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Metallic ferroelectric-ferromagnetic multiferroics in strained EuTiO$_{3-x}$H$_x$

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.