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

Thermal hysteresis in a micron-size superconducting quantum interference device ($\mu$-SQUID), with weak links as Josephson junctions, is an obstacle for improving its performance for magnetometry. Following the “hot-spot” model of Skocpol et al. [J. Appl. Phys. 45, 4054 (1974)] and by incorporating the temperature dependence of the superconductor thermal conductivity under a linear approximation, we find a much better agreement with the observed temperature dependence of the retrapping current in short superconducting Nb-based weak links and $\mu$-SQUIDs. In addition, using the temperature dependence of the critical current, we find that above a certain temperature hysteresis disappears. We analyze the current-voltage characteristics and the weak link temperature variation in both the hysteretic and nonhysteretic regimes. We also discuss the effect of the weak link geometry in order to widen the temperature range of hysteresis-free operation.

Figure 1

(a) Scanning electron micrograph (SEM) image of a WL. (b) Sketch of the sample geometry with the three regions discussed in the text.

Figure 2

(Color online) Comparison between exact [Eq. (11), black] and approximate [Eq. (12), red] solution for the reduced temperature profile for $x>x_0$. The respective calculated slopes at the origin are $-5.1$ and $-4.80$. Parameters are $t_b=0.5$ and $x_0=0.015$.

Figure 3

(Color online) (a) Plot of Eq. (14) right-hand side as a function of $x_0$ at different bath temperatures. The minima shown by arrows define the retrapping current $i_r$. (b) Variation in the retrapping current as a function of the bath temperature. (c) $I\text{−}V$ characteristics near $i_r$ at three different bath temperatures as indicated in the figure. The current and the voltage axes are normalized with respect to $I_0$ and $I_0{R}_c$, respectively. All the curves are plotted for $\beta =3.5$ and $x_1=0.015$.

Figure 4

(Color online) Temperature (normalized) evolution with radial distance (normalized) in a WL biased at its retrapping current at different bath temperatures indicated in the figures. The parameters are $x_1=0.015$ and (a) $\beta =3.5$, (b) $\beta =1.5$. (c) and (d) are magnifications near the normal region, corresponding to (a) and (b), respectively. The intersection with the dotted lines at $t=1$ indicate the N-S interface position.

Figure 5

(Color online) Variation in the N-S interface position $r_0$ in units of $r_1$ at the retrapping current as a function of (a) bath temperature $t_b$ for different values of $x_1$ and $\beta$, (b) parameter $\beta$, at $t_b=0.15$ for different values of $x_1$. (c) and (d) show the variation in the WL temperature at retrapping as a function of bath temperature for (c) $\beta =3.5$ and (d) $\beta =1.5$ for $x_1=0.015$ and 0.02. The arrows show the maxima.

Figure 6

Resistance vs temperature curve for sample S2 down to 4.2 K at a bias current of 0.1 mA. Inset shows the SEM image of a typical $\mu$-SQUID with a loop area $3.5\times 3.5\mu {\text{m}}^2$.

Figure 7

(Color online) (a) $I\text{−}V$ curve of sample S1 at four different bath temperatures. The plots of 1.5 K, 4.5 K, and 5.5 K have been shifted upward by 5 mV, 10 mV, and 15 mV, respectively, for clarity. (b) Temperature dependence of the critical and retrapping currents for the same sample.

Figure 8

(Color online) (a)–(d) Experimental (black) and numerical fit (red) of normalized $I\text{−}V$ curves for S1, S2, S3 at $T_b=300\text{mK}$ and WL1 at $T_b=4.2\text{K}$. (e) Variation in the normalized retrapping current with the normalized bath temperature for the six samples. The plots of S2, S3, S4, WL1, and WL2 have been shifted upward by multiples of 0.075 for clarity. The black dots are the data and the red curves are fits based on Eq. (14). The fit parameters and the values of $\beta$ and $x_1$ are listed in Table . The blue curve in (e) for S2 is the fit with Skocpol et al. (Ref. 11) [Eq. (14)] prediction, i.e., $i_r=0.18{\left(1-t\right)}^{1/2}$.

Figure 9

(Color online) Variation in critical current with the bath temperature in the high temperature regime for sample S1. The blue line is a straight line fitting $I_c=I_{c0}\left(1-T_b/T_c\right)$ with $I_{c0}=0.84\text{mA}$ and $T_c=5.05\text{K}$.

Figure 10

(Color online) Temperature dependence of $i_c$ (red) following Eq. (15) and of $i_r$ (black) calculated from our model, near the critical temperature. The parameters used for the plot are given in the figure. At $t_b=t_h$, the two curves cross each other, so that hysteresis disappears for higher temperatures.

Figure 11

(Color online) Variation in the hysteresis crossover temperature $t_h$ with the parameter $\beta$ for three different values of $x_1$ (i.e., WL width). The top axis represents $\ell /w$, calculated using the formula: $\beta =\frac{\pi }{2}\left(1+\frac{\ell }{w}\right)$.

Figure 12

(Color online) Calculation results with parameters $\beta$ and $x_1$ equal to 3.5 and 0.015, respectively, which give $t_h=0.83$. (a) Calculated temperature distribution at a bath temperature $t_b=0.85$ above the threshold temperature $t_h$ for normalized bias currents of 0.024, 0.054 and 0.076. Here $i_r$ is 0.093. (b( and (c) Variation in WL temperature (b) and $i\text{−}v$ curve (c) as a function of bias current for bath temperatures above the threshold $t_b=0.75$ (black) and below $t_b=0.9$ (red). The blue arrows indicate the direction of current sweep.