Dependence of energy gap on magnetic field in semiconductor nano-scale quantum rings

We study the electron and hole energy states for a complete three-dimensional (3D) model of semiconductor nano-scale quantum rings in an external magnetic field. In this study, the model formulation includes: (i) the position dependent effective mass Hamiltonian in non-parabolic approximation for electrons, (ii) the position dependent effective mass Hamiltonian in parabolic approximation for holes, (iii) the finite hard wall confinement potential, and (iv) the Ben Daniel–Duke boundary conditions. To solve this 3D non-linear problem, we apply the non-linear iterative method to obtain self-consistent solutions. We find a non-periodical oscillation of the energy band gap between the lowest electron and hole states as a function of external magnetic fields. The result is useful in describing magneto-optical properties of the nano-scale quantum rings.