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-Schr\"odinger self-consistent calculation, we determine all the intra- and inter-subband Rashba ($\alpha_1$, $\alpha_2$, $\eta$) and Dresselhaus ($\beta_1$, $\beta_2$, $\Gamma$) 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 $\alpha_1$ is essentially independent of $w$ in contrast to the decreasing linear Dresselhaus coefficient $\beta_1$. When we widen the well so that the second subband can also be populated, we observe that $\alpha_2$ decreases and $\alpha_1$ increases, both almost linearly with $w$. Interestingly, we find that in the parameter range studied (i.e., very asymmetric wells) $\alpha_2$ can attain zero and change its sign, while $\alpha_1$ is always positive. In this double-occupancy regime of $w$'s, $\beta_1$ is mostly constant and $\beta_2$ decreases with $w$ (similarly to $\beta_1$ for the single-occupancy regime). On the other hand, the intersubband Rashba coupling strength $\eta$ decreases with $w$ while the intersubband Dresselhaus $\Gamma$ 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.