Phonon heat capacity and self-heating normal domains in NbTiN nanostrips

Self-heating normal domains in thin superconducting NbTiN nanostrips with the granular structure were characterized via steady-state hysteretic current–voltage characteristics measured at different substrate temperatures. The temperature dependence and the magnitude of the current, which sustains a domain in equilibrium at different voltages, can only be explained with a phonon heat capacity noticeably less than expected for 3D Debye phonons. This reduced heat capacity coincides with the value obtained earlier from magnetoconductance and photoresponse studies of the same films. The rate of heat flow from electrons at a temperature T e to phonons in the substrate at a temperature T B is proportional to ( T e p − T B p ) with the exponent p ≈ 3, which differs from the exponents for heat flows mediated by the electron–phonon interaction or by escaping of 3D Debye phonons via the film/substrate interface. We attribute both findings to the effect of grains on the phonon spectrum of thin NbTiN films. Our findings are significant for understanding the thermal transport in superconducting devices exploiting thin granular films.