  Ferroelectric (FE) metals have been attracting attention as they possess both
metallicity and ferroelectricity, the two seemingly incompatible physical
properties. An important problem for both fundamental research and potential
applications is how to realize a strong FE metal. One strategy is to dope a
strong FE insulator into an FE metal provided that the induced free electrons
do not destroy FE distortion. Strained EuTiO$_{3}$ film is a strong FE
ferromagnet and a promising candidate for such a strategy. Here, we calculate
the structural, electronic, and polarization properties of Nb-doped strained
EuTiO$_{3}$ film using the hybrid density-functional theory. The results show
that the strained EuTi$_{0.875}$Nb$_{0.125}$O$_{3}$ film is a strong FE metal
as the Nb doping induces metallicity without weakening the FE distortion. The
underlying atomistic mechanism of the coexistence of metallicity and strong
ferroelectricity is discussed. The findings show that combining doping and
strain engineering is a promising way to realize new EuTiO$_{3}$-based strong
FE metals and may be used in other materials as well.
