  First-principles calculations to study the electronic and magnetic properties
of bulk, oxygen-deficient SrTiO$_3$ (STO) under different doping conditions and
densities have been conducted. The appearance of magnetism in oxygen-deficient
STO is not determined solely by the presence of a single oxygen vacancy but by
the density of free carriers and the relative proximity of the vacant sites. We
find that while an isolated vacancy behaves as a non-magnetic double donor,
manipulation of the doping conditions allows the stability of a single donor
state, with emergent local moments coupled ferromagnetically by carriers in the
conduction band. Strong local lattice distortions enhance the binding of this
state. The energy of the in-gap local moment can be further tuned by
orthorhombic strain. Consequently we find that the free-carrier density and
strain are fundamental components to obtaining trapped spin-polarized electrons
in oxygen-deficient STO, which may have important implications in the design of
optical devices.
