  We develop a minimal multiorbital tight-binding model with realistic hopping
parameters. The model breaks the symmetry of the tetragonal point group by
lowering it from $C_4$ to $D_{2d}$, which accurately describes the Fermi
surface evolution of the electron-doped BaFe$_{2-x}$Co$_x$As$_2$ and hole-doped
Ba$_{1-y}$K$_y$Fe$_2$As$_2$ compounds. An investigation of the phase diagram
with a mean-field $t$-$U$-$V$ Bogoliubov-de Gennes Hamiltonian results in
agreement with the experimentally observed electron- and hole-doped phase
diagram with only one set of $t$, $U$ and $V$ parameters. Additionally, the
self-consistently calculated superconducting order parameter exhibits
$s^\pm$-wave pairing symmetry with a small d-wave pairing admixture in the
entire doping range, % The superconducting $s^\pm + d$-wave order parameter
which is the subtle result of the weakly broken symmetry and competing
interactions in the multiorbital mean-field Hamiltonian.
