phenomenological modeling to provide a more detailed and complete picture of the electronic reconstruction at ideal LaAlO$_3$/SrTiO$_3$ interfaces. We predict the behavior of the sheet carrier density as a function of the LaAlO$_3$ thickness by constructing a continuous model which approximates the LaAlO$_3$ as a homogeneous media. We also use both DFT calculations and model calculations to give a simple yet quantitative picture of the external field effect that has been experimentally realized at the $n$-type interface [@Mannhart-Science-2006; @Triscone-Nature-2008]. In addition, we propose a new class of quantum well systems, based on the polar catastrophe mechanism, at which the spatial extent of the two-dimensional electron gas can be manually controlled.

The remainder of the paper is organized as follows. In Section \[method\], we discuss the technical details of our DFT calculations. We present our first-principles results in Section \[DFT\]. Section \[DFT-symmetric\] gives a brief discussion on the symmetric superlattices. In \[DFT-polar\], we briefly describe the polar catastrophe at various types of interfaces, and we introduce the new quantum well systems. Section \[DFT-bound\] discusses the computation of the on-site and hopping matrix elements and a tight-binding model. An exceptionally large Ti-La hopping is found at the $n$-type interface and its origin 