  The transport in complex multiple quantum well heterostructures is
theoretically described. The model is focused on quantum cascade detectors,
which represent an exciting challenge due to the complexity of the structure
containing 7 or 8 quantum wells of different widths. Electronic transport can
be fully described without any adjustable parameter. Diffusion from one subband
to another is calculated with a standard electron-optical phonon hamiltonian,
and the electronic transport results from a parallel flow of electrons using
all the possible paths through the different subbands. Finally, the resistance
of such a complex device is given by a simple expression, with an excellent
agreement with experimental results. This relation involves the sum of
transitions rates between subbands, from one period of the device to the next
one. This relation appears as an Einstein relation adapted to the case of
complex multiple quantum structures.
