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

Abstract

Recently, the nickel-based superconductor $T_c$ record was updated to $96\ \text{K}$ in bilayer $\mathrm{La_{3-x}Sm_{x}Ni_{2}O_{7-δ}}$ (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 $\mathrm{YBa_{2}Cu_{3}O_{7-δ}}$ (YBCO). Based on the formula $T_c = Λ/ξ^{2}$ (Planck ground-state quantum well oscillator hypothesis, $ξ$ = 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 $ξ$ ($3.6629\ Å$ vs $3.6720\ Å$) 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 $\sim100\ \text{K}$.

Figures (5)

• Figure 1: Planckian Quantum Wells and Localized Electrons in Unconventional Superconductors. (a) Octahedral quantum wells in cuprates (YBCO, Bi-2223): their geometry and density dictate Fermi surface structure and symmetry, with well depth $\xi$ governing optimal $T_{c}$. (b) Polyhedral quantum wells of 1111/111-type iron-based superconductors ($R$: substitutable elements). Unlike cuprates, they possess two distinct well types ($\alpha$, $\beta$) with depths $\xi_{\alpha}$, $\xi_{\beta}$ and corresponding dual superconducting phases. Lattice mirror symmetry ensures paired upward/downward quantum wells ($\alpha^{\pm}$, $\beta^{\pm}$). Since $\xi_{\alpha}<\xi_{\beta}$, hence $T_{c}^{\alpha}>T_{c}^{\beta}$. Pressure reduces $\xi_{\beta}$, driving originally localized $\beta$-pairing electrons in the Se plane into the high-$T_{c}$$\alpha$-Fe plane via enhanced Coulomb repulsion—this is the microscopic mechanism of dual-superconducting-phase transition in iron-based superconductors sun2012reemerging.
• Figure 2: Octahedral Quantum Wells and Localized Electrons in Nickel-based Superconductors. (a) Tetragonal and orthorhombic bilayer nickel-based superconductors. (b) Undoped monoclinic nickel-based superconductors (monoclinic angle $\theta$): full quantum well symmetry screens localized electrons, resulting in an insulating state. (c) Ionic substitution breaks quantum well mirror symmetry along the $Z$-axis (octahedron vertex connecting direction), generating electron-hole dipoles and inducing the insulator-to-superconductor phase transition.
• Figure 3: Mirror Symmetry Breaking and Insulator-to-Superconductor Transition Induced by Pressure, Oxygen Doping and Ionic Substitution (interna pressure). (a) Symmetric quantum well of the Mott insulator: localized electrons are screened (as illustrated in the inset at the bottom right, and by the requirement of electroneutrality, the octahedral quantum well is equivalent to a hole) to an insulating state (NSC) by symmetry. (b) Pressure does not alter symmetry or insulating property. (c) Oxygen doping breaks mirror symmetry, as shown in the schematic at the bottom right, generates electron-hole dipoles (e-h pairs) and triggers the insulator-to-superconductor transition (SC). (d) Ionic substitution similarly induces mirror symmetry breaking and superconducting transition. (e)-(f) Intrinsically mirror-asymmetric orthorhombic $\mathrm{La_3Ni_2O_7}$ achieves superconducting transition via pressure alone. (g)-(h) Conversely, intrinsically symmetric monoclinic $\mathrm{La_3Ni_2O_7}$ requires both ionic substitution and pressure to realize symmetry breaking and superconducting transition.
• Figure 4: Comparison of Polyhedral Quantum Well Structures between Cuprate YBCO and Nickelate LSNO. (a) Projection of YBCO (Fig. \ref{['figure1']}(a)) along the $[110]$ direction; (b)-(c) Projections of monoclinic LSNO (Fig. \ref{['figure2']}(c)) along $[100]$ and tetragonal LSNO (Fig. \ref{['figure2']}(a)) along $[110]$, respectively. They share nearly identical octahedral quantum well structures with $\xi_a \approx \xi_b > \xi_c$, indicating analogous superconducting origins. Based on localized quantum well superconductivity theory, their $T_c$ values are 93 K, 93.4 K and 97.1 K, in excellent agreement with experimental data (93 K, 92 K, 96 K).
• Figure 5: Relationship between Element Substitution and Optimal $T_c$ in Ni-based Superconductor Series. The blue line is the experimental $c$-axis lattice parameters of $\mathrm{La_{3-x}R_xNi_2O_{7-\delta}}$ ($\mathrm{R} = \text{La}$ to $\text{Er}$) from Li et al.. The purple line denotes the estimated optimal $T_c$ values calculated via experimental data and Eq. (\ref{['dTc']}). Results show element substitution cannot significantly enhance $T_c$, with the upper limit of $100\ \text{K}$ for existing Ni-based superconductors corresponding to a quantum well depth of $\xi = 3.54\ \text{\AA}$. The orange line presents Fe-based superconductors' experimental element substitution vs. $T_c$ results, confirming the limitation of element substitution in improving $T_c$.