A Unified Understanding of the Experimental Controlling of the T${}_{\text{c}}$ of La${}_3$Ni${}_2$O${}_7$

Recently, a series of experiments have been conducted which control the superconducting T${}_{\text{c}}$ of the bilayer nickelates La${}_3$Ni${}_2$O${}_7$ through tuning the oxygen stoichiometry, the element substitution, the pressure or strain, catching great interests. Here, we provide a unified understanding toward these experiments based on the previously proposed effective $d_{x^2-y^2}$-orbital bilayer $t-J_{\parallel }-J_{\perp }$ model with model parameters input from first-principle calculations. This model exhibits a T${}_{\text{c}}$-controlling behavior well analogous to the hole-doped overdoped cuprates, due to near quarter-filling of the $d_{x^2-y^2}$ orbital. For doping dependence, this mode exhibits a particle-hole asymmetry: The hole (electron) doping makes the system more (less) heavily overdoped and suppresses (enhances) T${}_{\text{c}}$.This character well explains the experimental finding that hole doping introduced through increasing oxygen stoichiometry or alkaline-earth Ca${}^{2+}$/Sr${}^{2+}$ substitution of La suppresses T${}_{\text{c}}$, and the half-dome'' behavior in the oxygen-stoichiometry controlling. For interaction dependence, T$_\text{c}$ follows the variation of $J_\perp$, which well explains the enhancement of bulk T$_\text{c}$ under pressure by Sm/Nd substitution of La, the right-triangle’’ shaped bulk T${}_{\text{c}}$-pressure relation and the enhancement of T${}_{\text{c}}$ with compressive strain in the film. In comparison with weak-coupling theories in which T${}_{\text{c}}$ mainly relies on the density of states and the $d_{z^2}$-orbital dominated pairing mechanism in which T${}_{\text{c}}$ scales with $d_{z^2}$ hole density, our model provides a more natural and unified understanding toward experiments. We propose that electron doping implemented through approaches without inducing disorder, e.g. substitution of La by element with higher valence, can enhance T${}_{\text{c}}$.