Revisiting Rashba and Dresselhaus spin–orbit interactions in electric and magnetic fields: Exact energy results and approximations

In this paper one deals with the derivation of approximations as well as of exact results concerning the energy of a planar electron subjected to both Rashba and Dresselhaus spin–orbit interactions under the influence of a transversal magnetic field and of an additional in-plane electric field. One begins by applying quickly tractable large nn-approximations, where nn stands for the oscillator quantum number. Reordering leading terms, we found that the energies characterizing combined spin–orbit interactions proceed specifically in terms of concrete selections of the couplings between spin-up and spin-down states. In addition, interpolations between the exact energies of Rashba and Dresselhaus systems can also be proposed. The derivation of exact bound-state energies in magnetic fields proceeds in turn by selecting spin-up and spin-down states in a suitable manner. This amounts to solving cubic equations presented before, but now the interpretations are rather different. Switching on the electric field leads to reasonably accurate energies proceeding in terms of a ten order polynomial equation. Both energy approximations and exact results serve a deeper understanding, as well as for related comparisons.