  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.
