the SrTiO$_3$ valence band states.

However, the binding of the electrons at the $n$-type interface cannot be explained by the above arguments. As no polar fields are present in this system to create a potential that will bind the electrons, what mechanism overcomes the kinetic cost inherent in localizing them to the interface? Whatever mechanism is present is absent for the holes, whose spatial profile is relatively uniform in both materials. We provide the answer to this fundamental question in Section \[DFT-bound\], when we discuss the tight-binding hopping matrix elements across both interfaces.

Before we move onto the next section, we would like to give a short discussion on the valence band offset (VBO) of LaAlO$_3$/SrTiO$_3$ interfaces. The VBO of the $n$-type interface has been calculated by several groups [@Albina-PRB-2007; @Lee-MRS-2007; @Pickett-PRB-2008; @Cen-NatMat-2008; @Satpathy-PRL-2008]. Although all of the computed values place the conduction band edge of LaAlO$_3$ significantly higher than that of SrTiO$_3$ at the $n$-type interface, the large range of computed VBO values (shown in Table I), is puzzling. To investigate this issue, we use two different approaches to calculate the VBO of the $n$-type interface and the $p$-type interface. The valence band offset is defined by:

$$\label{vbo}
E_{\textrm{VBO}}=E_{\textrm{V}}^{\textrm{LAO}}-E_{\textrm{V}}^{\textrm{STO}}$$

