Geometry-related magnetic interference patterns in long $SNS$ Josephson junctions

We have measured the critical current dependence on the magnetic flux of two long $SNS$ junctions differing by the normal wire geometry. The samples are made by a Au wire connected to W contacts, via focused ion beam assisted deposition. We could tune the magnetic pattern from the monotonic Gaussian-like decay of a quasi-one-dimensional (1D) normal wire to the Fraunhofer-like pattern of a square normal wire. We explain the monotonic limit with a semiclassical 1D model, and we fit both field dependencies with numerical simulations of the two-dimensional Usadel equations. Furthermore, we observe both integer and fractional Shapiro steps. The magnetic flux dependence of the integer steps reproduces as expected that of the critical current $I_c$, while fractional steps decay slower with the flux than $I_c$.

Figure 1

Scanning electron microscope images of samples WAu-Sq (a) and WAu-N (b). One can see as a clear halo the W contamination around the superconducting contacts. (c) Scheme of a $SIS$ Josephson junction. (d) Scheme of the 1D model developed.

Figure 2

Main frame: sample WAu-N normalized critical current vs normalized flux at $T=60$ mK. Light blue line: numerical simulation of the 2D Usadel equations, where the flux has been rescaled by a factor of 2.5 to take into account an imperfect $S/N$ interface. Inset: theoretical predictions for the aspect ratio of junction WAu-N and perfect interfaces: the analytical result of the Usadel equation in the 1D limit $L\gg w$ (red line), the numerical simulation of the 2D Usadel equation (blue line), the semiclassical model for a 1D diffusive normal wire (dotted line), and a Gaussian curve with $\sigma =1$ in the notation of Eq. (7) (dashed line). The Gaussian best fit of the experimental data, which is also a good approximation of the numerical Usadel result, is obtained for $\sigma =0.5$ [as defined in Eq. (7)]. $I_{c,\max }=2.4\mu$A.

Figure 3

Sample WAu-Sq normalized critical current vs normalized flux at $T=60$ mK (blue dots). A trapped flux of 4.3 G has been subtracted. Red line: numerical simulation of the 2D Usadel equation for a junction with aspect ratio $L/w=0.7$ and W wires inductance $\mathcal{L}=11.5$ $p$H. Inset: raw data for $I_c\left(B\right)$, with $I_{c,\max }=38.8\mu$A.

Figure 4

(a) Differential resistance $dV/dI$ of sample WAu-Sq in the presence of an irradiation at frequency $f=2.8$ GHz. (b) dc voltage vs Shapiro step order for sample WAu-Sq; the voltage was directly deduced from the measured $V\left(I\right)$ curves. Lines show the predicted dependence $V=h/2enf$. (c) Normalized flux dependence of Shapiro steps current amplitude for $n=1$ and $n=1/2$ (symbols) compared to the critical current dependence under microwaves, which cause the reduction in $I_c$ amplitude (continuous lines); the trapped flux of $4.3\text{G}$, corresponding to $0.7{\Phi }_0$, is partially compensated by the inductance-related tilt of the $I_c\left(\Phi \right)$ central peak.