arXiv:1807.07720v2 [cond-mat.supr-con] 14 May 2019 Inductive Shunt Eliminates Thermal Bistability in Superconducting Weak Links Sourav Biswas, 1 Clemens B. Winkelmann, 2 Herv´ e Courtois, 2 Thierry Dauxois, 3 Hillol Biswas, 1 and Anjan K. Gupta 1, ∗ 1 Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India 2 Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´ eel, Grenoble, France 3 Univ Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France (Dated: May 15, 2019) The quantum phase-coherent behavior of superconducting weak link (WL) is often masked by the heat dissipation and related thermal hysteresis. The latter can be reduced by improving heat evacuation and reducing the critical current, so that a phase-dynamic regime is obtained; however over a narrow bias-current and temperature range. Here we demonstrate that an inductive shunt with well-chosen shunt parameters introduces an unusual nonlinear dynamics that destabilizes an otherwise stable fixed point in the dissipative branch. This leads to a nonhysteretic behavior with large voltage oscillations in intrinsically hysteretic WL-based micron-size superconducting quantum interference devices. A dynamic thermal model describes quantitatively our observations and further allows us to elaborate on the optimal shunting conditions. Superconducting weak links (WL) [1] acting as Joseph- son junctions are of great interest for a range of quantum applications. A WL is usually probed with a dc current bias in the phase dynamic state [2] so that a dc voltage is measured. In particular, a WL-based micron-size su- perconducting quantum interference device (µ-SQUID) features then a flux-sensitive voltage [3] and can reach a magnetic moment resolution better than 1 µ B [4], which makes it an ultimate probe for quantum nanomag- netism [5–8]. The main limitation to µ-SQUIDs opera- tion resides in the (thermal) hysteresis of their current- voltage characteristics (IVCs) at low temperatures, due to poor heat evacuation from the WL to the bath [9– 12]. A time-dependent Ginzburg-Landau approach cap- turing non-equilibrium effects on the order-parameter re- laxation can model the hysteresis and the phase dynamic regime [13–15] in WLs. A more simple dynamic thermal model (DTM) successfully describes the same behavior by considering both the phase dynamics and the Joule heat evacuation [3, 16]. A resistive shunt [17–19] placed close to a WL can remove thermal hysteresis down to cer- tain temperature. However, for very low temperatures the required small resistor makes the voltage modula- tion in a µ-SQUID miniscule. Further, shunts with a large inductance lead to relaxation oscillations [20] due to substantial delay in current switching. In this Letter, we report on the striking effect of induc- tive shunting on the behavior of a WL-based µ-SQUID. When we use a shunt resistor with an adequate induc- tance in series, we observe reversible IVCs with large volt- age modulations down to 1.3 K. The dynamic retrapping current increases with the inductance, leading to an in- crease in the temperature range of the reversible regime, up to a limiting value above which relaxation oscillations appear. The dynamic thermal model incorporating the nonlinear dynamics of temperature and current, quanti- tatively explains the observations. It is possible to observe a finite voltage state with phase-correlation across a WL such that the bias cur- rent across it is dynamically shared between a normal and super-current. In this dynamic regime, the WL does heat up above the bath temperature T b , due to dissipa- tion from periodic phase slips, but stays below its critical temperature T c [16]; thanks to the efficient heat conduc- tion to the bath. This regime spans over a current range between the dynamic and static retrapping currents, i.e. I dyn r and I h , respectively. For I<I dyn r , the dynamic state is unstable, while for I>I h the WL tempera- ture T WL exceeds T c , leading to a loss of phase corre- lation across the WL [3, 16]. In the DTM, the thermal heat loss from the WL to the substrate is described by k(T WL − T b ). A dimensionless parameter β = I 0 c 2 (T b )RN k(Tc −T b ) then determines the accessibility of the dynamic regime at a given T b . Here I 0 c , R N and k are the zero-field criti- cal current, normal resistance and heat loss coefficient of the WL, respectively. We consider a WL that is resistively and inductively shunted with a resistance R S and an inductance L in se- ries, as shown in Fig. 1(b) inset. The time dependence of the shunt current I sh is described by the equation: L dI sh dt + I sh R S = Φ0 2π dϕ dt with ϕ as the phase difference across the WL. Writing, in addition, an RSJ-type equa- tion and the heat balance in the WL, one obtains the full set of dimensionless equations determining the dynamics of phase, temperature and shunt current [3, 16] ˙ φ = i − (1 − p) sin(2πγφ) − i sh (1) ˙ p = − γ α p + β γ α ˙ φ 2 (2) ˙ i sh = −i sh + r ˙ φ. (3) The relevant time scales are the thermal time τ th = C WL /k, the Josephson time τ J =Φ 0 /I 0 c (T b )R N , and the inductive time τ L = L/R S . Here C WL is the WL heat capacity and I 0 c (T b ) is assumed to be linear with T b . We also introduce the parameters r = R N /R S , γ = τ L /τ J and α = τ th /τ J together with the time unit τ = tγ/τ J , the reduced phase φ = ϕ/(2πγ ) and the reduced tempera-