  Using angle-resolved photoemission spectroscopy (ARPES), we investigate the
electronic band structure and Fermi surface of ferromagnetic
La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ ($x=0.38$). Besides the expected two hole
pockets and one electron pocket of majority-spin $e_g$ electrons, we show an
extra electron pocket around the $\Gamma$ point. A comparison with
first-principles spin-polarized band-structure calculations shows that the
extra electron pocket arises from $t_{2g}$ electrons of minority-spin
character, indicating this compound is not a complete half-metallic
ferromagnet, with similar expectations for lightly-doped cubic manganites.
However, our data suggest that a complete half-metallic state is likely to be
reached as long as the bandwidth is mildly reduced. Moreover, the band-resolved
capability of ARPES enables us to investigate the band structure effects on
spin polarization for different experimental conditions.
