Controlling Hysteresis in Superconducting Weak Links and $\mu $-Superconducting Quantum Interference Devices.*

We have fabricated and studied the current-voltage characteristics of a number of niobium film based weak-link devices and $\mu $-SQUIDs showing a critical current and two re-trapping currents. We have proposed a new understanding for the re-trapping currents in terms of thermal instabilities in different portions of the device. We also find that the superconducting proximity effect and the phase-slip processes play an important role in dictating the temperature dependence of the critical current in the non-hysteretic regime. The proximity effect helps in widening the temperature range of hysteresis-free characteristics. Finally we demonstrate control on temperature-range with hysteresis-free characteristics in two ways: 1) By using a parallel shunt resistor in close vicinity of the device, and 2) by reducing the weak-link width. Thus we get non-hysteretic behavior down to 1.3 K temperature in some of the studied devices.