  Half-metallicity in materials has been a subject of extensive research due to
its potential for applications in spintronics. Ferromagnetic manganites have
been seen as a good candidate, and aside from a small minority-spin pocket
observed in La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$ $(x=0.38)$, transport
measurements show that ferromagnetic manganites essentially behave like half
metals. Here we develop robust tight-binding models to describe the electronic
band structure of the majority as well as minority spin states of
ferromagnetic, spin-canted antiferromagnetic, and fully antiferromagnetic
bilayer manganites. Both the bilayer coupling between the MnO$_2$ planes and
the mixing of the $|x^2 - y^2>$ and $|3z^2 - r^2>$ Mn 3d orbitals play an
important role in the subtle behavior of the bilayer splitting. Effects of
$k_z$ dispersion are included.
