Theoretical designing of multiband Nickelate and Palladate superconductors with $d^{8+\delta}$ configuration

In a previous study, we proposed a possibility of high $T_c$ superconductivity in mixed-anion nickelates with $d^{8+\delta}$ electron configuration. The theory was based on the fact that the two-orbital Hubbard model, when all the intra- and interorbital interactions have the same magnitude, is equivalent to the bilayer Hubbard model, which has been suggested to exhibit high $T_c$ superconductivity. The energy level offset $\Delta E$ in the two-orbital model is transformed to twice the interlayer hopping in the bilayer model, and hence appropriately large $\Delta E$ is favorable for superconductivity in the former. Extending this idea to multiorbital systems, we previously suggested materials with large energy level offset between $d_{x^2-y^2}$ and other $d$ orbitals, such as Ca$_2$NiO$_2$Cl$_2$, to be good candidates for high $T_c$ superconductivity, but such materials have not been synthesized to our knowledge. In the present study, we first focus on Sr$_2$NiO$_2$Cl$_2$, which has been synthesized in the past but has small $\Delta E$, and study the effect of applying pressure, which enhances $\Delta E$. We also study a 4d analogue of Sr$_2$NiO$_2$Cl$_2$, namely, Sr$_2$PdO$_2$X$_2$ ($X=$ Cl, F, H) , in which $\Delta E$ turns out to be large. The analysis using fluctuation exchange approximation suggests possibility of superconductivity in these systems with large $\Delta E$. We also study the effect of electron doping of these material, which is expected to enhance superconductivity, within the virtual crystal approximation.