  The origin of the two-dimensional electron gas (2DEG) in the interface
between $\gamma$-Al$_2$O$_3$ (GAO) and SrTiO$_3$ (STO) (GAO/STO) as well as the
reason for the high mobility of the 2DEG is still in debate. In this paper, the
electronic structures of [001]-oriented GAO/STO heterostructures with and
without oxygen vacancies are investigated by first-principle calculations based
on the density functional theory. The calculation results show that the
necessary condition for the formation of 2DEG is that the GAO/STO
heterostructure has the interface composed of Al and TiO$_2$ layers. For the
heterostructure without oxygen vacancy on the GAO side, the 2DEG originates
from the polar discontinuity near the interface, and there is a critical
thickness for the GAO film, below which the 2DEG would not present and the
heterostructure exhibits insulator characteristics. For the case that only the
GAO film contains oxygen vacancies, the polar discontinuity near the interface
disappears, but the 2DEG still exists. In this situation, the critical
thickness of the GAO film for 2DEG formation does not exist either. When the
GAO film and STO substrate both contain oxygen vacancies, it is found that the
2DEG retains as long as the oxygen vacancies on the STO side are not very close
to the interface. The low-temperature mobilities of the 2DEGs in these GAO/STO
heterostructures are considered to be governed by the ionized impurity
scattering, and $\sim$3 to $\sim$11 times as large as that in
LaAlO$_3$/SrTiO$_3$ heterojunction. The high mobility of the 2DEG is mainly due
to the small electron effective mass in GAO/STO heterostructure.
