  We study Josephson current in superconductor / diffusive ferromagnet
/superconductor junctions by using the recursive Green function method. When
the exchange potential in a ferromagnet is sufficiently large as compared to
the pair potential in a superconductor, an ensemble average of Josephson
current is much smaller than its mesoscopic fluctuations. The Josephson current
vanishes when the exchange potential is extremely large so that a ferromagnet
is half-metallic. Spin-flip scattering at junction interfaces drastically
changes the characteristic behavior of Josephson current. In addition to
spin-singlet Cooper pairs, equal-spin triplet pairs penetrate into a half
metal. Such equal-spin pairs have an unusual symmetry property called
odd-frequency symmetry and carry the Josephson current through a half metal.
The penetration of odd-frequency pairs into a half metal enhances the low
energy quasiparticle density of states, which could be detected experimentally
by scanning tunneling spectroscopy. We will also show that odd-frequency pairs
in a half metal cause a nonmonotonic temperature dependence of the critical
Josephson current.
