  We theoretically study the superconducting properties of multi-band
two-dimensional transition metal oxide superconductors by analyzing not only
the role played by conventional singlet pairings, but also by the triplet order
parameters, favored by the spin-orbit couplings present in these ma- terials.
In particular, we focus on the two-dimensional electron gas at the (001)
interface between LaAlO3 and SrTiO3 band insulators where the low electron
densities and the sizeable spin-orbit couplings affect the superconducting
features. Our theoretical study is based on an extended su- perconducting
mean-field analysis of the typical multi-band tight-binding Hamiltonian, as
well as on a parallel analysis of the effective electronic bands in the
low-momentum limit, including static on-site and inter-site intra-band
attractive potentials under applied magnetic fields. The presence of triplet
pairings is able to strongly reduce the singlet order parameters which, as a
result, are no longer a monotonic function of the charge density. The interplay
between the singlet and the triplet pairings affects the dispersion of
quasi-particle excitations in the Brillouin zone and also induces anisotropy in
the superconducting behavior under the action of an in-plane and of an out-
of-plane magnetic fields. Finally, non-trivial topological superconducting
states become stable as a function of the charge density, as well as of the
magnitude and of the orientation of the magnetic field. In addition to the
chiral, time-reversal breaking, topological superconducting phase, favored by
the linear Rashba couplings and by the on-site attractive potentials in the
presence of an out- of-plane magnetic field, we find that a time-reversal
invariant topological helical superconducting phase is promoted by not-linear
spin-orbit couplings and by the inter-site attractive interactions in the
absence of magnetic field.
