Effect of isotropy and anisotropy of the confinement potential on the Rashba spin-orbit interaction for an electron in a two-dimensional quantum dot system

We theoretically study the energetics of the Rashba spin-orbit interaction (SOI) in the two-dimensional (2D) system by comparing the numerical calculation of the exact diagonalization with the analytical calculation based on the perturbation approach and also with the unitary transformed effective Hamiltonian method. The Rashba SOI consists of ls-like and Zeeman-like components, and the out-of-plane application of the external electric field generates the ls-like component, whereas the in-plane application generates the Zeeman-like part. Accordingly, we can separate them by tuning the direction of the applied external electric field. Interestingly, these features do not change provided the confinement is isotropic. The unitary transformation of the total Hamiltonian and the Liouvillian operator expansion technique demonstrate that the Rashba SOI energetics is represented fully in terms of the six orders, in the Rashba coupling. Consequently, the second-order perturbation approach satisfactorily describes the inherent features. When anisotropy is introduced in the confinement, the angular momentum is no longer a good quantum number. The resulting energetics of the Rashba SOI is then unified into the isotropic ground-state type (l = 0).