Superconducting gap symmetry of 2DEG at (111)-oriented LaAlO$_3$/SrTiO$_3$ interface

We investigate the superconducting properties of the two-dimensional electron gas at the (111) LaAlO$_3$/SrTiO$_3$ interface. Using a multiorbital tight-binding model defined on a hexagonal lattice, we analyze the emergence of superconductivity driven by both interlayer (nearest-neighbor) and intralayer (next-nearest-neighbor) pairing interactions, with a particular focus on the symmetry of the superconducting gap. We demonstrate that, in both pairing scenarios, the superconducting gap transforms according to the $A_1$ irreducible representation of the $C_{6v}$ point group. Within the interlayer pairing scenario, the superconducting phase is characterized by a fully gapped quasiparticle excitation spectrum exhibiting extended $s$-wave symmetry, accompanied by an enhancement of the superconducting gap magnitude in the vicinity of the van Hove singularity. Conversely, the intralayer pairing channel produces a distinctive double-dome structure in the superconducting phase diagram, with the gap symmetry evolving from a fully gapped, extended $s$-wave at low carrier densities to a nodal extended $s$-wave state at higher electron concentrations. The qualitative agreement with experimentally observed nonmonotonic behavior of the critical temperature $T_c(V_g)$ suggests that intralayer next-nearest-neighbor pairing may play a dominant role in the superconductivity of the (111) LAO/STO interface.