Hole subbands in strained quantum-well semiconductors in [hhk] directions

We give a general formulation for both the Luttinger and the Bir-Pikus Hamiltonians with a spin quantization parallel to the [hhk] direction. This allows one to obtain the hole subbands for any growth direction parallel to [hhk]. The results are given explicitly in the [001], [111], [110], and [112] directions. We show that the axial approximation is exact if the wave vector parallel to the plane (hhk) is equal to zero for (hhk)=(001) (a well-known result) or (111) but that this is not strictly true for (110) or (112). We clarify the link between the axial mass and the cyclotron mass. We then present a method of numerical calculation that is suited to any asymmetric quantum well. We discuss the efficiency and the limits of the Broido-Sham transformation and we show that this transformation is not applicable for the [112] direction even as a rough approximation. Finally, we give the hole subband dispersion in strained ${\mathrm{Cd}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te-CdTe-${\mathrm{Cd}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Mn}}_{\mathit{x}}$Te quantum wells where, according to the growth direction, the quantum well may or may not be piezoelectric.