Role of background impurities in the single-particle relaxation lifetime of a two-dimensional electron gas

We re-examine the quantum ${\ensuremath{\tau}}_{q}$ and transport ${\ensuremath{\tau}}_{t}$ scattering lifetimes due to background impurities in two-dimensional systems. We show that the well-known logarithmic divergence in the quantum lifetime is due to the nonphysical assumption of an infinitely thick heterostructure and demonstrate that the existing nondivergent multiple scattering theory can lead to unphysical quantum scattering lifetimes in high quality heterostructures. We derive a nondivergent scattering lifetime for finite thickness structures, which can be used both with lowest order perturbation theory and the multiple scattering theory. We calculate the quantum and transport lifetimes for electrons in generic GaAs-AlGaAs heterostructures and find that the correct ``rule of thumb'' to distinguish the dominant scattering mechanisms in GaAs heterostructures should be ${\ensuremath{\tau}}_{t}/{\ensuremath{\tau}}_{q}\ensuremath{\lesssim}10$ for background impurities and ${\ensuremath{\tau}}_{t}/{\ensuremath{\tau}}_{q}\ensuremath{\gtrsim}10$ for remote impurities. Finally we compare theoretical results for ${\ensuremath{\tau}}_{q}$ and ${\ensuremath{\tau}}_{t}$ with experimental data from a GaAs 2DEG in which only background impurity scattering is present. We obtain excellent agreement between the calculations and experimental data and are able to extract the background impurity density in both the GaAs and AlGaAs regions.