Nonmonotonic size dependence of the dark/bright exciton splitting in GaAs nanocrystals

The dark/bright exciton splitting Delta(X) in semiconductor nanocrystals is usually caused by electron-hole exchange interactions. Since the electron-hole wave-function overlap is enhanced by quantum confinement, it is generally assumed that Delta(X) increases monotonically as the quantum-dot size decreases. Using atomistic pseudopotential calculations, we show that in GaAs nanocrystals Delta(X) scales nonmonotonically with the nanocrystal size. By analyzing the nanocrystal wave functions in terms of contributions from different k points in the bulk Brillouin zone, we identify the origin of such nonmonotonic behavior in a transition of the lowest conduction-band wave function from Gamma like to X like as the nanocrystal radius decreases below 19 A. The nonmonotonicity arises because the long-range component of the electron-hole exchange interaction all but vanishes when the electron wave function becomes X like. We also show that the direct/indirect transition induced in GaAs nanocrystals by external pressure results in a sudden reduction in Delta(X).