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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.

Polar metals are defined by the coexistence of metallicity and polar crystal structure. They have potential applications in non-linear optics, ferroelectric devices, and quantum devices. Meanwhile, ferroelectric-ferromagnetic (FE-FM) multiferroics display simultaneous ferroelectricity and ferromagnetism, leading to new technologies in information storage. It remains an open question whether metallicity, ferroelectricity, and ferromagnetism can coexist in a single domain of a material. EuTiO$_3$ is actively studied for potential applications in magnetic sensors, memories, magneto-optical devices, and energy conversion devices. It stabilizes at a multi-critical equilibrium and exhibits a rich range of intriguing properties. Here, using the results from hybrid density functional theory calculations, we report metallic FE-FM multiferroics in strain-engineered epitaxial EuTiO$_3$ with H doping. The emergence of the magnetism in polar metals provides a new degree of freedom to control these materials in applications. The underlying mechanism for the coexistence of metallicity, ferroelectricity, and ferromagnetism is discussed. The ferromagnetism in metallic EuTiO$_{3-x}$H$_x$ is explained by the Ruderman-Kittel-Kasuya- Yosida (RKKY) interaction, which agrees with experiments. The coexistence of metallicity and ferroelectricity is allowed because the electrons at the Fermi level are weakly coupled to the ferroelectric distortion. Our results suggest that the combined effect of strain and doping is responsible for achieving EuTiO$_3$-based metallic FE-FM multiferroics and may provide a new way for obtaining metallic FE-FM multiferroics in other materials.