  We study the zero temperature ground state of a two-dimensional atomic Fermi
gas with chemical potential and population imbalance in the mean-field
approximation. All calculations are performed in terms of the two-body binding
energy $\epsilon_B$, whose variation allows to investigate the evolution from
the BEC to the BCS regimes. By means of analytical and exact expressions we
show that, similarly to what is found in three dimensions, at fixed chemical
potentials, BCS is the ground state until the critical imbalance $h_c$ after
which there is a first-order phase transition to the normal state. We find that
$h_c$, the Chandrasekhar-Clogston limit of superfluidity, has the same value as
in three dimensional systems. We show that for a fixed ratio
$\epsilon_B/\epsilon_F$, where $\epsilon_F$ is the two-dimensional Fermi
energy, as the density imbalance $m$ is increased from zero, the ground state
evolves from BCS to phase separation to the normal state. At the critical
imbalance $m_c$ phase separation is not supported and the normal phase is
energetically preferable. The BCS-BEC crossover is discussed in balanced and
imbalanced configurations. Possible pictures of what may be found
experimentally in these systems are also shown. We also investigate the
necessary conditions for the existence of bound states in the balanced and
imbalanced normal phase.
