Charge disproportionation driven polar magnetic metallic double-layered perovskite Sr$_3$Co$_2$O$_7$

Strong coupling among spontaneous structural symmetric breaking, magnetism and metallicity in an intrinsic polar magnetic metal can give rise to novel physical phenomena and holds great promise for applications in spintronics. Here, we elucidate the mechanism of metallic ferroelectricity in the recently discovered polar metal Sr$_3$Co$_2$O$_7$. Our first-principles calculations reveal that both the spontaneous ferroelectric displacements and the metallicity originate from charge disproportionation of Co ions. This is characterized by an inverted ligand-field splitting of the Co $t_2g$ orbitals at one site, while the metallic behavior is preserved by the t$_2g$ orbitals at both sites. The charge disproportionation stabilizes the asymmetric phase Within the framework of the on-site Hubbard U interaction. We thus propose that in related transition metal oxides, charge disproportionation within specific orbitals can concurrently drive metallicity and ferroelectricity, enabling strong coupling between these properties. More remarkably, this mechanism allows for the coexistence of magnetism, as evidenced in Sr$_3$Co$_2$O$_7$. Our findings highlight a promising avenue for realizing polar magnetic metals and provide a new design principle for exploring multifunctional materials.