Effect of electrostatic confinement on the dome-shaped superconducting phase diagram at the LaAlO$_3$/SrTiO$_3$ interface

The two-dimensional electron gas (2DEG) at the LaAlO$_3$/SrTiO$_3$ (LAO/STO) interface exhibits gate-tunable superconductivity with a dome-like shape of critical temperature as a function of electron concentration. This behavior has not been unambiguously explained yet. Here, we develop a microscopic model based on the Schr"odinger-Poisson approach to determine the electronic structure of the LAO/STO 2DEG, which we then apply to study the principal characteristics of the superconducting phase within the real-space pairing mean-field approach. For the electron concentrations reported in the experiment, we successfully reproduce the dome-like shape of the superconducting gap. According to our analysis such behavior results from the interplay between the Fermi surface topology and the gap symmetry, with the dominant extended s-wave contribution. Similarly as in the experimental report, we observe a bifurcation effect in the superconducting gap dependence on the electron density when the 2DEG is electrostatically doped either with the top gate or the bottom gate. Our findings explains the dome-shaped phase diagram of the considered heterostucture with good agreement with the experimental data which, in turn, strongly suggest the appearance of the extended s-wave symmetry of the gap in 2DEG at the LAO/STO interface.