Quantum-confinement-induced Γ→Xtransition in GaAs/AlGaAs quantum films, wires, and dots

Large GaAs domains embedded in an ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As matrix act as potential wells for both electrons and holes, resulting in a direct band-gap system. When the GaAs domains become small, however, quantum-confinement effects may push the Γ-like conduction-band state localized on GaAs above the X-like conduction-band state of the ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As alloy, leading to an indirect band-gap system. Using a pseudopotential band-structure method, as well as the conventional one-band effective-mass approximation, we investigate the nature of the direct→indirect (Γ→X) transition in GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum films, wires, and dots. In the case of an isolated GaAs quantum structure embedded in AlAs, we find that the critical size for the onset of the Γ→X transition increases from ∼31 Å in a two-dimensional film through ∼56 Å in a one-dimensional cylindrical wire to ∼80 Å in a zero-dimensional spherical dot. The interaction between GaAs quantum structures tends to reduce the critical size for the Γ→X transition. We further study the effect of the alloy composition on the Γ→X transition, finding that the critical size decreases when the Ga concentration of the alloy increases. In the case of spherical GaAs quantum dots embedded in an ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As alloy, we show that, as a function of the dot radius and the alloy composition, different alignments of the band-edge states lead to different regimes of the lowest-energy optical transition.