Effects of reduced dimensionality, crystal field, electron-lattice coupling, and strain on the ground state of a rare-earth nickelate monolayer

Motivated by the potential for cupratelike superconductivity in monolayer rare-earth nickelate superlattices, we study the effects of crystal field splitting, lattice distortions, and strain on the charge, magnetic, and orbital order in undoped two-dimensional (2D) nickelate monolayers $R\mathrm{Ni}{\mathrm{O}}₃$. We use a two-band Hubbard model to describe the low-energy electron states, with correlations controlled by an effective Hubbard $U$ and Hund's $J$. The electrons are coupled to the octahedral breathing-mode lattice distortions. Treating the lattice semiclassically, we apply the Hartree-Fock approximation to obtain the phase diagram for the ground state as a function of the various parameters. We find that the 2D confinement leads to strong preference for the planar ${d}_{{x}²\ensuremath{-}{y}²}$ orbital even in the absence of a crystal-field splitting. The ${d}_{{x}²\ensuremath{-}{y}²}$ polarization is enhanced by adding a crystal field splitting, whereas coupling to breathing-mode lattice distortions weakens it. However, the former effect is stronger, leading to ${d}_{{x}²\ensuremath{-}{y}²}$ orbital and antiferromagnetic order at reasonable values of $U,J$ and thus to the possibility to realize cupratelike superconductivity in this 2D material upon doping. We also find that the application of tensile strain enhances the cupratelike phase and phases with orbital polarization in general, by reducing the ${t}₂/{t}₁$ ratio of next-nearest to nearest neighbor hopping. On the contrary, systems with compressive stress have an increased hopping ratio and consequently show a preference for ferromagnetic (FM) phases, including, unexpectedly, the out-of-plane ${d}_{3{z}²\ensuremath{-}{r}²}$ FM phase.