Numerical calculation of electron energy states for nanoscopic InAs/GaAs quantum rings

In this paper we study the electron energy states for nanoscopic semiconductor quantum rings. The effective one-band Hamiltonian approximation and the Ben Daniel-Duke boundary conditions are simultaneously considered in our three-dimensional (3D) model. The rectangular and ellipsoidal torus-shaped rings have been investigated with the 3D model. The proposed model is numerically solved with nonlinear iterative method. This computational method calculates the solution without any fitting parameters and is robust for all simulation cases. For InAs/GaAs quantum rings, it is found that (1) there is a significant energy difference between the 2D and 3D models; (2) the electron energy state depends strongly on the ring shape and size; and (3) the dependency of the energy state on an external magnetic field is different from conventional 1D/2D periodical result. We find the electron energy state nonperiodically oscillates versus the applied magnetic field which is in agreement with the experimental observation.