Band Structure and Pairing Nature of La$_3$Ni$_2$O$_7$ Thin Film at Ambient Pressure

Recently, evidences of superconductivity (SC) with onset $T_c$ above the McMillan limit have been detected in the La$_3$Ni$_2$O$_7$ ultrathin film grown on the LaSrAlO$_4$ substrate at ambient pressure. This progress opens a new era in the field of the nickelate superconductors. Here we perform a density-functional-theory (DFT) based calculation for the band structure of this material. The obtained DFT+$U$ band structure has the feature that the bonding $d_{z^2}$ band crosses the Fermi level, forming the hole pocket $\gamma$, consistent with the angle-resolved photoemission spectrum (ARPES). Taking the low-energy Ni-$(3d_{z^2},3d_{x^2-y^2})$ orbitals placed on the tetragonal lattice structure, we construct a 2D bilayer four-band tight-binding model which well captures the main features of the DFT+$U$ band structure. Then considering the multi-orbital Hubbard interaction, we adopt the random-phase approximation (RPA) approach to investigate the pairing nature. The obtained pairing symmetry is $s^{\pm}$ or $d_{xy}$ for the hole-doping level $\delta$ below or above 0.12, induced by the different Fermi surface nesting situations. For the realistic $\delta=0.21$ measured by the ARPES, our RPA calculations obtain the next-nearest-neighbor pairing $d_{xy}$-wave SC dominated by the $d_{z^2}$ orbital, consistent with the experimental observation that the $T_c$ enhances with the shrinking of the in-plane lattice constants. This pairing state is induced by the nesting between the different patches within the $\gamma$ pocket. Our results appeal for experimental verifications.