Pairing mechanism and superconductivity in pressurized La$_5$Ni$_3$O$_{11}$

The discovery of superconductivity (SC) with critical temperature $T_c$ above the boiling point of liquid nitrogen in pressurized La$_3$Ni$_2$O$_{7}$ has sparked a surge of exploration of high-$T_c$ superconductors in the Ruddlesden-Popper (RP) phase nickelates. More recently, the RP phase nicklate La$_5$Ni$_3$O$_{11}$, which hosts layered structure with alternating bilayer and single-layer NiO$_2$ planes, is reported to accommodate SC under pressure, exhibiting a dome-shaped pressure dependence with highest $T_c\approx 64$ K, capturing a lot of interests. Here, using density functional theory (DFT) and random phase approximation (RPA) calculations, we systematically study the electronic properties and superconducting mechanism of this material. Our DFT calculations yield a band structure including two nearly decoupled sets of sub-band structures, with one set originating from the bilayer subsystem and the other from the single-layer one. RPA-based analysis demonstrates that SC in this material occurs primarily within the bilayer subsystem exhibiting an $s^\pm$ wave pairing symmetry similar to that observed in pressurized La$_3$Ni$_2$O$_{7}$, while the single-layer subsystem mainly serves as a bridge facilitating the inter-bilayer phase coherence through the interlayer Josephson coupling (IJC). Since the IJC thus attained is extremely weak, it experiences a prominent enhancement under pressure, leading to the increase of the bulk $T_c$ with pressure initially. When the pressure is high enough, the $T_c$ gradually decreases due to the reduced density of states on the $\gamma$-pocket. In this way, the dome-shaped pressure dependence of $T_c$ observed experimentally is naturally understood.