  Metallic LaAlO$_3$/SrTiO$_3$ (LAO/STO) interfaces attract enormous attention,
but the relationship between the electron mobility and the sheet electron
density, $n_s$, is poorly understood. Here we derive a simple expression for
the three-dimensional electron density near the interface, $n_{3D}$, as a
function of $n_s$ and find that the mobility for LAO/STO-based interfaces
depends on $n_{3D}$ in the same way as it does for bulk doped STO. It is known
that undoped bulk STO is strongly compensated with $N \simeq 5 \times
10^{18}~\rm{cm^{-3}}$ background donors and acceptors. In intentionally doped
bulk STO with a concentration of electrons $n_{3D} < N$ background impurities
determine the electron scattering. Thus, when $n_{3D} < N$ it is natural to see
in LAO/STO the same mobility as in the bulk. On the other hand, in the bulk
samples with $n_{3D} > N$ the mobility collapses because scattering happens on
$n_{3D}$ intentionally introduced donors. For LAO/STO the polar catastrophe
which provides electrons is not supposed to provide equal number of random
donors and thus the mobility should be larger. The fact that the mobility is
still the same implies that for the LAO/STO the polar catastrophe model should
be revisited.
