Numerical Calculation of Electronic Structure for Three-Dimensional Nanoscale Semiconductor Quantum Dots and Rings

In this paper the electronic structure of nanoscale ellipsoid-torus-shaped semiconductor quantum dot and quantum ring is investigated of utilizing a unified model. This three-dimensional model considers the effective one-band Hamiltonian, the position- and energy-dependent effective mass approximation and Landé factor, the finite hard wall confinement potential, and the Ben Daniel-Duke boundary conditions. It is solved numerically without any fitting parameters by using a computationally cost effective nonlinear iterative method. It is found that the penetration of magnetic fields into non-simply connected topology of structures leads to substantial difference in the transition energy between InAs/GaAs quantum dot and quantum ring. The quantum ring exhibits non-periodical electron-hole transition energies when the magnetic field increases. Contrary to the one-dimensional periodical argument on the ring's energy spectra, our examination into the nanoscale semiconductor quantum ring agrees with the experimental result. The energy band gap of quantum dot is an increasing function of the magnetic field. For quantum rings the energy band gap oscillates non-periodically and the oscillation period is strongly controlled by the inner radius of structures. The magnetization of quantum ring not only jumps non-periodically but also saturates eventually when the magnetic field increases.