of the short-wavelength rotonlike excitations
of the AB mode instead due to pair-breaking. We determine the critical
velocities as functions of the interaction strength across the BCS-BEC
crossover regime.
