Magnetization and magnetic susceptibility in nanoscale vertically coupled semiconductor quantum rings

In this paper we computationally examine the magnetization and the magnetic susceptibility for vertically coupled quantum rings (VCQRs) under applied magnetic fields. The theoretical model of VCQRs considers a three-dimensional (3D) effective one-electronic-band Hamiltonian with the position- and energy-dependent effective mass, the finite hard-wall confinement potential, and the Ben Daniel-Duke boundary condition. The nonlinear iterative method is applied to solve the problem in the structure of VCQRs. For the structure formed with nanoscale disk-shaped InAs/GaAs quantum rings, the the tunable states of structure as well as the electron transition energy is dominated by the radius of ring (R) and the inter-distance (d) between quantum rings. The electron energy oscillates non-periodically among the lowest electron states as a function of external magnetic fields due to the penetration of magnetic fields into the inter-regions of VCQRs. The magnetization of VCQRs at zero temperature is non-periodical oscillation and the period of jump is governed by R. Therefore, the differential susceptibility of VCQRs has delta-like paramagnetic peaks. When d is increased, the peak is decreased which is contrary to conventional mesoscopic arguments. Our investigation is constructive for studying the magneto-optical phenomena of the nanoscale semiconductor artificial molecules.