  We study the heat relaxation in current biased metallic films in the regime
of strong electron-phonon coupling. A thermal gradient in the direction normal
to the film is predicted, with a spatial temperature profile determined by the
temperature-dependent heat conduction. In the case of strong phonon scattering
the heat conduction occurs predominantly via the electronic system and the
profile is parabolic. This regime leads to the linear dependence of the noise
temperature as a function of voltage bias, in spite of the fact that all the
dimensions of the film are large compared to the electron-phonon relaxation
length. This is in stark contrast to the conventional scenario of relaxation
limited by the electron-phonon scattering rate. A preliminary experimental
study of a 200 nm thick NbN film indicates the relevance of our model for
materials used in superconducting nanowire single-photon detectors.
