conduction (valence) band of LaAlO$_3$, which is shared evenly by the two interfaces. Therefore each interface is doped by 0.5 electrons ($n$-type) or 0.5 holes ($p$-type) per two-dimensional unit cell. These values are precisely those needed to fully compensate the polar field of LaAlO$_3$ [@Hwang-NatMat-2006], so the symmetric supercell approach is equivalent to studying the properties of the interfaces when the LaAlO$_3$ film is very thick (infinite thickness limit).

We briefly present results on the symmetric interfaces and highlight some key observations and questions that we will answer in later sections. We show in Fig. \[fig:chargespatial\]a the $xy$-plane integrated *conduction* electron density for the double $n$-type superlattice, and in Fig \[fig:chargespatial\]b the hole density for the double $p$-type superlattice. The method of how to calculate the transferred charge density is detailed in Appendix \[appendix-density\]. The integrated densities are 0.49 electrons per $n$-type interface and 0.49 holes per $p$-type interface, showing that the ionic picture for LaAlO$_3$ is highly accurate, and that one should expect 0.5 electrons (holes) at the fully compensated $n$-type ($p$-type) interface for very thick LaAlO$_3$ films.

The density profiles in Fig. \[fig:chargespatial\] display an interesting asymmetry. One can characterize the electrons in the double $n$-type system as 