O-$p$ states eventually becomes higher than that of the Ti-$d$ states of the embedded SrTiO$_3$ film, and hence an insulating-to-metallic transition occurs. The difference between the quantum well systems and the $n$-type interface is that once the insulating-to-metallic transition happens, the conduction electrons get trapped in the Ti-$d$ states of the embedded SrTiO$_3$ film rather than those of the SrTiO$_3$ substrate. Therefore, by changing the thickness of the embedded SrTiO$_3$ film, we can manually control the spatial extent of the electron gas while holding the number of transferred electrons constant. Consequently, we have two parameters to vary: 1) the thickness of embedded SrTiO$_3$ film to control the spatial extent of electron gas, and 2) the thickness of LaAlO$_3$ capping layer to control the insulating-to-metallic transition. For simplicity, in the following, we focus on one particular subclass in which the embeded SrTiO$_3$ is narrowed down to a single TiO$_2$ layer. The thickness of the LaAlO$_3$ capping layer will be varied. The quantum wells (QW) in the following refer to this particular subclass.

Like the $p$-type interface, at this new class of quantum well systems, the energy gap is not the most direct quantity to monitor for the insulating-to-metallic transition. As the 