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

Abstract

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 intersubband Rashba (${\ensuremath{\alpha}}_{1}$, ${\ensuremath{\alpha}}_{2}$, $\ensuremath{\eta}$) and Dresselhaus (${\ensuremath{\beta}}_{1}$, ${\ensuremath{\beta}}_{2}$, $\ensuremath{\Gamma}$) coupling strengths. For relatively narrow wells with only one subband occupied, our results are consistent with the data of Koralek et al. [Nature (London) 458, 610 (2009)], i.e., the Rashba coupling ${\ensuremath{\alpha}}_{1}$ is essentially independent of $w$ in contrast to the decreasing linear Dresselhaus coefficient ${\ensuremath{\beta}}_{1}$. When we widen the well so that the second subband can also be populated, we observe that ${\ensuremath{\alpha}}_{2}$ decreases and ${\ensuremath{\alpha}}_{1}$ increases, both almost linearly with $w$. Interestingly, we find that in the parameter range studied (i.e., very asymmetric wells), ${\ensuremath{\alpha}}_{2}$ can attain zero and change its sign, while ${\ensuremath{\alpha}}_{1}$ is always positive. In this double-occupancy regime of $w$'s, ${\ensuremath{\beta}}_{1}$ is mostly constant and ${\ensuremath{\beta}}_{2}$ decreases with $w$ (similarly to ${\ensuremath{\beta}}_{1}$ for the single-occupancy regime). On the other hand, the intersubband Rashba coupling strength $\ensuremath{\eta}$ decreases with $w$, while the intersubband Dresselhaus $\ensuremath{\Gamma}$ 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 multisubband wells.