Band-gap renormalization in semiconductor quantum wells

We develop a theory for the many-body electron-electron– and electron-phonon–interaction–induced band-gap renormalization of quasi-two-dimensional electron systems by calculating the electron and hole dynamical self-energy corrections at the band edges of semiconductor quantum-well structures. The self-energy calculations are performed within the random-phase approximation (RPA) by treating the electron-electron Coulomb interaction and the electron–optical-phonon Fröhlich interaction on an equal footing. We calculate the band-gap renormalization as a function of the carrier density and the quantum-well width and obtain good agreement with existing experimental results. We find that, when the carrier density and the well width are expressed in terms of effective dimensionless variables by scaling them with the effective Bohr radius for the material, the dimensionless–band-gap renormalization expressed in units of effective rydberg shows a universality, independent of the band-structure details of the material and dependent only on ${\mathit{r}}_{\mathit{s}}$, the dimensionless interparticle separation, and on the dimensionless well width. We show that this two-parameter universality can be reduced to an approximate one-parameter universality by choosing suitable quasi-two-dimensional Bohr radius and effective rydberg as effective length and energy scaling units, respectively. Finally, our complete RPA calculation shows that the popular and easy-to-use plasmon-pole approximation may not be quantitatively accurate in quasi-two-dimensional systems whereas the usual ${\mathrm{\epsilon }}_0$ approximation employed to include approximately the effects of Fröhlich interaction in the theory works extremely well for the weakly polar III-V compound semiconductor materials studied in this paper.