Theoretical Prediction of optimal $T_c$ for Nickelate $\mathrm{La_{3-x}Sm_{x}Ni_{2}O_{7-\delta}}$

Abstract

Recently, the nickel-based superconductor $T_c$ record was updated to $96\ \text{K}$ in bilayer $\mathrm{La_{3-x}Sm_{x}Ni_{2}O_{7-\delta}}$ (LSNO) under pressure, raising a critical question: Can its $T_c$ exceed the 164 K benchmark of copper-based superconductors? We find that both monoclinic and tetragonal LSNO have an octahedral quantum well structure (determining $T_c$) nearly identical to $\mathrm{YBa_{2}Cu_{3}O_{7-\delta}}$ (YBCO). Based on the formula $T_c = \Lambda/\xi^{2}$ (Planck ground-state quantum well oscillator hypothesis, $\xi$ = lattice parameter-determined quantum well depth), we predict Sm-doped nickelate $T_c$ values of $93.4\ \text{K}$ (monoclinic) and $97.1\ \text{K}$ (tetragonal), in excellent agreement with experimental data ($92\ \text{K}$ and $96\ \text{K}$). Notably, despite distinct composition and symmetry (LSNO: $P2_1/m$; YBCO: $Pmmm$), their $\xi$ ($3.6629\ \AA$ vs $3.6720\ \AA$) and $T_c$ ($92\ \text{K}$ vs $93\ \text{K}$) are nearly identical. This validates the proposed superconducting formula and unifies copper-based and nickel-based superconductors at the angstrom-scale octahedral quantum well. Further predictions indicate the maximum achievable $T_c$ for lanthanide-based nickelates (regardless of layer number) is $\sim100\ \text{K}$.