LAUSR

We build a theoretical model for the electronic properties of the two-dimensional (2D) electron gas that forms at the interface between insulating ${\mathrm{SrTiO}}_{3}$ and a number of polar cap layers, including ${\mathrm{LaTiO}}_{3}$, ${\mathrm{LaAlO}}_{3}$, and ${\mathrm{GdTiO}}_{3}$. The model treats conduction electrons within a tight-binding approximation and the dielectric polarization via a Landau-Devonshire free energy that incorporates strontium titanate's strongly nonlinear, nonlocal, and temperature-dependent dielectric response. The self-consistent band structure comprises a mix of quantum 2D states that are tightly bound to the interface and quasi-three-dimensional (3D) states that extend hundreds of unit cells into the ${\mathrm{SrTiO}}_{3}$ substrate. We find that there is a substantial shift of electrons away from the interface into the 3D tails as temperature is lowered from 300 K to 10 K. This shift is least important at high electron densities $(\ensuremath{\sim}{10}^{14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2})$ but becomes substantial at low densities; for example, the total electron density within 4 nm of the interface changes by a factor of two for 2D electron densities $\ensuremath{\sim}{10}^{13}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$. We speculate that the quasi-3D tails form the low-density high-mobility component of the interfacial electron gas that is widely inferred from magnetoresistance measurements.