Computer simulation of magnetization for vertically coupled nanoscale quantum rings

In this paper we computationally investigate the energy spectra and magnetization for a system consisting of vertically coupled nanoscale semiconductor quantum rings (VCNSQRs) under an external magnetic field B. We use the three-dimensional (3D) effective one-band Hamiltonian, the energy- and position-dependent quasi-particle effective mass approximation, and the Ben Daniel-Duke boundary conditions. For a system consisting of vertically 2-coupled nanoscale InAs/GaAs quantum rings, our 3D simulation finds that its magnetization (M) is non-periodical oscillating function of B due to penetration of B into the torus and vertically coupled regions. It depends not only on the ring´s radius (e.g., base, inner, and outer radius) but also the inter-distance (d) of stacked layers. Jumping period of M is non-periodical and the jumping magnitude is gradually weakened when B is increased. Numerical results provide interesting information for exploring the energy shell structure of vertically coupled nanoscale semiconductor quantum rings. We believe that the study is useful for optoelectronics, spintronics, and quantum Q-bit applications using these structures.