Weak coupling studies on pairing mechanism and related properties of Ruddlesden-Popper phase layered nickelate based superconductors

<sec>The discovery of superconductivity in Ruddlesden-Popper (RP) phase layered nickelates under high pressure has opened a new avenue for exploring nontraditional pairing mechanisms beyond cuprates and iron-based superconductors. In particular, La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> exhibits a superconducting transition temperature (<inline-formula><tex-math id="M9">\begin{document}$ T_{\rm c} $\end{document}</tex-math></inline-formula>) as high as 80 K at ~15 GPa, making it the second class of oxides that achieve liquid-nitrogen temperature superconductivity. Subsequent experiments have extended superconductivity to related compounds such as La<sub>4</sub>Ni<sub>3</sub>O<sub>10</sub> and La<sub>5</sub>Ni<sub>3</sub>O<sub>11</sub>, as well as epitaxially grown thin films at ambient pressure. These findings have motivated extensive theoretical efforts to elucidate the microscopic pairing mechanism.</sec><sec>This review summarizes recent progress from the perspective of weak-coupling theories, including random phase approximation (RPA), functional renormalization group (FRG), and fluctuation-exchange (FLEX) approaches. Density functional theory (DFT) calculations reveal that the low-energy degrees of freedom are dominated by Ni <inline-formula><tex-math id="M11">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ 3{\rm d}_{x^2-y^2} $\end{document}</tex-math></inline-formula> orbitals. In La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>, pressure-induced metallization of the bonding <inline-formula><tex-math id="M13">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> band produces the <i>γ</i> pocket, enhancing spin fluctuations, and stabilizing superconductivity. These fluctuations support superconductivity through interlayer <inline-formula><tex-math id="M15">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> pairing characterized by an <inline-formula><tex-math id="M16">\begin{document}$ \rm s^{\pm} $\end{document}</tex-math></inline-formula> gap. Hole doping or substitution may restore the <i>γ</i> pocket and achieve bulk superconductivity at ambient pressure.</sec><sec>For La<sub>4</sub>Ni<sub>3</sub>O<sub>10</sub>, theoretical calculations indicate predominantly <inline-formula><tex-math id="M18">\begin{document}$ \rm s^{\pm} $\end{document}</tex-math></inline-formula> pairing from interlayer <inline-formula><tex-math id="M19">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> orbitals, with weaker strength than La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>, explaining its lower <inline-formula><tex-math id="M20">\begin{document}$ T_{\rm c} $\end{document}</tex-math></inline-formula> and showing little sensitivity to band structure. In La<sub>5</sub>Ni<sub>3</sub>O<sub>11</sub>, composed of alternating single-layer and bilayer units, superconductivity mainly arises from the bilayer subsystem, again dominated by <inline-formula><tex-math id="M21">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> orbitals. Interestingly, the interplay between inter-bilayer Josephson coupling and the suppression of density of states leads to a dome-shaped <inline-formula><tex-math id="M22">\begin{document}$ T_{\rm c} $\end{document}</tex-math></inline-formula>-pressure phase diagram, differing from the monotonic behavior of La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub>.</sec><sec>Epitaxial (La, Pr)<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> thin films display superconductivity above 40 K at ambient pressure. Theory predicts doping-dependent pairing: <inline-formula><tex-math id="M23">\begin{document}$ \rm s^{\pm} $\end{document}</tex-math></inline-formula> symmetry is favored at low doping levels, while d<sub><i>xy</i></sub> pairing emerges at higher doping, in agreement with experimental indications of both nodeless and nodal gap behaviors.</sec><sec>In addition to superconductivity, the experiments reveal the spin-density-wave (SDW) sequence in bulk La<sub>3</sub>Ni<sub>2</sub>O<sub>7</sub> and La<sub>4</sub>Ni<sub>3</sub>O<sub>10</sub> at ambient pressure. Weak-coupling calculations confirm that these SDWs are driven by Fermi surface nesting and that their suppression under pressure gives rise to strong spin fluctuations which act as the glue for Cooper pairing. This highlights the intimate connection between the density-wave parent states and high-pressure superconductivity in nickelates.</sec><sec>In summary, weak-coupling theories provide a unified framework for RP nickelates, highlighting the key roles of <inline-formula><tex-math id="M24">\begin{document}$ 3{\rm d}_{z^2} $\end{document}</tex-math></inline-formula> orbitals, interlayer pairing, and spin fluctuations. They suggest that pressure, doping, substitution, and epitaxial strain can optimize superconductivity and potentially achieve high-<inline-formula><tex-math id="M25">\begin{document}$ T_{\rm c} $\end{document}</tex-math></inline-formula> phases at ambient pressure. Key challenges remain in clarifying orbital competition, the SDW-superconductivity interplay, and strong-correlation effects, requiring close collaboration between advanced experiments and multi-orbital many-body theory.</sec>