Effect of boundary conditions on the energy spectra of semiconductor quantum dots calculated in the effective mass approximation

We consider semiconductor quantum dots shaped as a rectangular prism and a sphere, in the framework of the effective mass approximation. The Schrödinger equation was solved assuming a specular reflection of a particle from the boundary of the quantum dot, which will set the equivalent values for the particle wave function for an arbitrary original point inside the quantum dot and its image in the quantum dot wall. It is shown that the results obtained in this approximation for some classical problems coincide with those based on traditional approaches using “impenetrable walls” or periodic Born-von Karman boundary conditions. Additionally, several problems that are difficult to treat using the traditional approaches could be easily resolved with the suggested mirror-type boundary conditions. It is also shown that this “mirror walls” approach is favorable for effective mass approximation. The comparison of the calculated energy spectra with existing experimental data shows reasonable agreement between the theory and the experiment.