The origin of the 2-dimensional electron system (2DES) appearing at the (001) interface of band insulators $rm SrTiO_3$ and $rm LaAlO_3$ has been rationalized in the framework of a polar catastrophe scenario. This implies the existence of a critical thickness of polar $rm LaAlO_3$ overlayer ($4~rm u.c.$) for the appearance of the 2DES: polar catastrophe for thick $rm LaAlO_3$ overlayer is avoided either through a Zener breakdown or a stabilization of donor defects at the $rm LaAlO_3$ surface, both providing electrons to dope the substrate. The observation of a critical thickness is observed in experiments, supporting these hypotheses. Yet, there remains an open debate about which of these possible mechanisms actually occurs first. Using hybrid functional Density Functional Theory, we re-examine these mechanisms at the same level of approximation. Particularly, we clarify the role of donor defects in these heterostructures, and argue that, under usual growth conditions, electric-field driven stabilization of oxygen vacancies and hydrogen adsorbates at the LAO surface occur at a smaller LAO thickness than required for Zener breakdown.
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