  The superfluid phase diagrams of a two-dimensional cold polarized Fermi gas
in the BCS-BEC crossover are systematically and analytically investigated. In
the BCS-Leggett mean field theory, the transition from unpolarized superfluid
phase to normal phase is always of first order. For a homogeneous system, the
two critical Zeeman fields and the critical population imbalance are
analytically determined in the whole coupling parameter region, and the
superfluid-normal mixed phase is shown to be the ground state between the two
critical fields. The density profile in the presence of a harmonic trap
calculated in the local density approximation exhibits a shell structure, a
superfluid core at the center and a normal shell outside. For weak interaction,
the normal shell contains a partially polarized cloud with constant density
difference surrounded by a fully polarized state. For strong interaction, the
normal shell is totally in fully polarized state with a density profile
depending only on the global population imbalance. The di-fermion bound states
can survive in the whole highly imbalanced normal phase.
