The onset of exciton absorption in modulation doped GaAs quantum wells

We study the evolution of the absorption spectrum of a modulation doped GaAs/AlGaAs semiconductor quantum well with decreasing the carrier density. We find that there is a critical density which marks the transition from a Fermi edge singularity to a hydrogen-like behavior. At this density both the lineshape and the transitions energies of the excitons change. We study the density dependence of the singularity exponent $\alpha $ and show that disorder plays an important role in determining the energy scale over which it grows.