  We present thermopower measurements on a high electron mobility
two-dimensional electron gas (2DEG) in a thin suspended membrane.We show that
the small dimension of the membrane substantially reduces the thermal
conductivity compared to bulk material so that it is possible to establish a
strong thermal gradient along the 2DEG even at a distance of few micrometers.
We find that the zero-field thermopower is significantly affected by the micro
patterning. In contrast to 2DEGs incorporated in a bulk material, the diffusion
contribution to the thermopower stays dominant up to a temperature of 7 K until
the phonon-drag becomes strong and governs the run of the thermopower. We also
find that the coupling between electrons and phonons in the phonon-drag regime
is due to screened deformation potentials, in contrast to piezoelectric
coupling found with bulk phonons.
