Universal density scaling of disorder-limited low-temperature conductivity in high-mobility two-dimensional systems

We theoretically consider the carrier density dependence of low-temperature electrical conductivity in highquality and low-disorder two-dimensional (2D) "metallic" electronic systems such as 2D GaAs electron or hole quantum wells or doped/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 similar to n(beta(n)) as a function of the 2D carrier density n where the exponent beta(n) is a smooth, but nonmonotonic, function of density n 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 similar to 1, where k(F) similar to root 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) <= 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 a occurring around k(F)d similar to 1. We discuss experimental scenarios for the verification of our theory.