Phase diagram and high density two-dimensional electron gas at the LaAlO$_{3}$/La$_{0.5}$Sr$_{0.5}$TiO$_{3}$/SrTiO$_{3}$ heterostructures

We report a two dimensional electron gas with a high carrier density at the LaAlO$_{\mathrm{3}}$/La$_{\mathrm{0.5}}$Sr$_{\mathrm{0.5}}$TiO$_{\mathrm{3}}$/SrTiO$_{\mathrm{3}}$ heterostructures, reaching a value of about five times higher than that observed at the LaAlO$_{\mathrm{3}}$/SrTiO$_{\mathrm{3}}$ interface. The La$_{\mathrm{0.5}}$Sr$_{\mathrm{0.5}}$TiO$_{\mathrm{3}}$ polar layer is introduced to preserve the degeneracy of the Ti t$_{\mathrm{2g}}$ orbitals and minimize the disorder at the La$_{\mathrm{0.5}}$Sr$_{\mathrm{0.5}}$TiO$_{\mathrm{3}}$/SrTiO$_{\mathrm{3}}$ interface. Various thickness combinations of La$_{\mathrm{0.5}}$Sr$_{\mathrm{0.5}}$TiO$_{\mathrm{3}}$and LaAlO$_{\mathrm{3}}$ layers are used for tuning the total internal potential of the polar layer responsible for the charge transfer. Experimental data showed that the carrier density increases by raising the total internal potential, and this is in a good agreement with a simple electrostatic model. A complete metal-insulator phase diagram is obtained, which shows that at least 3.15 eV polar potential is needed to form the metallic interface at the SrTiO$_{\mathrm{3}}$, providing an estimate for the critical thickness needed for the metallic phase. Nonlinear Hall effect was observed below 60 K which can be understood by multiple filling of the degenerated orbitals responsible for multiple band electronic conductions.