Strong magnetic interaction in localized two-dimensional electron gas in LaAlO$_3$/SrTiO$_3$/NdGaO$_3$ heterostructures*

Via reducing the thickness of SrTiO$_3$ layer, strong localization of two-dimensional electron gas is artificially introduced at LaAlO$_3$/SrTiO$_3$ interface grown on NdGaO$_3$ (110) substrates. This localization is characterized by the low carrier density, robust insulating ground state and variable range hopping behavior at low temperature. Given the spatially-limited conducting channel in the thin SrTiO$_3$ layer, the degeneration of Ti 3d orbitals at the interface should be responsible for this strong localization. Moreover, the typical butterfly-like hysteresis loop and unusual anisotropic features in magnetoresistance observed in the thinner SrTiO$_3$ samples indicate the enhanced magnetic interaction in this strongly localized two-dimensional electron gas.