Lower-lying states of hydrogenic impurity in lens-shaped and semi-lens-shaped quantum dots

The lower- lying states of a hydrogenic impurity, located at the centre of an infinite barrier lens- shaped quantum dot ( LSQD), are calculated analytically in parabolic rotational coordinates. The solutions are obtained directly using the Frobenius method and by transforming the separated differential equations into the Whittaker equation. Results are given for both symmetric and asymmetric LSQDs. It is found that the energy states of the system are positive for a very small LSQD and decrease as the size of the dot increases. They become negative as the size increases, and approach the energy states of a free hydrogen atom. Also symmetric and antisymmetric eigenfunctions have been constructed for a hydrogenic impurity in a symmetric LSQD. Antisymmetric eigenfunctions can be used as eigenfunctions of a hydrogenic impurity located at the surface of a semi- LSQD.