  We experimentally study for the first time the self-heating normal domain in
magnetic field using nanowires made of granular NbTiN films with sub-10~nm
thicknesses. Specifically, we found that at temperatures below 10 K, the heat
capacity of phonons in such films is reduced with respect to predictions of the
Debye model. Moreover, as the temperature decreases, the phonon heat capacity
reduces quicker than the Debye prediction. These effects strengthen when the
film thickness decreases. We attribute the observed reduction in the heat
capacity to the size effect, which arises from the confinement of phonon modes
within the grains; a phenomenon that has been predicted but not yet observed.
These findings hold great importance in understanding the role of heat
transport in superconducting electronic devices and have the potential for
practical applications in mid-infrared photon sensing with superconducting
nanowire detectors.
