  We study the low-temperature thermopower of micron sized, free-standing
membranes containing a two-dimensional electron system. Suspended membranes of
320 nm thickness including a high electron mobility structure in Hall bar
geometry of 34 {\mu}m length are prepared from GaAs/AlGaAs heterostructures
grown by molecular beam epitaxy. Joule heating on the central region of the
membrane generates a thermal gradient with respect to the suspension points
where the membrane is attached to cold reservoirs. Temperature measurements on
the membrane reveal strong thermal gradients due to the low thermal
conductivity. We measure the zero-field thermopower and find that the
phonon-drag contribution is suppressed at low temperatures up to 7 K.
