Temperature profile of hotspots in narrow current-biased superconducting strips

The one-dimensional heat flow equation controlling the temperature of a current-driven hotspot (HS) in a long superconducting microbridge is reexamined in all its components. The resulting nonlinear differential system, which admits temperature-dependent thermal conductivities, and a blackbody-like phonon radiation into the substrate, is solved numerically. In this work, the phonon escape rate is not the outcome of a best-fitting procedure, but rather is derived from the dependence, in a pulse experiment, of the HS nucleation time upon the current intensity. As a result, the temperature profile of a self-heating HS in a niobium strip can be computed without any adjustable parameter for each choice of the bath temperature. One notes a severe limitation of the HS temperature as compared to previous models. The minimum current sustaining a stable HS thus determined is in close agreement with direct measurements even far from the critical temperature. The method is applied to a NbN filament typical of the superconducting single photon detectors.