Spin-orbit splitting of electronic states in semiconductor asymmetric quantum wells

The spin-orbit splitting in the dispersion relation for electrons in III-V semiconductor asymmetric quantum wells is studied within the standard envelope-function formalism starting from the eight-band Kane model for the bulk. The Rashba spin-orbit splitting in the different subbands is obtained for both triangular and square asymmetric quantum wells. It is shown, for example, that the Rashba splitting in AlAs/GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Al}}_{\mathrm{x}}$As square quantum wells is of the order of 1 meV and presents a maximum as a function of the well width. The splitting of the excited subband in square and triangular quantum wells is shown to be bigger and smaller than the splitting in the first subband, respectively. A simple single-band approach, employing spin-dependent boundary conditions and approximate coupling parameters, is also introduced and its range of validity assessed. The discussion presented clarifies the treatment of abrupt interfaces, the Ando argument against the splitting, and the use of common approximations such as neglecting the barrier penetration or the energy-dependent corrections to the parameters. Good agreement is found with available experimental data.