  The fundamental limits of the microwave noise performance of high electron
mobility transistors (HEMTs) are of scientific and practical interest for
applications in radio astronomy and quantum computing. Self-heating at
cryogenic temperatures has been reported to be a limiting mechanism for the
noise, but cryogenic cooling strategies to mitigate it, for instance using
liquid cryogens, have not been evaluated. Here, we report microwave noise
measurements of a packaged two-stage HEMT amplifier immersed in normal and
superfluid $^4$He baths and in vacuum from 1.6 - 80 K. We find that these
liquid cryogens are unable to mitigate the thermal noise associated with
self-heating. Considering this finding, we examine the implications for the
lower bounds of cryogenic noise performance in HEMTs. Our analysis supports the
general design principle for cryogenic HEMTs of maximizing gain at the lowest
possible power.
