Shallow-Donor States in Spherical Quantum Dots with Parabolic Confinement

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 [V b (eV) ??1.247 x for r >] or parabolic [Vb (r) ??β2?r2] 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.