Screening in modulation-doped quantum wells: Finite-thickness correction

The electrostatic potential of a point charge screened by quantum-confined free carriers in a semiconductor heterostructure is calculated analytically, with inclusion of a spatial extension of the screening charge along the growth axis. The resulting expression is tested by computing numerically the binding energy of excitons between E2 and H1 subbands in a one-side modulation-doped quantum well, giving values in good agreement with experiment. The finite-thickness correction is 2.1 meV for a 150-Å quantum well with a carrier density of ${10}^{11}$ ${\mathrm{cm}}^{\mathrm{-}2}$. It is found that the screening effect can be significantly reversed in wide wells with large carrier concentration due to the band-bending effect.