Electronic structure of three-dimensional triangular torus-shaped quantum rings under external magnetic fields

In this paper, we calculate the electron-hole energy states and the magnetization for InAs/GaAs triangular torus-shaped (TTS) quantum rings in a magnetic field. Our three-dimensional (3D) model considers (i) the effective one-band Hamiltonian approximation, (ii) the position- and energy-dependent quasi-particle effective mass approximation, (iii) the finite hard wall confinement potential, and (iv) the Ben Daniel-Duke boundary conditions. This model is solved numerically with the nonlinear iterative method to obtain the "self-consistent" solutions. We investigate the electron-hole energy spectra versus magnetic field for two different ring widths: R₀ = 20 and 50 nm, and find that they strongly depend on the ring shape and size. Since the magnetic field penetrates into the inside region of the nonsimply connected ring, the electron (hole) transition energy between the lowest states versus magnetic field oscillates nonperiodically and is different from that of quantum dots. We find the magnetization at zero temperature is a negative function, saturates, and oscillates nonperiodically when the magnetic field increases.