of the holes resembles that of a particle in a box, implying that the holes are very weakly bound to the interface. As elucidated in our previous work [@Chen-PRB-2009], the reason for this asymmetry is that the conducting electron gas is trapped at the $n$-type interface due to a large interfacial hopping matrix element. We will discuss this phenomenon further in Section \[DFT-bound\].

### The quantum well system {#sec:quantumwell}

Though the conduction electrons are bound to the $n$-type interface, they still spread over several nanometers. The observed thickness of the electron gas at the $n$-type interface varies widely in experiments, from nanometers to microns, depending on the growth conditions [@Chen-Adv-2010]. Thus it would be desirable to be able to manually control the thickness of the electron gas. Based on the polar catastrophe mechanism, we propose a new class of quantum well systems that will allow this functionality. The quantum well system, which is illustrated in Fig. \[fig:interface\]d, has the following composition: [SrTiO$_3$ ]{}substrate/2 u.c. [LaAlO$_3$ ]{}/$m$ u.c. [SrTiO$_3$ ]{}/[LaAlO$_3$ ]{}capping layer, where $m$ can be varied to control the thickness of the conducting electron gas, as we discuss below. Along the (001) direction, this configuration results in three interfaces: one 