The processing of weak electromagnetic signals at radio frequencies, as needed in radio astronomical receivers or in qubit read-out, demands for low-noise amplifiers cooled to cryogenic temperatures. This work describes the characterization, modeling, and optimization of high-electron-mobility transistor amplifiers for cryogenic ultra-low-noise operation. As a result, a monolithic 4 – 8 GHz LNA with a noise temperature of 3.6 K is demonstrated.