Energy states and magnetization in nanoscale quantum rings

In this paper we calculate electron energy states and magnetization for torus shaped nanoscale quantum rings with external magnetic fields. We use the three-dimensional effective one-band Hamiltonian, the energy and position-dependent quasiparticle effective mass approximation, and the Ben Daniel-Duke boundary conditions. The dependence of the energy spectrum on the sizes and shapes of the quantum rings was calculated and the result agrees with experimental observations. Penetration of the magnetic field into torus region results in an aperiodic oscillation of magnetization at zero temperature. It saturates with the increasing of the magnetic field strength.