A Two-Fluid Model for the Transition Shape in Transition-Edge Sensors

Superconducting microcalorimeters based on transition-edge sensors (TESs) are being successfully used in applications ranging from optical photon counting to gamma-ray and alpha particle spectroscopy. Practical instruments often require a complex optimization among speed, linearity and energy resolution. However, a lack of understanding of the superconducting transition limits our ability to predict the behavior of a new TES design. Specifically, there is an unmet need for a model that predicts the current and temperature dependent resistance surface that describes the transition: R(I,T). This paper describes the predictions of a two-fluid model for the resistance of a TES based on a Ginzburg-Landau form of the critical current. We compare the predictions of the model for the logarithmic derivatives of resistance with temperature and current (alpha and beta) to measurements of TESs used in x-ray and gamma spectrometers. The model shows excellent qualitative agreement that provides useful insight into the dependence of alpha and beta on the current density and bias point of the TES.