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A thermal model with AC Josephson effect for a shunted superconducting weak-link

Superconducting weak-link (WL), behaving like a Josephson junction (JJ), is fundamental to many superconducting devices such as nanoSQUIDs, single-photon detectors, and bolometers. The interplay between unique nonlinear dynamics and inevitable Joule heating in a JJ leads to new characteristics. Here, we report a time-dependent model incorporating thermal effect in the AC Josephson regime for a Josephson WL shunted by a resistor together with an inductor to investigate the dynamics as well as the resulting current-voltage characteristics. We find that the dynamic regime where phase and temperature oscillate simply widens due to a pure resistive shunt. However, a significant inductive time-scale in the shunt loop, competing with the thermal time-scale, introduces high-frequency relaxation oscillations in the dynamic regime. Based on numerical analysis, we present state diagrams for different parameter regimes. Our model is a guide for better controlling the parameters in the experiments of WL-based devices.