Possible superconductivity in $d^{n} (n\neq 9)$ platforms ?

Superconductivity in square BX$_2$-plane-based materials currently is only found in $d^9$ configuration. This raises a question that whether other configurations $d^{n}$ ($n\neq 9$) of transition metals in the prototype of square BX$_2$ plane can also host superconductivity. We systematically explore this question via analyzing the electronic structure of materials from $d^{8}$ to $d^5$ platforms using density-functional calculation, including existing materials SrFeO$_2$ and SrFeO$_2$F, and proposed ones LaNiO$_2$F and LaCoO$_2$F. Results show a good commonality between these materials and the cuprate and nickelate superconductors in their high-energy electronic structure, namely dominant low-energy states by ligand $p$ and $d_{x^2-y^2}$ orbitals. Other $d$ orbitals are all in the high-energy channel due to strong intra-atomic repulsion resulting in similar low-energy effective Hamiltonian except for the different number of local spins. These results hopefully suggest the possible superconductivity besides the prototype of $d^9$. Superconducting phases found in these sets of materials will be highly valuable to understand the high-temperature superconductivity and even to find better superconducting families than the cuprates.