  We theoretically consider the carrier density dependence of low-temperature
electrical conductivity in high-quality and low-disorder two-dimensional (2D)
`metallic' electronic systems such as 2D GaAs electron or hole quantum wells or
gated graphene. Taking into account resistive scattering by Coulomb disorder
arising from quenched random charged impurities in the environment, we show
that the 2D conductivity \sigma(n) varies as \sigma ~ n^{\beta(n)} as a
function of the 2D carrier density n where the exponent \beta(n) is a smooth,
but non-monotonic, function of density with possible nontrivial extrema. In
particular, the density scaling exponent \beta(n) depends qualitatively on
whether the Coulomb disorder arises primarily from remote or background charged
impurities or short-range disorder, and can, in principle, be used to
characterize the nature of the dominant background disorder. A specific
important prediction of the theory is that for resistive scattering by remote
charged impurities, the exponent \beta can reach a value as large as 2.7 for
k_F d ~ 1, where k_F ~\sqrt{n} is the 2D Fermi wave vector and d is the
separation of the remote impurities from the 2D layer. Such an exponent \beta
(>5/2) is surprising because unscreened Coulomb scattering by remote impurities
gives a limiting theoretical scaling exponent of \beta = 5/2, and naively one
would expect \beta(n) \le 5/2 for all densities since unscreened Coulomb
scattering should nominally be the situation bounding the resistive scattering
from above. We find numerically and show theoretically that the maximum value
of \alpha (\beta), the mobility (conductivity) exponent, for 2D semiconductor
quantum wells is around 1.7 (2.7) for all values of d (and for both electrons
and holes) with the maximum \alpha occurring around k_F d ~ 1. We discuss
experimental scenarios for the verification of our theory.
