  Considering the increasing number of experimental results in the
manufacturing process of quantum dots with different geometries, and the fact
that most numerical methods that can be used to investigate quantum dots with
non-trivial geometries require large computational capacities the finite
element method becomes an incredibly attractive tool for modeling semiconductor
QDs. In the current article, the authors have used the FEM to obtain the first
twenty-six probability densities and energy values for the following GaAs
structures: rectangular, spherical, cylindrical, ellipsoidal, spheroidal, and
conical QDs, quantum rings, nanotadpoles, and nanostars. The results of the
numerical calculations were compared with the exact analytical solutions and a
good deviation was obtained. The ground states energies dependence on the
element size was obtained to find the optimal parameter for the investigated
structures. The abovementioned calculation results were used to obtain valuable
insight into the effects of the size quantization s dependence on the shape of
the QDs. Additionally, the wavefunctions and energies of spherical CdSe/CdS
quantum dots were obtained while taking into account the diffusion effects on
the potential depth with the use of a piecewise Woods-Saxon potential. The
diffusion of the effective mass and the dielectric permittivity is obtained
with the use of a normal Woods-Saxon potential. A structure with a
quasi-type-II band alignment was obtained at the core size of 2.2nm. This
result is consistent with the experimental data.
