Spin splitting of holes in symmetric GaAs/GaxAl1−xAs quantum wells

The in-plane Zeeman splitting of holes in symmetric GaAs/Ga${}_{x}$Al${}_{1\ensuremath{-}x}$As quantum wells with various orientations is studied using the Luttinger model. Our calculations extend previous theories by taking fully into account the wave-vector and magnetic-field dependences of the spin $g$ factor as well as an effect of bulk inversion asymmetry. The spin-orbit coupling in two-dimensional systems is known to lock the hole spin parallel to the growth direction, resulting in a weak spin response to the in-plane magnetic field $\mathbf{B}$. We demonstrate that, for higher values of the wave vector $\mathbf{k}$, the hole spin rotates toward the well plane and becomes pinned to the momentum. As a consequence, the in-plane Zeeman splitting strongly increases and is anisotropic with respect to relative orientations of $\mathbf{B}$ and $\mathbf{k}$. We also show that, for low-symmetry growth directions, the $g$ factor at moderate fields is highly asymmetric under the inversion of $\mathbf{B}$. This asymmetry results entirely from the properties of the Luttinger Hamiltonian and has no analog in nonmagnetic systems. The bulk inversion asymmetry is shown to influence the spin splitting mainly at low fields and to produce an additional field-inversion asymmetry of $g$, sensitive to the value of the Dresselhaus parameter ${C}_{k}$. Our calculations of the heavy-hole $g$ factor are in satisfactory agreement with the few available experimental data. We propose additional experiments to verify our theoretical findings.