Band-gap renormalization in semiconductor quantum wells containing carriers

A theoretical calculation is presented of the so-called ‘‘gap renormalization’’ due to free carriers for the quasi-two-dimensional (2D) electrons or holes confined in a semiconductor quantum well. A general theory of the effect is developed assuming parabolic subbands, the Hubbard approximation (random-phase approximation) for the correlation energy, and a model potential containing the well thickness for the effective 2D Coulomb interaction. Results are presented for gap renormalization versus carrier density for GaAs wells of 81 and 217 Å thickness. An experimental measurement of gap renormalization is presented which is based on an analysis of the excitation and luminescence spectra of a p-type modulation-doped Ga(${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$)As multilayer sample of well width 107 Å and hole density 5.3×${10}^{10}$ ${\mathrm{cm}}^{\mathrm{-}2}$. The calculated value is in excellent agreement with the experimental value (6.3 meV) in this case.