Effect of bulk inversion asymmetry on [001], [110], and [111] GaAs/AlAs quantum wells

The spin splitting of the conduction band due to the inversion asymmetry of the microscopic crystal potential is predicted to depend linearly on the components of the parallel wave number ${\mathrm{k}}_{\parallel }$ for small $k_{\parallel }$ in undoped GaAs/AlAs quantum wells. The linear dependence is due to the quantization of the confinement levels; the bulk spin splitting depends cubically on $\mathrm{k}$. We have calculated the splitting both numerically and analytically for GaAs/AlAs quantum wells grown along the [001], [110], and [111] directions. In the linear regime the average spin splitting is shown to be about a factor of 2 smaller for [110] quantum wells than for [001]- and [111]-grown quantum wells. The splitting is calculated to be $\Delta E=4\mathrm{}$ meV at $k_{\parallel }=0.07\left(\frac{2\pi }{a}\right)$ for a 25-Å [001]-grown GaAs/AlAs quantum well.