Influence of longitudinal and lateral confinements on excitons in cylindrical quantum dots of semiconductors

The ground-state energy of an exciton in a cylindrical quantum dot with finite potential barriers is calculated by the variation method within the envelope-function approximation. The computations are performed in the cases of infinite and finite band offsets. The limit of a strictly two-dimensional quantum disk is discussed. Our numerical results are particularized to the GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Al}}_{\mathit{x}}$As system with 0.10≤x≤0.30. In this case, the effect of the quantum confinement is the highest for dot dimensions near 50 Å.