  The recent experimental realization of spin-orbit coupling for ultra-cold
atoms has generated much interest in the physics of spin-orbit coupled
degenerate Fermi gases. Although recently the BCS-BEC crossover in
three-dimensional (3D) spin-orbit coupled Fermi gases has been intensively
studied, the corresponding two-dimensional (2D) crossover physics has remained
unexplored. In this paper, we investigate, both numerically and analytically,
the BCS-BEC crossover physics in 2D degenerate Fermi gases in the presence of a
Rashba type of spin-orbit coupling. We derive the mean field gap and atom
number equations suitable for the 2D spin-orbit coupled Fermi gases and solve
them numerically and self-consistently, from which the dependence of the ground
state properties (chemical potential, superfluid pairing gap, ground state
energy per atom) on the system parameters (e.g., binding energy, spin-orbit
coupling strength) is obtained. Furthermore, we derive analytic expressions for
these ground state quantities, which agree well with our numerical results
within a broad parameter region. Such analytic expressions also agree
qualitatively with previous numerical results for the 3D spin-orbit coupled
Fermi gases, where analytic results are lacked. We show that with an increasing
SOC strength, the chemical potential is shifted by a constant determined by the
SOC strength. The superfluid pairing gap is enhanced significantly in the BCS
limit for strong SOC, but only increases slightly in the BEC limit.
