being bound to the interface. As the figure shows, they are confined in the [SrTiO$_3$ ]{}, with the density decreasing away from the interface. On the other hand, in the double $p$-type system, the holes reside in both the LaAlO$_3$ and SrTiO$_3$, do not show a strong preference for the interface itself, and have a slightly higher amplitude in the LaAlO$_3$.

Our calculated band offsets place the LaAlO$_3$ valence band maximum $\simeq$ 0.1 eV above the SrTiO$_3$ valence band maximum ($p$-type) and the LaAlO$_3$ conduction band minimum $\sim$ 2 eV above the SrTiO$_3$ conduction band minimum ($n$-type) [@Albina-PRB-2007; @Pickett-PRB-2008]. [*A priori*]{}, one would expect that any introduced carriers would occupy the most energetically favorable band edge available, [*i.e.*]{} holes would migrate to the LaAlO$_3$ valence band edge and electrons to the SrTiO$_3$ conduction band edge. Furthermore, the minimization of kinetic energy would lead to a relatively uniform distribution of carriers. In Fig. \[fig:chargespatial\], we see that, overall, the holes prefer to be in the LaAlO$_3$ at the $p$-type interface, while the electrons reside in SrTiO$_3$ at the $n$-type interface. For the $p$-type interface, the small value of the band offset and the large hole density result in some “leakage” into 