  The sensitivity of direct terahertz detectors based on self-mixing of terahertz electromagnetic wave in field-effect transistors is being improved with noise-equivalent power close to that of Schottky-barrier-diode detectors. Here we report such detectors based on AlGaN/GaN two-dimensional electron gas at 77~K are able to sense broadband and incoherent terahertz radiation. The measured photocurrent as a function of the gate voltage agrees well with the self-mixing model and the spectral response is mainly determined by the antenna. A Fourier-transform spectrometer equipped with detectors designed for 340, 650 and 900~GHz bands allows for terahertz spectroscopy in a frequency range from 0.1 to 2.0~THz. The 900~GHz detector at 77~K offers an optical sensitivity about $1~\mathrm{pW/\sqrt{Hz}}$ being comparable to a commercial silicon bolometer at 4.2~K. By further improving the sensitivity, room-temperature detectors would find applications in active/passive terahertz imaging and terahertz spectroscopy. 
Detection of incoherent broadband terahertz light using antenna-coupled   high-electron-mobility field-effect transistors