Electrical control of the spin-orbit coupling in GaAs from single to double and triple wells

We consider a realistic GaAs/Al0.3Ga0.7As well, inside which there are either one or two additional AlxGa1 − xAs barriers embedded, with two occupied electron subbands ν = 1, 2. By varying the Al content x in the AlxGa1 − xAs layer, we investigate the electrical control of the spin-orbit (SO) interaction, i.e., intrasubband Rashba (Dresselhaus) αν (βν) and intersubband Rashba (Dresselhaus) η (Γ), in the course of the transition of our system from single to double and triple wells. At x = 0 (single well), the scenario of SO terms is usual, e.g., α1 and α2 have the same sign and both change almost linearly as functions of an external gate voltage Vg. In contrast, when x away from zero and only one AlxGa1 − xAs barrier embedded (double well), α1 and α2 tend to have opposite signs, and α2 first increases with Vg, while peaks at some point depending on x. For a larger value of x, α2 increases with Vg more abruptly till it peaks. As opposed to α1 and α2, the intrasubband Dresselhaus terms βν have a relatively weak dependence on Vg, and β1 and β2 become close as x increases. As for the intersubband SO terms, at x = 0, the Rashba coupling η remains essentially constant, while the Dresselhaus Γ changes almost linearly with Vg. When x is nonzero, on the one hand, both η and Γ have a sensitive dependence on Vg near the symmetric configuration; on the other hand, right at the symmetric configuration η exhibits the highest while Γ vanishes. In the case of our system having two additional AlxGa1 − xAs barriers (triple well), we find that the gate dependence of SO terms becomes more smooth and βν becomes more stronger. The persistent-spin-helix symmetry of the two subbands is also discussed. These results are expected to be important for a broad control of the SO interaction in semiconductor nanostructures.