Temperature dependence of THz detection by field effect transistors was investigated in a wide range of temperatures from 275 K down to 5 K. The important increase of the photoresponse following 1/T functional dependence was observed when cooling from room temperature down to 30 K. At the temperatures below $30 K, the THz response saturated and stayed temperature independent. Similar behavior was observed for GaAs, GaN, and Si based field effect transistors. The high temperature data were successfully interpreted using recent theory of overdamped plasma excitation in field effect transistors. The low temperature saturation of the photoresponse was tentatively explained by the change of the transport regime from diffusive to ballistic or traps governed one. Our results clearly show that THz detectors based on field effect transistors may improve their responsivity with lowering temperature but in the lowest temperatures (below $30 K) further improvement is hindered by the physics of the electron transport itself. V
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