is to use a symmetric superlattice approach [@Pickett-PRB-2006; @Pickett-PRB-2008; @Freeman-PRB-2006; @Albina-PRB-2007; @Gemming-Mater-2006; @Tsymbal-JAP-2008; @Tsymbal-PRL-2009; @Satpathy-PRL-2008; @Kelly-EPL-2008], as exemplified by Fig. \[fig:symif\], which shows a simulation cell without vacuum in which a number of [SrTiO$_3$ ]{}unit cells are adjacent to a number of [LaAlO$_3$ ]{}unit cells and periodic boundary conditions are imposed. Therefore, a non-stoichiometric supercell contains two identical [SrTiO$_3$/LaAlO$_3$ ]{}interfaces: [*e.g.*]{} in Fig. \[fig:symif\]a, we have an additional TiO$_2$ layer in [SrTiO$_3$ ]{}and an additional LaO layer in LaAlO$_3$, and in Fig. \[fig:symif\]b, there is an additional SrO layer in [SrTiO$_3$ ]{}and an additional AlO$_2$ in LaAlO$_3$. In this way, the superlattice contains two $n$-type interfaces (TiO$_2$/LaO) or two $p$-type interfaces (SrO/AlO$_2$).

The advantage of the superlattice approach is that no vacuum is needed in the simulation cell, making computation easier. However, due to the imposed symmetry and non-stoichiometry of the LaAlO$_3$ film, this geometry does not result in a polar field, so the evolution of the polar catastrophe can not be modeled. In addition, the non-stoichiometry of the LaAlO$_3$ also imposes a fixed carrier doping: in the ionic limit, an LaO (AlO$_2$) layer has a charge of +1 (-1), and there is one extra electron (hole) present in the 