  We report a first-principles study of Bi-based 3$d$-5$d$ ordered double
perovskite oxides (A$_2$BB$^\prime$O$_6$) with a 3$d$ atom (Fe) at the B-site
and 5$d$ atoms (Re,Ir) at the B$^\prime$-site while keeping highly polarizable
ions (Bi$^{3+}$) at the A-site. We find that, under coherent heteroepitaxy,
Bi$_2$FeReO$_6$} exhibits a strain-driven anti-ferromagnetic insulator to
ferrimagnetic semi-metal transition, while Bi$_2$FeIrO$_6$ shows correlation
driven ferromagnetic insulator to ferrimagnetic half-metal transition with
calculated magnetic moments of 5 $\mu_B$/f.u. and 3 $\mu_B$/f.u., respectively.
These properties along with the low band gaps in the insulating phases make the
compounds appealing for spintronics applications. Furthermore, in
Bi$_2$FeIrO$_6$, the conduction and valence states are localized on different
transition metal sublattices implying more efficient electron-hole separation
upon photoexcitation, a desirable feature for photovoltaic applications.
