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Thermal hysteresis in a micron-size Superconducting Quantum InterferenceDevice ($\mu$-SQUID), with weak links as Josephson junctions, is an obstaclefor improving its performance for magnetometery. Following the "hot-spot" modelof Skocpol et al. [J. Appl. Phys. {\bf 45}, 4054 (1974)] and by incorporatingthe temperature dependence of thermal conductivity of superconductor using alinear approximation, we find a much better agreement with the observedtemperature dependence of the retrapping current in short superconductingNb-based weak links and $\mu$-SQUIDs. In addition, using the temperaturedependence of the critical current, we find that above a certain temperaturehysteresis disappears. We analyze the current-voltage characteristics and theweak link temperature variation in both the hysteretic and non-hystereticregimes. We also discuss the effect of the weak link geometry in order to widenthe temperature range of hysteresis-free operation.

Hysteresis in superconducting short weak links and $\mu$-SQUIDs