Two distinct regimes for the electrical control of the spin–orbit interaction in GaAs wells

•Linear or nonlinear dependence on a gate voltage.•Same or opposite signs of the Rashba couplings.•Persistent-spin-helix symmetry points with two subbands.

We investigate the electrical control of the spin–orbit (SO) interaction in GaAs wells, involving both one- and two-subband electron occupations altered by a gate potential VgVg, over a wide range of well widths w  's. Through the self-consistent Schrödinger–Poisson calculation, we determine all the intrasubband Rashba αναν(ν=1,2)(ν=1,2) and Dresselhaus βνβν, and also the intersubband Rashba η and Dresselhaus Γ   couplings. We observe two distinct regimes marked off around w=wc=30−35nm. In the first regime with wwcw>wc, we observe that α1 and α2 can either have the same sign or opposite signs depending on VgVg. In addition, even though α1 sensitively depends on VgVg, we find that α2 remains essentially constant within a certain gate voltage range. As for the Dresselhaus couplings in this second regime, the inequality β1<β2β1<β2 (valid in the first regime) only holds near the symmetric configuration of our wells, but otherwise the inequality tends to be reversed. On the other hand, we find that both the intersubband Rashba η and Dresselhaus Γ   couplings change almost linearly with VgVg in the first regime while in the second one a maximum of |η||η| occurs near the symmetric configuration. We also determined the persistent-spin-helix symmetry points of the two subbands, where the Rashba and the renormalized (due to cubic corrections) Dressehaus couplings are matched. We find that the helix symmetry for the second subband retains over a broad range of VgVg's, thus possibly facilitating its locking in practice. Our results should be essential for experiments pursuing an universal electrical control of the SO interaction in semiconductor nanostructures.