  Understanding heating and cooling mechanisms in mesoscopic
superconductor-semiconductor hybrid devices is crucial for their application in
quantum technologies. Owing to the poor thermal conductivity of typical
devices, heating effects can drive superconducting-to-normal phase transitions
even at low applied bias, observed as sharp conductance dips through the loss
of Andreev excess currents. Tracking such dips across magnetic field, cryostat
temperature, and applied microwave power, which constitutes Joule spectroscopy,
allows to uncover the underlying cooling bottlenecks in different parts of a
device. By applying this technique, we analyze heat dissipation in devices
based on full-shell InAs-Al nanowires and reveal that superconducting islands
are strongly susceptible to heating as their cooling is limited by the rather
inefficient electron-phonon coupling, as opposed to grounded superconductors
that primarily cool by quasiparticle diffusion. Our measurements indicate that
powers as low as 50-150 pW are able to fully suprpress the superconductivity of
an island. Finally, we show that applied microwaves lead to similar heating
effects as DC signals, and explore the interplay of the microwave frequency and
the effective electron-phonon relaxation time.
