Full electrostatic control of the band structure of the LaAlO$_{3}$-SrTiO$_{3}$ interface two-dimensional electron system

The two-dimensional electron system at the interface between LaAlO$_{3}$ and SrTiO$_{3}$ has several unique properties that can be tuned by an externally applied gate voltage. Our magnetotransport data on top-gated Hall bars reveals a $d_{yz,xz}$ Lifshitz transition at a carrier density of 2.9 x 10$^{13}$ cm$^{-2}$ and a surprising reduction of $d_{xy}$-type carrier density with gate voltage above this transition. These observations indicate a gate-tunable band structure, which is controlled by the electrostatic confinement. This is supported by self-consistent Schr\"{o}dinger-Poisson calculations, which reproduce the observed reduction of $d_{xy}$-type charge carrier density by including interband electronic correlations. In combination with back-gating, we show that the top-gated $d_{yz,xz}$ Lifshitz transition can be tuned by a back-gate voltage, establishing full electrostatic control of the band structure and confinement of the system. The expected effect of either a top- or back-gate voltage on the boundary conditions of the Schr\"{o}dinger-Poisson model is confirmed by the experimental results.