The evidence of a parabolic potential well in quantum wires and dots was reported in the literature, and a parabolic potential is often considered to be a good representation of the “barrier” potential in semiconductor quantum dots. In the present work, the variational and fractionaldimensional space approaches are used in a thorough study of the binding energy of on-center shallow donors in spherical GaAs-Ga1-xAlxAs quantum dots with potential barriers taken either as rectangular [Vb (eV) = 1.247 x for r R] or parabolic [Vb (r) = β2r2] isotropic barriers. We define the parabolic potential with a β parameter chosen so that it results in the same E0 groundstate energy as for the spherical quantum dot of radius R and rectangular potential in the absence of the impurity. Calculations using either the variational or fractional-dimensional approaches both for rectangular and parabolic potential result in essentially the same on-center binding energies provided the dot radius is not too small. This indicates that both potentials are alike representations of the quantum-dot barrier potential for a radius R quantum dot provided the parabolic potential is defined with β chosen as mentioned above.
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Duque, C.A., Porras-Montenegro, N., de Dios-Leyva, M. et al. Shallow-donor states in spherical quantum dots with parabolic confinement. MRS Online Proceedings Library 692, 9201 (2001). https://doi.org/10.1557/PROC-692-H9.20.1