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

Phonon heat capacity and disorder: new opportunities for performance enhancement of superconducting devices