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abstract: 'We present a first-principles study of the electronic structures and properties of ideal (atomically sharp) LaAlO$_3$/ SrTiO$_3$ (001) heterointerfaces and their variants such as a new class of quantum well systems. We demonstrate the insulating-to-metallic transition as a function of the LaAlO$_3$ film thickness in these systems. After the phase transition, we find that conduction electrons are bound to the $n$-type interface while holes diffuse away from the $p$-type interface, and we explain this asymmetry in terms of a large hopping matrix element that is unique to the $n$-type interface. We build a tight-binding model based on these hopping matrix elements to illustrate how the conduction electron gas is bound to the $n$-type interface. Based on the ‘polar catastrophe’ mechanism, we propose a new class of quantum wells at which we can manually control the spatial extent of the conduction electron gas. In addition, we develop a continuous model to unify the LaAlO$_3$/SrTiO$_3$ interfaces and quantum wells and predict the thickness dependence of sheet carrier densities of these systems. Finally, we study the external field effect on both LaAlO$_3$/SrTiO$_3$ interfaces and quantum well systems. Our systematic study of the electronic reconstruction of LaAlO$_3$/SrTiO$_3$ interfaces may serve as a 