Recently, evidence of superconductivity (SC) with onset $T_c$ above the McMillan limit has been detected in the ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ ultrathin film grown on the ${\mathrm{LaSrAlO}}_4$ substrate at ambient pressure. This progress opens another 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 $\mathrm{DFT}+U$ 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 $\mathrm{DFT}+U$ band structure. Then considering the multiorbital 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 are appeal for experimental verifications.

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• Pairing mechanisms
• Superconducting order parameter
• Superconductivity
• High-temperature superconductors
• Strongly correlated systems
• Unconventional superconductors
• Random phase approximation