Superconductivity in the spin-state crossover materials: Nickelates with planar-coordinated low-spin Ni$^{2+}$ ions

We theoretically study quasi-two-dimensional nickel compounds, where the nickel ions assume Ni$^{2+}$ $d^8$ valence state and feature a low-spin $S=0$ ground state quasidegenerate with $S=1$ ionic excitations. Such a level structure is supported by square-planar coordination of nickel ions or a suitable substitution of apical oxygens. We construct the corresponding singlet-triplet exchange model and explore its phase diagram and excitation spectrum. By hole doping, we further introduce mobile Ni$^{3+}$ $d^7$ ionic configurations with effective spins $S=1/2$, and analyze their interactions with the $d^8$ singlet-triplet background. The interplay with the triplet excitations in the $d^8$ sector is found to have a deep impact on the propagation of the doped hole-like charge carriers and is identified as a powerful source of Cooper pairing among them.