  Effective spin-orbit (SO) Hamiltonians for conduction electrons in wurtzite
heterostructures are lacking in the literature, in contrast to zincblende
structures. Here we address this issue by deriving such an effective
Hamiltonian valid for quantum wells, wires, and dots with arbitrary confining
potentials and external magnetic fields. We start from an 8$\times$8 Kane model
accounting for the $s$--$p_z$ orbital mixing important to wurtzite structures,
but absent in zincblende, and apply both quasi-degenerate perturbation theory
(L\"owdin partitioning) and the folding down approach to derive an effective
2$\times$2 electron Hamiltonian. We obtain the usual $k$-linear Rashba term
arising from the structural inversion asymmetry of the wells and, differently
from zincblende structures, a bulk Rashba-type term induced by the inversion
asymmetry of the wurtzite lattice. We also find linear- and cubic-in-momentum
Dresselhaus contributions. Both the bulk Rashba-type term and the Dresselhaus
terms originate exclusively from the admixture of $s$- and $p_z$-like states in
wurtzites structures. Interestingly, in these systems the linear Rashba and the
Dresselhaus terms have the same symmetry and can in principle cancel each other
out completely, thus making the spin a conserved quantity. We determine the
intrasubband (intersubband) Rashba $\alpha_\nu$ ($\eta$) and linear Dresselhaus
$\beta_\nu$ ($\Gamma$) SO strengths of GaN/AlGaN single and double wells with
one and two occupied subbands ($\nu=1,2$). We believe our general effective
Hamiltonian for electrons in wurtzite heterostructures put forward here, should
stimulate additional theoretical works on wurtzite quantum wells, wires, and
dots with variously defined geometries and external magnetic fields.
