  Using the first-principles density functional approach, we investigate
Ca$_2$FeOsO$_6$, a material of double perovskite structure synthesized
recently. According to the calculations, Ca$_2$FeOsO$_6$ is a ferrimagnetic
Mott-insulator influenced by the cooperative effect of spin-orbit coupling
(SOC) and Coulomb interactions of Fe-3$d$ and Os-5$d$ electrons, as well as the
crystal field. When Fe is replaced with Ni, the system exhibits half metallic
(HM) states desirable for spintronic applications. In
[Ca$_2$Fe$_{1-x}$Ni$_x$OsO$_6$]$_2$, HM ferrimagnetism is observed with
$\mu_{\rm tot}=2\mu_{\rm B}$ per unit cell for doping rate $x=0.5$, whereas HM
antiferromagnetism (HMAFM) with nearly zero spin magnetization in the unit cell
for $x=1$, respectively. It is emphasized that half metallicity is retained
even with SOC effect due to the large exchange-splitting between spin-up and
spin-down bands close to the Fermi level.
