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The discovery of high-$T_c$ superconductivity (SC) in La$_3$Ni$_2$O$_7$ (LNO)has aroused a great deal of interests. Previously, it was proposed that theNi-$3d_{z^2}$ orbital is crucial to realize the high-$T_c$ SC in LNO: Thepreformed Cooper pairs therein acquire coherence via hybridization with the$3d_{x^2-y^2}$ orbital to form the SC. However, we held a different viewpointthat the interlayer pairing $s$-wave SC is induced by the $3d_{x^2-y^2}$orbital, driven by the strong interlayer superexchange interaction. To includeeffects from both $E_g$-orbitals , we establish a two-orbital bilayer $t$-$J$model. Our calculations reveal that due to the no-double-occupancy constraint,the $3d_{x^2-y^2}$ band and the $3d_{z^2}$ bonding band are flattened by afactor of about 2 and 10, respectively, which is consistent with recentangle-resolved-photo-emission-spectroscopy measurements. Consequently, a hightemperature SC can be hardly induced in the $3d_{z^2}$-orbital due to thedifficulty to develop phase coherence. However, it can be easily achieved bythe $3d_{x^2-y^2}$ orbital under realistic interaction strength. With electrondoping, the $3d_{z^2}$-band gradually dives below the Fermi level, but $T_c$continues to enhance, suggesting that it is not necessary for the high-$T_c$ SCin LNO. With hole doping, $T_c$ initially drops and then rises, accompanied bythe crossover from the BCS to BEC-type superconducting transitions.

Interplay of two $E_g$ orbitals in Superconducting La$_3$Ni$_2$O$_7$ Under Pressure