  Emergent phases in the two-dimensional electron gas (2DEG) formed at the
interface between two insulating oxides have attracted great attention in the
past decade. We present ab-initio electronic structure calculations for the
interface between a Mott insulator GdTiO$_3$ (GTO) and a band insulator
SrTiO$_3$ (STO) and compare our results with those for the widely studied
LaAlO$_3$/SrTiO$_3$ (LAO/STO) interface between two band insulators. Our
GTO/STO results are in excellent agreement with experiments, but qualitatively
different from LAO/STO. We find an interface carrier density of 0.5$e^{-}$/Ti,
independent of GTO thickness in both superlattice and thin film geometries, in
contrast to LAO/STO. The superlattice geometry in LAO/STO offers qualitatively
the same result as in GTO/STO. On the other hand, for a thin film geometry, the
interface carrier density builds up only beyond a threshold thickness of LAO.
The positive charge at the vacuum surface that compensates the 2DEG at the
interface also exhibits distinct behaviors in the two systems. The top GTO
layer is found to be insulating due to correlation-driven charge
disproportionation, while the top LAO layer is metallic within band theory and
may become insulating due to surface disorder or surface reconstruction.
