Wannier excitons in low-dimensional microstructures: Shape dependence of the quantum size effect

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Abstract
The shape dependence of the quantum size effect of the Wannier exciton is clarified by a simple variational calculation for a model of microcrystal with cylindrical shape. According to the change of the ratio of the radius of the section and the length of the cylinder to the effective Bohr radius of the exciton, the motional state of the exciton changes from three-dimensional to quasi-two-, one-, and zero-dimensional. It is shown that when there is a large anisotropy in the shape of the microstructure, the spatial extension of the exciton wave function along the relatively free coordinate shrinks from the bulk value, reflecting the low dimensionality due to the strong confinement along the axis perpendicular to it.