Spin-Hall current and nonlocal transport in ferromagnet-free multiband models for SrTiO₃-based nanodevices in the presence of impurities

We compute the spin-Hall conductance in a multiband model describing the two-dimensional electron gas formed at a LaAlO₃/SrTiO₃ interface in the presence of a finite concentration of impurities. Combining linear response theory with a systematic calculation of the impurity contributions to the self-energy, as well as to the vertex corrections of the relevant diagrams, we recover the full spin-Hall vs sheet-conductance dependence of LaAlO₃/SrTiO₃ as reported in Trier et al. [Nano Lett. 20, 395 (2020)1530-698410.1021/acs.nanolett.9b04079], finding a very good agreement with the experimental data below and above the Lifshitz transition. In particular, we demonstrate that the multiband electronic structure leads to only a partial, instead of a complete, screening of the spin-Hall conductance, which decreases with increasing the carrier density. Our method can be generalized to other two-dimensional systems characterized by a broken inversion symmetry and multiband physics.