  We investigate the single-particle and collective excitations of a Rashba
spin-orbit coupled atomic Fermi gas with attractive interaction, loaded in a
two dimensional (2D) square optical lattice, in the presence of an effective
out-of-plane Zeeman field. Our numerical calculations show that the many body
physics of the Bardeen-Cooper-Schrieffer (BCS) side is strongly modified
compared to the Fermi gases in the free space. The physics behind this
statement is in the fact that without a lattice structure, if the value of the
Zeeman field does not exceed some threshold value, the minimum of the
single-particle ground state energy is infinitely degenerate and occurs along a
ring in $(k_x,k_y)$ space. This reduces the effective dimensionality; the
single-particle density of states is a constant at low energies, and the
molecular pairing is strongly enhanced. In contrast to the continuum, in the 2D
lattice case there are only four degenerate minima, and because of that, the
spin-orbit coupling (SOC) in optical lattices gives rise to unusual properties
entirely different from the continuum. For example, in the continuum, the
pairing gap as well as the condensate fraction are strongly enhanced by the SOC
strength on the BCS side. In the presence of lattice geometry the gap and the
condensate fraction increase as a function of the SOC strength only at the
small fillings and weak attraction limit. Moreover, we found that the speed of
sound also exhibits different behavior: in the 3D and 2D continuum, the slope
of the Goldstone mode decreases as a function of the SOC strength, while in the
lattice case the speed of sound increases monotonically with SOC strength.
