Tunable spin and orbital Edelstein effect at (111) LaAlO$_3$/SrTiO$_3$ interface

Converting charge current into spin current is one of the main mechanisms exploited in spintronics. One prominent example is the Edelstein effect, namely the generation of a magnetization in response to an external electric field, which can be realized in systems with lack of inversion symmetry. If a system has electrons with an orbital angular momentum character, an orbital magnetization can be generated by the applied electric field giving rise to the so-called orbital Edelstein effect. Oxide heterostructures are the ideal platform for these effects due to the strong spin-orbit coupling and the lack of inversion symmetries. Beyond a gate-tunable spin Edelstein effect, we predict an orbital Edelstein effect an order of magnitude larger then the spin one at the (111) LaAlO$_3$/SrTiO$_3$ interface. We model the material as a bilayer of $t_{2g}$ orbitals using a tight-binding approach, while transport properties are obtained in the Boltzmann approach. We give an effective model at low filling which explains the non-trivial behaviour of the Edelstein response, showing that the hybridization between the electronic bands crucially impacts the Edelstein susceptibility.