great promise for the development of novel applications in nano-scale oxide electronics [@Cen-NatMat-2008; @Cen-Science-2009]. However, though significant efforts have been made in both theory [@Pickett-PRB-2006; @Pickett-PRB-2008; @Freeman-PRB-2006; @Albina-PRB-2007; @Tsymbal-PRL-2009; @Kelly-EPL-2008; @Demkov-PRB-2008] and experiment [@Willmott-PRL-2007; @Kala-PRB-2007; @Herranz-PRL-2007; @Siemons-PRL-2007; @Sing-PRL-2009; @Salluzzo-PRL-2009; @Thiel-PRL-2009], the origin of the new phases emerging at the LaAlO$_3$/SrTiO$_3$ interface has not been completely resolved yet. In part, this is due to the fact that more than one mechanism may play a role in determining the behavior, and furthermore, the dominant mechanism can depend on external conditions [@Eckstein-NatMat-2007].

One mechanism that is generally believed to play a role in the formation of the two-dimensional electron gas is known as the polar catastophe. This mechanism is a direct result of the charge discontinuity that occurs at an abrupt interface between a nonpolar (SrTiO$_3$) and a polar (LaAlO$_3$) material. Both materials are perovskite oxides with an $ABO_3$ structure that forms alternating planes of $AO$ and $BO_2$ stacked along the (001) direction. Consequently, there are two types of LaAlO$_3$/SrTiO$_3$ interfaces along this direction: TiO$_2$/LaO (known as the $n$-type interface) and SrO/AlO$_2$ (the $p$-type interface) [@Hwang-Nature-2004; @Hwang-NatMat-2006]. In the ionic limit, SrTiO$_3$ is composed of charge neutral atomic layers (SrO)$^0$ and (TiO$_2$)$^0$, while 