Proximity dc squids in the long-junction limit

We report the design and measurement of superconducting/normal/superconducting (SNS) proximity dc squids in the long-junction limit, i.e., superconducting loops interrupted by two normal metal wires, which are roughly a micrometer long. Thanks to the clean interface between the metals, a large supercurrent flows through the device at low temperature. The dc squidlike geometry leads to an almost complete periodic modulation of the critical current through the device by a magnetic flux, with a flux periodicity of a flux quantum $h/2e$ through the SNS loop. In addition, we examine the entire field dependence, notably the low and high field dependences of the maximum switching current. In contrast to the well-known Fraunhofer-type oscillations typical of wide junctions, we find a monotonous Gaussian extinction of the critical current at high field. As shown by Cuevas and co-workers [Phys. Rev. B 76, 064514 (2007); Phys. Rev. Lett. 99, 217002 (2007)], this monotonous dependence is typical of long and narrow diffusive junctions. We also find in some cases a puzzling reentrance at low field. In contrast, the temperature dependence of the critical current is well described by the proximity effect theory, as found by Dubos et al. [Phys. Rev. B 63, 064502 (2001); Phys. Rev. Lett. 87, 206801 (2001)] on SNS wires in the long-junction limit. The switching current distributions and hysteretic $IV$ curves also suggest interesting dynamics of long SNS junctions with an important role played by the diffusion time across the junction.

Figure 1

SEM images of the two Nb/Au/Nb SNS dc squids Nb750 and Nb1200. Insufficient etching of the Nb in the central part of the longest SNS squid (right image) causes the normal region to have a length of 1200 nm instead of 1800 nm.

Figure 2

(Color online) SEM pictures of Al/Au/Al SNS wires and SNS dc squids. From top to bottom wires Alw1250 and Alw1300 and from left to right: SNS dc squids Alsq1500, Alsq900, and Alsq1900. The scale bar is $2\mu \text{m}$.

Figure 3

(Color online) Resistance of the two Nb/Au/Nb proximity dc squids versus temperature during cool down with no applied magnetic field. The configuration is a four-wire measurement. The first transition around 9 K is the superconducting transition of niobium. The transition at the lower temperature (around 5 K for the shortest wire and 2 K for the longest one) is the transition of the gold wire from normal to a proximity-induced superconducting state. Temperatures are only approximate above 4 K due to poor calibration of the thermometer in this range.

Figure 4

(Color online) Resistance of three Al/Au/Al wires as a function of the temperature during cool down with no applied magnetic field. Sample Alw1250 is measured in a four-wire configuration, Alw1300 and Alw900 were measured in a three-wire configuration, and constant resistances of 79 and $72\Omega$, respectively, corresponding to the resistance of one lead, have been subtracted. The double transition at 1.6 and 1 K for samples Alw900 and Alw1300 is due to their geometry, in which part of the superconducting wire is made of pure Al with a critical temperature of 1.6 K and part is made of an Al/Au bilayer with a critical temperature of 1 K. Inset: Zoom of the transition to the zero-resistance state of Alw1250 and Alw1300.

Figure 5

(Color online) Typical $V\left(I\right)$ curves of SNS dc squids at two magnetic fields, one in which the interference between both branches is constructive $\left(\Phi /{\Phi }_0=n\right)$ and the other in which the interference is destructive $\left(\Phi /{\Phi }_0=n+1/2\right)$, showing maximum and minimum switching currents. In both cases, the hysteresis is clearly visible and characterized by a large ratio (roughly 3) between switching current $I_s$ and retrapping current $I_r$. The sample measured here is the shortest Au/Nb dc squid Nbsq750. The $V\left(I\right)$ curves for the SNS wires are similar. Inset: $IV$ curve of the same sample at 1.2 K, where there is no hysteresis. The shape of the curve corresponds to the square-root dependence expected in the overdamped regime, i.e., $Q<1$ (continuous line).

Figure 6

(Color online) Differential resistance of the shortest Au/Nb dc squid (Nbsq750) at maximum and minimum switching currents at 60 mK.

Figure 7

(Color online) Left panel: Switching current $I_s$ and width of the histogram [standard deviation (SD)] as a function of temperature for the short Nb dc squid (Nbsq750) at $\Phi =0$. Right panel: ratio of $\text{SD}/I_s$, compared to a linear and a $T^{2/3}$ dependence. Fitting functions (continuous lines) are ${10}^3\text{SD}/I_s=0.1+T^{2/3}$ and ${10}^3\text{SD}/I_s=1.18\left(T+T_0\right)$, with $T_0=128\text{mK}$. Inset: Histogram of the switching current for $\Phi /{\Phi }_0=n+1/2$ for the shortest junction Nb/Au ring at $T=60\text{mK}$. The total number of counts is 2000.

Figure 8

(Color online) Switching and retrapping currents as functions of temperature for both Nb dc squids at the maximal switching current (comparison between experiment and theory). The filled dots and squares are the data points. The dashed lines are fits to the high temperature formula for the proximity effect [see Eq. (1) of Dubos et al. (Ref. 10), from which the Thouless energy is deduced]. The black lines are the numerical solutions to the Usadel equations using the Thouless energy deduced from the high temperature fit. The hysteresis disappears (i.e., the switching and retrapping currents are equal) at a temperature close to $11E_{Th}$.

Figure 9

(Color online) Switching current of Alsq1900 SNS dc squid as a function of the magnetic field at $T=16\text{mK}$ (circles) and fit to Eq. (3) (line) showing the dc squidlike behavior. Here, $I_1=2.1\mu \text{A}$ and $I_2=2.7\mu \text{A}$.

Figure 10

(Color online) Dots: Switching current over three magnetic field periods for the longest Nb dc squid Nbsq1200, which is measured at 60 mK. The lines are the simulated curves corresponding to the same maximum switching current of $70\mu \text{A}$: The black dotted curve corresponds to a noninductive squid with identical junctions (the same critical current), the red dashed-dotted curve corresponds to a noninductive squid with different junctions, and the green line corresponds to a squid with identical junctions but two inductive branches with identical inductance $L=6\text{pH}$.

Figure 11

(Color online) Sample Alsq900; comparison between the positive switching current in the up sweep (increasing current, red square symbols) and the negative switching current in the down sweep (decreasing current, black round sumbols). The black curve with small open dots is the transformation of the down switching curve via the time reversal operation $T\left[I_s^{\text{dn}}\left(H\right)\right]=-I_s^{\text{dn}}\left(-H\right)$ and is seen to be superimposed on $I_s^{\text{up}}\left(H\right)$, as expected from the global time- reversal symmetry of the experiment.

Figure 12

(Color online) Switching the current as a function of the magnetic field for Al SNS dc squid Alsq1500 at 17 mK. The line is a Gaussian. A flux quantum through the normal wire corresponds to a field $H_0=88\text{G}$.

Figure 13

(Color online) Switching the current of the short-junction Nb/Au dc squid Nbsq750 as a function of the magnetic field at 60 mK. The black line is a Gaussian. A flux quantum through the normal wire corresponds to a field $H_0=66\text{G}$.

Figure 14

(Color online) Field dependence of the normalized switching current for all samples at lowest temperatures and comparison with theory. For the samples with the squid geometry, the data points are the local maxima of the full (modulated) curves. The four panels correspond to four different aspect ratios of the normal wires. The continuous lines are the theoretical predictions of the Usadel equations at zero temperature for the exact aspect ratio of the junction. The prediction of the 1D semiclassical calculation [Eq. (6)] is the thin line. The vertical scale is logarithmic and the horizontal axis is the square of the magnetic field normalized by the field $H_0$, which corresponds to a flux quantum through the wire surface.

Figure 15

(Color online) Field dependence of the switching current, as a function of temperature. Left panel: Maximum switching current (extracted from the modulated curves) as a function of the magnetic field for Al SNS dc squid Alsq1500, at temperatures of (from top to bottom) 16, 55, 110, 168, 233, and 304 mK (symbols). The vertical scale is logarithmic, and the horizontal axis is the square of the reduced magnetic field. The dashed lines correspond to a Gaussian decay. Right panel: Full symbols, temperature dependence of the coefficient of the Gaussian decay [coefficient $a$ of Eq. (7)] extracted from the data, compared to the prediction of the Usadel equations (empty symbols).

Figure 16

(Color online) Switching current as a function of the magnetic field for the three Al/Au SNS wires at 17 mK. Switching the current of wire Alw900 has been divided by 2 for better comparison with the two other samples. Two of the three wires show a reentrant behavior.