Electron transition energy for vertically coupled InAs/GaAs semiconductor quantum dots and rings

We investigate the transition energy of vertically coupled quantum dots and rings (VCQDs and VCQRs) with a three-dimensional (3D) model under an applied magnetic field. The model formulation includes (1) the position-dependent effective mass Hamiltonian in the nonparabolic approximation for electrons, (2) the position-dependent effective mass Hamiltonian in the parabolic approximation for holes, (3) the finite hard-wall confinement potential, and (4) the Ben Daniel-Duke boundary conditions. We explore small VCQDs and VCQRs with disk (DI) and conical (CO) shapes. For small VCQDs and VCQRs, the electron-hole transition energy is dominated by the interdistance d which plays a crucial role in the tunable states of structures. Under zero magnetic field, there is about 25% variation in the electron ground state energy for both InAs/GaAs DI-shaped VCQDs and VCQRs with d varying from 0.4 nm to 4.8nm. The energy spectra of the CO-shaped VCQDs are the most stable against the structure interdistance deviations (among dots and rings of the same volume). For a fixed d, VCQDs show diamagnetic shift; contrarily, VCQRs imply a nonperiodical transition among the lowest electron energy states. The energy band gap of VCQRs oscillates nonperiodically between the lowest electron and holes states as a function of external magnetic fields. Our investigation is constructive for studying the magneto-optical phenomena of the nanoscale semiconductor artificial molecules.