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 high-quality 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 σ(n) varies as σ∼nβ⁽n⁾ as a function of the 2D carrier density n where the exponent β(n) is a smooth, but nonmonotonic, function of density n with possible nontrivial extrema. In particular, the density scaling exponent β(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 β can reach a value as large as 2.7 for k_Fd∼1, where k _F∼√n is the 2D Fermi wave vector and d is the separation of the remote impurities from the 2D layer. Such an exponent β (>5/2) is surprising because unscreened Coulomb scattering by remote impurities gives a limiting theoretical scaling exponent of β=5/2, and naively one would expect β(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 α (β), 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 α occurring around k_Fd∼1. We discuss experimental scenarios for the verification of our theory.