In-Plane Ni-O-Ni Bond Angles as Structural Fingerprints of Superconductivity in Layered Nickelates: Effects of Pressure, Strain, Layering, and Correlations

We investigate the structural and electronic conditions conducive to superconductivity in layered nickelates using density functional theory with Hubbard corrections (DFT+$U$). For both the bilayer and 1-3 polymorphs of La$_3$Ni$_2$O$_7$, we find that the in-plane Ni-O-Ni bond angles under pressure strongly correlate with the experimentally observed superconducting transition temperature ($T_c$) dome, and may serve as a reasonable proxy. Under compressive strain, the bond angles straighten, peaking near 2\% strain-consistent with experimental reports of superconductivity in strained bilayer thin films. However, the bond angles at this strain are more bent than those achieved under hydrostatic pressure, correlating with a lower $T_c$. We show that increasing the number of NiO$_2$ layers, as in La$_4$Ni$_3$O$_{10}$, or substituting heavier rare-earth elements (e.g., Pr) raises the pressure required to reach the structural configuration associated with superconductivity. Our results indicate that these systems require higher external pressure to achieve in-plane bond straightening. Varying the on-site Coulomb interaction $U$ reveals that stronger electronic correlations delay the structural transition and favor high-spin states. This suggests that moderate correlation strength may be optimal for superconductivity, with stronger correlation preventing the formation of favorable bond geometries. Electronic structure analysis shows that the Ni $e_g$ orbitals dominate near the Fermi level and shift downward with pressure, enhancing Ni-O hybridization. These results highlight how pressure and strain tune structural features that may be essential for engineering high-$T_c$ phases in nickelate superconductors.