is likely that the lattice constants have minute differences (due to different pseudopotentials), which causes a variation in strain and probably affects the alignment of valence band edges. Another possibility is that we, as well as a few other groups [@Albina-PRB-2007; @Lee-MRS-2007; @Pickett-PRB-2008; @Satpathy-PRL-2008], use symmetric nonstoichiometric superlattices, in which the macroscopic average potentials are flat in both LaAlO$_3$ and SrTiO$_3$, to perform the calculations. However, if a stoichiometric LaAlO$_3$ slab is employed [@Cen-NatMat-2008] so that a net internal electric field results, the determined VBO will likely be different from the symmetric nonstoichiometric case. Although our results for the $n$-type interface agree well with the available experiments, the value as well as variations of the VBO for the $n$-type interface is not a closed subject and needs further work. In contrast, the computed values of the VBO for the $p$-type interface are in general agreement and $\simeq$ 0.1 eV.

The polar catastrophe {#DFT-polar}
---------------------

### $n$-type, $p$-type and $np$-type interfaces

As discussed in the Introduction, there are two types of LaAlO$_3$/SrTiO$_3$ interfaces: the $n$-type (TiO$_2$/LaO) interface and the $p$-type (SrO/AlO$_2$) interface. If a stoichiometric LaAlO$_3$ layer on an SrTiO$_3$ substrate also has a SrTiO$_3$ capping layer, then we have both 