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Spin-orbit interaction in GaAs wells: From one to two subbands

We investigate the Rashba and Dresselhaus spin-orbit (SO) couplings in GaAs quantum wells in the range of well widths $w$ allowing for a transition of the electron occupancy from one to two subbands. By performing a detailed Poisson-Schrödinger self-consistent calculation, we determine all the intra- and inter-subband Rashba ($α_1$, $α_2$, $η$) and Dresselhaus ($β_1$, $β_2$, $Γ$) coupling strengths. For relatively narrow wells with only one subband occupied, our results are consistent with the data of Koralek \emph{et al.} [Nature \bfs{48}, 610 (2009)], i.e., the Rashba coupling $α_1$ is essentially independent of $w$ in contrast to the decreasing linear Dresselhaus coefficient $β_1$. When we widen the well so that the second subband can also be populated, we observe that $α_2$ decreases and $α_1$ increases, both almost linearly with $w$. Interestingly, we find that in the parameter range studied (i.e., very asymmetric wells) $α_2$ can attain zero and change its sign, while $α_1$ is always positive. In this double-occupancy regime of $w$'s, $β_1$ is mostly constant and $β_2$ decreases with $w$ (similarly to $β_1$ for the single-occupancy regime). On the other hand, the intersubband Rashba coupling strength $η$ decreases with $w$ while the intersubband Dresselhaus $Γ$ remains almost constant. We also determine the persistent-spin-helix symmetry points, at which the Rashba and the renormalized (due to cubic corrections) linear Dresselhaus couplings in each subband are equal, as a function of the well width and doping asymmetry. Our results should stimulate experiments probing SO couplings in multi-subband wells.