is explained. Section \[oxygen-repulsion\] studies the behavior of oxygen vacancies at the interfaces. Section \[DFT-thickness\] examines the thickness dependence of the sheet carrier density. Simulations of the external field effect are presented in \[DFT-external\]. In Section \[model\], we present a continuous model which gives a simple yet quantitative description of both the sheet carrier density and the external field effect. We conclude in Section \[conclusion\].

Method Details {#method}
==============

Our calculations are performed using density functional theory within the *ab initio* supercell plane-wave approach [@Payne-RMP-1992], with the code PWscf in the Quantum-ESPRESSO package [^1]. We employ the local density approximation (LDA) [@Kohn-PR-1965] and ultrasoft pseudopotentials [@Vanderbilt-PRB-1990], which include semi-core states in Sr, Ti and La. The reference configurations of the pseudopotentials are: Sr $4s^24p^65s^2$ ($r^s_c=2.0 a_0,
r^p_c=1.8 a_0$), Ti $3s^23p^63d^14s^2$ ($r^s_c=r^p_c=r^d_c=1.8
a_0$), La $5s^25p^65d^16s^{1.5}6p^{0.5}$ ($r^s_c=r^d_c=2.2 a_0,
r^p_c=2.0 a_0)$, Al $3s^23p^1$ ($r^s_c=1.8 a_0, r^p_c=1.82 a_0$) and O $2s^22p^4$ ($r^s_c=r^p_c=1.3 a_0$), where $a_0$ is the Bohr radius. The plane wave basis energy cutoff and charge cutoff are 30 Ry and 180 Ry, respectively. We use a Gaussian smearing width of 5 mRy when sampling the Brillouin zone. The $k$-grid sampling of the Brillouin zone is $10\times10\times 1$ where the $z$-axis is orthogonal 