Spin-orbit Interaction and Energy States in Nanoscale Semiconductor Quantum Rings

We study the e ect of spin-orbit iteration on the electron energy spectra in tree-dimensional (3D) nanoscale semiconductor quantum rings. Ultrasmall InAs quantum ring embedded in GaAs matrix is numerically solved with the e ective one electronic band Hamiltonian, the energyand position-dependent electron e ective mass approximation, and the spin-dependent Ben Daniel-Duke boundary conditions. The multishift QR algorithm is implemented in the nonlinear iterative method for solving the corresponding nonlinear eigenvalue problem. It is found that the spin-dependent boundary conditions lead to a spin-splitting of the electron energy states with non-zero angular momentum. The splitting is strongly dependent on the ring dimension. Meanwhile, it is larger than that of quantum dot and demonstrates an experimentally measurable quantity for relatively small quantum rings.