LaAlO$_3$ consists of positively charged (LaO)$^{+}$ and negatively charged (AlO)$^{-}$ atomic layers. When LaAlO$_3$ is grown epitaxially on the SrTiO$_3$ substrate along the (001) direction, an internal electric field directed from (LaO)$^+$ to (AlO$_2$)$^-$ is formed through LaAlO$_3$, resulting in a potential difference that, mathematically, diverges with increasing LaAlO$_3$ thickness. In reality, to offset the diverging potential, an electronic reconstruction is expected to occur, with electrons transferring from the film surface (or a $p$-type interface) to the $n$-type interface. The polar catastrophe mechanism [@Hwang-Nature-2004; @Hwang-NatMat-2006] has been confirmed in density functional theory (DFT) calculations [@Demkov-PRB-2008; @Chen-PRB-2009; @Pickett-PRL-2009; @Son-PRB-2009], and could be responsible for the presence of the two-dimensional electron gas at the atomically sharp $n$-type interface (i.e., with no defects).

However, the polar catastrophe mechanism alone can not explain all of the electronic properties of the LaAlO$_3$/SrTiO$_3$ interface, such as the observed confinement of the transferred electrons within several nanometers of the $n$-type interface [@Basletic-NatMat-2008]. In previous work [@Chen-PRB-2009], we have shown that the electrons are bound to the $n$-type interface as a result of the chemical environment of the interface, which produces a large hopping matrix element between La and Ti.

In this paper, we use both DFT and 