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.
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