  Cryogenic low noise amplifiers based on high electron mobility transistors
(HEMTs) are widely used in applications such as radio astronomy, deep space
communications, and quantum computing, and the physical mechanisms governing
the microwave noise figure are therefore of practical interest. In particular,
the contribution of thermal noise from the gate at cryogenic temperatures
remains unclear owing to a lack of experimental measurements of thermal
resistance under these conditions. Here, we report measurements of gate
junction temperature and thermal resistance in a HEMT at cryogenic and room
temperatures using a Schottky thermometry method. At temperatures $\sim 20$ K,
we observe a nonlinear trend of thermal resistance versus power that is
consistent with heat dissipation by phonon radiation. Based on this finding, we
consider heat transport by phonon radiation at the low-noise bias and liquid
helium temperatures and estimate that the thermal noise from the gate is
several times larger than previously assumed owing to self-heating. We conclude
that without improvements in thermal management, self-heating results in a
practical lower limit for microwave noise figure of HEMTs at cryogenic
temperatures.
