  Recently, we have found an additional spin-orbit (SO) interaction in quantum
wells with two subbands [Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term
is non-zero even in symmetric geometries, as it arises from the intersubband
coupling between confined states of distinct parities, and its strength is
comparable to that of the ordinary Rashba. Starting from the $8 \times 8$ Kane
model, here we present a detailed derivation of this new SO Hamiltonian and the
corresponding SO coupling. In addition, within the self-consistent Hartree
approximation, we calculate the strength of this new SO coupling for realistic
symmetric modulation-doped wells with two subbands. We consider gated
structures with either a constant areal electron density or a constant chemical
potential. In the parameter range studied, both models give similar results. By
considering the effects of an external applied bias, which breaks the
structural inversion symmetry of the wells, we also calculate the strength of
the resulting induced Rashba couplings within each subband. Interestingly, we
find that for double wells the Rashba couplings for the first and second
subbands interchange signs abruptly across the zero bias, while the
intersubband SO coupling exhibits a resonant behavior near this symmetric
configuration. For completeness we also determine the strength of the
Dresselhaus couplings and find them essentially constant as function of the
applied bias.
