  We present an in-depth study of the spin-orbit (SO) interactions occurring in
inversion-asymmetric two-dimensional hole gases at semiconductor
heterointerfaces. We focus on common semiconductors such as GaAs, InAs, InSb,
Ge, and Si. We develop a semi-analytical variational method to quantify SO
interactions, accounting for both structure inversion asymmetry (SIA) and bulk
inversion asymmetry (BIA). Under certain circumstances, using the
Schrieffer-Wolff (SW) transformation, the dispersion of the ground state heavy
hole subbands can be written as $E(k) = A k^2 - B k^4 \pm C k^3$ where $A$,
$B$, and $C$ are material- and structure-dependent coefficients. We provide a
simple method of calculating the parameters $A$, $B$, and $C$, yet demonstrate
that the simple SW approximation leading to a SIA (Rashba) spin splitting
$\propto k^3$ frequently breaks down. We determine the parameter regimes at
which this happens for the materials above and discuss a convenient
semi-analytical method to obtain the correct spin splitting, effective masses,
Fermi level, and subband occupancy, together with their dependence on the
charge density, dopant type, and dopant concentration for both inversion and
accumulation layers. Our results are in good agreement with fully numerical
calculations as well as with experimental findings. They suggest that a naive
application of the simple cubic Rashba model is of limited use in common
heterostructures, as well as quantum dots. Finally, we find that for the single
heterojunctions studied here the magnitudes of BIA terms are always much
smaller than those of SIA terms.
