Superconductivity in nickelate and cuprate superconductors with strong bilayer coupling

The discovery of superconductivity at 80 K under high pressure in La$_3$Ni$_2$O$_7$ presents the groundbreaking confirmation that high-$T_c$ superconductivity is a property of strongly correlated materials beyond cuprates. We use density functional theory (DFT) calculations of the band structure of La$_3$Ni$_2$O$_7$ under pressure to verify that the low-energy bands are composed almost exclusively of Ni 3$d_{x^2-y^2}$ and O 2$p$ orbitals. We deduce that the Ni 3$d_{z^2}$ orbitals are essentially decoupled by the geometry of the high-pressure structure and by the effect of the Ni Hund coupling being strongly suppressed, which results from the enhanced interlayer antiferromagnetic interaction between $d_{z^2}$ orbitals and the strong intralayer hybridization of the $d_{x^2-y^2}$ orbitals with O 2$p$. By introducing a tight-binding model for the Fermi surfaces and low-energy dispersions, we arrive at a bilayer $t$-$t_\perp$-$J$ model with strong interlayer hopping, which we show is a framework unifying La$_3$Ni$_2$O$_7$ with cuprate materials possessing similar band structures, particularly the compounds La$_2$CaCu$_2$O$_6$, Pb$_2$Sr$_2$YCu$_3$O$_8$, and EuSr$_2$Cu$_2$NbO$_8$. We use a renormalized mean-field theory to show that these systems should have ($d$+$is$)-wave superconductivity, with a dominant $d$-wave component and the high $T_c$ driven by the near-optimally doped $\beta$ band, while the $\alpha$ band adds an $s$-wave component that should lead to clear experimental signatures.