Origin of the dome-shaped superconducting phase diagram in $\mathrm{SrTiO}_3$-based interfaces

A dome-shaped phase diagram of superconducting critical temperature upon doping is often considered as a hallmark of unconventional superconductors. This behavior, observed in two-dimensional electron gases in $\mathrm{SrTiO}_3$-based interfaces whose electronic density is controlled by field effect, has not been explained unambiguously yet. Here, we elaborate a generic scenario for the superconducting phase diagram of these oxide interfaces based on Schr\"odinger-Poisson numerical simulations of the quantum well and transport experiments on a double-gate field-effect device. We propose that the optimal doping point of maximum $T_c$ marks the transition between a single-band and a fragile two-gap s$\pm$-wave superconducting state involving $t_{2g}$ bands of different orbital character. At the optimal doping point, we predict and observe experimentally a bifurcation in the dependence of $T_c$ on the carrier density, which is controlled by the details of the doping execution. Where applying a back-gate voltage triggers the filling of a high-energy $d_\mathrm{xy}$ subband and initiates the overdoped regime, doping with a top-gate delays the filling of the subband and maintains the 2-DEG in the single-band superconducting state of higher $T_c$.