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

Abstract

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.