  We investigate the conductivity of doped graphene in the semiclassical
Boltzmann limit, as well as the conductivity minimum within the self-consistent
transport theory. Using the hard-disk model for a two-dimensional distribution
of impurities gives rise to both strong increase in the slope of conductivity
at low charge carrier densities in graphene and a strongly sub-linear behavior
of the conductivity at high charge carrier densities when the correlation
distance between the impurities is large. On the other hand, we find that a
super-linear dependence of the conductivity on charge carrier density in
heavily doped graphene may arise from increasing the distance of impurities
from graphene or allowing their clustering, whereas the existence of a electric
dipole impurities may give rise to an electron-hole asymmetry in the
conductivity. We show that finite thickness of a dielectric layer in the top
gating configuration, as well as the existence of non-zero air gap(s) between
graphene and the dielectric(s) exert strong influences on the conductivity and
its minimum. While a decrease in the dielectric thickness is shown to increase
the conductivity in doped graphene and even gives rise to finite conductivity
in neutral graphene for a 2D distribution of impurities. We find that an
increase in the dielectric thickness gives rise to a super-linear behavior of
the conductivity when impurities are homogeneously distributed throughout the
dielectric. Moreover, the dependence of graphene's mobility on its charge
carrier density is surprisingly strongly affected, quantitatively and
qualitatively, by the graphene-dielectric gap(s) when combined with the precise
position of a 2D distribution of charged impurities. Finally, we show the
changes in the conductivity minimum in neutral graphene as the correlation
distance between the impurities and the dielectric thickness is altered.
