Density scaling and optical properties of semiconductor parabolic and square quantum wells

Using self-consistent density-functional calculations of the electronic structure of GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As square and parabolic quantum wells as a function of the well width a, curvature α, and electron concentration ${\mathit{N}}_{\mathit{s}}$, we examine the transition of the density profile in going from the electric quantum limit to multiple subband occupancy. We show that variation of ${\mathit{N}}_{\mathit{s}}$ produces a similar effect on the electronic structure to variation of a (α) for a square (parabolic) well, and we demonstrate a scaling behavior of the density profile (in the square-well case) with respect to these two parameters. We provide a simple formula for ${\mathit{N}}_{\mathit{s}}$ and a values at which the second subband of a square well begins filling. We compute the optical spectrum for the parabolic and square wells. We show that, for a square well, level mixing is small and resonances generally have single-particle character. We show that, for parabolic wells, mode mixing produces a single collective resonance in response to long-wavelength excitations at the bare well frequency. We show that this manifestation of the generalized Kohn theorem is affected by the finite boundaries of a real parabolic well, which allow the electromagnetic wave to access other modes of the interacting system. In particular, we examine this symmetry breaking as the electron concentration is increased and subband filling proceeds.