Double loop dc-SQUID as a tunable Josephson diode Open Access

In recent years, there has been a growing interest in the development of superconducting electronics.^1–4 This technology has several interesting applications, such as low energy consumption and extremely high-frequency capabilities.^5–10 Additionally, superconducting electronics is a useful platform for studying fundamental physics related to condensed matter.^11–16 An example of this is the Supercurrent Diode Effect (SDE), which is the property of superconducting two-terminal devices to transfer dissipationless current with different magnitudes in two directions. This phenomenon originates from a multitude of physical effects, including systems with ferromagnetic layers¹⁷ and hybrid semiconductor-superconductor junctions,^18–23 superconducting quantum interference devices (SQUIDs),^24–31 metallic constrictions,^32–34 and metallic strips.^35,36 It has been demonstrated that in SQUID-like supercurrent diodes, the harmonic content of the weak links forming the interferometer plays a significant role in the appearance of SDE when there is no substantial self-inductance.^21,29 This finding opens up possibilities for exploring these nonlinearities through SDE, providing a practical tool for investigating the physics of weak links beyond the single-harmonic Current-to-Phase Relation (CPR).

In this work, we introduce a non-reciprocal supercurrent element, which is realized through a double-loop SQUID (DL-SQUID), and present its theoretical model and experimental implementation. The device can be controlled by magnetic fluxes that thread a double-loop geometry, allowing interference among its three metallic weak links and tuning of the overall CPR. The double loop structure allows us to break the inversion symmetry of the device, enabling non-reciprocal superconducting transport from reciprocal elements. The weak links are realized using superconductor-normal metal-superconductor (SNS) Josephson junctions, which carry a large amount of dissipationless current compared to conventional JJs due to their tunnel-barrier-free structure. The presented DL-SQUID realizes an all-metallic version of the SD studied in Ref. 21, with SQUIDs based on hybrid semiconductor-superconductor Josephson junctions. In that work, the authors can modify the Josephson coupling of their SQUIDs through electrostatic gating. This is an advantage with respect to our device since there they have the possibility of in situ modifying the asymmetry among the junctions, which greatly impacts the rectification performance. On the other hand, in Ref. 21, the double-loop geometry is achieved by an elaborated 4-terminal design, including a total number of six Josephson junctions. Our 2-terminal device realizes the same physical principle by only three SNS junctions, fabricated with a single-step lithography and evaporation process, at expenses of a fixed SQUID asymmetry. We demonstrate theoretically that the presented device can achieve the Josephson Diode Effect (JDE) by breaking both inversion and time-reversal symmetry using external magnetic fields only, producing a theoretical sizable rectification even for a perfectly symmetric device. In our measurement, we can fully explore the parameter space by tuning the magnetic flux of each loop singularly.

Figure 1 displays $η(Φ1,Φ1)$ for four different values of $α1,2$⁠. The reported asymmetry values are as follows: $α1=α2=1$ [panel (a)], $α1=α2=1/2$ [panel (b)], $α1=α2=2$ [panel (c)], and $α1=0.8$ and $α2=1.5$ [panel (d)]. This selection of values aims to represent some physical cases that can occur in a real device. For instance, panel (a) illustrates the rectification efficiency of a fully symmetric device, which means a device with three identical junctions. Notably, rectification is present even in a perfectly symmetrical device, with an apical value of approximately 25%. This is a remarkable fact since to achieve JDE or SDE, the inversion symmetry has to be broken, usually by introducing geometric or intrinsic asymmetries in the system. In the presented interferometer, the inversion symmetry can be broken, even for symmetrical weak links configurations, just through a non-uniform magnetic field by making $Φ1$ and $Φ2$ unequal. Furthermore, we notice two main symmetries of the system that can be found in the present plot (a). The first one is an odd symmetry with respect to the diagonal $Φ1=Φ2$⁠. This means that the transformation $Φ1→Φ2$ and $Φ2→Φ1$ produces a CPR, which is odd with respect to the initial one, hence $I(φ0,Φ1,Φ2)=−I(−φ0,Φ2,Φ1)$⁠. The odd rectification pattern $η(Φ1,Φ2)=−η(Φ2,Φ1)$ is a direct consequence of this fact. The second appreciable symmetry is an even symmetry with respect to the line $Φ1=−Φ2$⁠. Now, the transformation $Φ1→−Φ2Φ2→−Φ1$ produces an even CPR with respect to the initial one, which, in this case, leads to $I(φ0,Φ1,Φ2)=I(φ0,−Φ2,−Φ1)$⁠. The even symmetry rectification $η(Φ1,Φ2)=η(−Φ2,−Φ1)$ descends from this property. We observe a similar pattern in plots (b) and (c), where the lateral junctions are equal but have lower and higher critical current values, respectively, when compared to the central junction. In both cases, we see the same symmetries as in plot (a), but with different apical rectification values. Plot (b) shows a maximum achievable rectification of 36%, while plot (c) shows 24%. It is worth noting that the plots shown so far have been calculated for a symmetric device, where both time-reversal and inversion symmetries are achieved through specific configurations of the magnetic field. However, the last plot on the panel shows a case where all the junctions have different critical currents. This results in the loss of the above-mentioned symmetries but is closer to the physical case. It is important to note that, due to the geometrical asymmetry of the junctions, diode effects are visible even at $Φ1=Φ2$⁠. In the following section, we investigate the impact of these asymmetries on device performance by evaluating the apical rectification $ηMax=Maxϕ1,ϕ2[η]$ as a function of $α1$ and $α2$⁠.

The contour plot in Fig. 2 illustrates the apical rectification value $ηmax$ for a particular $α1$ and $α2$ configuration in the range $0<αi<4$⁠. $ηmax$ is calculated for each configuration of $α1$ and $α2$ by sampling $η(Φ1,Φ2)$ with 6400 points and by extracting the maximum value in the calculated table. From the plot, we notice that the system reaches rectification as high as $∼42%$ for particular asymmetry values, while for some $α1,2$ values, we obtain some areas vanishing rectification. We can see one for $α1=α2=0$⁠, which corresponds to the case where the lateral junctions have vanishing critical currents, representing the case of a single weak link. Other two vanishing rectification points can be found for $α1=0$ and $α2=1$ and vice versa, which corresponds to the case where one lateral junction has a vanishing critical current, while the other is identical to the central one. The latter case corresponds to the single-loop geometry treated in Ref. 29, where we show that with identical branches $η=0$⁠.

As already anticipated, the DL-SQUID shows the intriguing feature of being able to break both inversion and time-reversal symmetry using magnetic field biasing only. This feature is achieved due to the presence of the double-loop geometry, which, even in a perfectly symmetric weak-link configuration, can induce inversion symmetry breaking if $Φ1≠Φ2$⁠. It is hence interesting to investigate $ηmax$ on the symmetry line $α1=α2=1$⁠, hence where the side weak links are equal, and the central one can vary freely. This is displayed in the inset of Fig. 2. Here, the blue curve shows the raw numerical sampling operated on each weak link configuration with 10 000 points on the $η(Φ1,Φ2)$ function. The black dashed curve is then a moving average on three points superimposed on the raw data, to smooth out the sharp features coming from the raw numerical sampling. $ηmax$ vanishes at $α1,2=0$⁠, corresponding to a case where the side weak links are negligibly small with respect to the central one, then has an absolute maximum at $α1,2≈0.65$⁠, and, finally, a second peak at $α1,2≈1.65$⁠, then going monotonically to zero for higher asymmetry values corresponding to the limit of a single symmetric SQUID.

From this analysis, we can deduce some useful concepts. The JDE in this system stems from the high harmonic content of the CPRs used to model the weak links. As such, the harmonic interference of the three branches, together with their relative weights, determines the magnitude of $ηmax$ and the respective $α1,2$ at which the effect appears. Indeed, it can be shown that in the same system, by replacing a KO-1 CPR with a sinusoidal CPR, the JDE is lost regardless of the critical current relative values. By comparing the highest rectification value theoretically obtained in this work, $ηmax=42%$⁠, with the single-loop version investigated in Ref. 29, which reaches $ηmax=26%$⁠, we conclude that the JDE here achieved great benefits from the addition of another weak link. This can be understood by a much more efficient harmonic combination arising from the presence of the third weak link.

Motivated by these theoretical results, we fabricated and tested a physical implementation of the above-described DL-SQUID. The device was realized using the standard Niemeyer-Dolan technique, using a double-layer lithographic mask of MMA + PMMA A4. The deposition of the metal layers was performed using a UHV e-beam evaporator with a residual base pressure of $4.6×10−10$ Torr. During the first evaporation, we deposited a thin Cu layer of 24 nm at an angle of 35°, with a rate of 2 Å/s and a deposition pressure of $<10−11$ Torr. Afterward, a second deposition of 180 nm of Al at 0° and a deposition pressure of $10−8$ Torr were performed to realize the thick Al banks and loops. The Al deposition rate was kept at 1 Å/s for the first 30 nm and then raised to 2 Å/s until the end of the evaporation. The presented fabrication steps produce devices with normal state resistance of $≈10 Ω$⁠.

Figure 4 displays the comparison between the measured switching currents $Isw$ and the theoretically calculated values of the critical currents in our DL-SQUID. The panel shows on the left the experimental data representing $Isw+$ (a), $Isw−$ (c), and their relative color bar indicating the critical current in μA. On the right panels, the theoretically calculated values of $Ic+Ic+,0$ (b) and $Ic−Ic−,0$ (d) are reported for $α1=α2=1$⁠. There, $Ic+,0$ and $Ic−,0$ are the absolute maximum and minimum critical currents evaluated in the parameter space, respectively. To reproduce the experimental data shown in (a) and (c), we made use of Eqs. (6) and (7) to link $Φ1$ and $Φ2$ with the experimental control parameters $Icoil$ and $Iflux$⁠. The mutual inductance between the coil and the loops turns out to be $Mcoil=$ 16 pH, while the mutual inductances between the flux line and the two loops result in $M1=$ 0.9 pH and $M2=$ 0.6 pH, respectively. The latter was estimated from the analytical formula of the mutual inductance between a strip and a square spiral, while the former from the periodicity of the interference pattern of the device. By examining the theoretical and experimental results, we can notice some aspects that deserve deeper consideration. First of all, the predictions for $Ic+Ic+,0$ and $Ic−Ic−,0$ reveal a theoretical maximum suppression of the critical currents of about $Ic+−Ic+−,0≈90%$⁠. This is not the case for the experimental data, where we can record a maximum suppression of about $35%$⁠. The reason for this inefficiency in suppressing the critical current may be related to the length of the weak links, which, for the present device, spans from 90 to 110 nm. The length of the weak link (L) for a SNS junction, indeed, determines the harmonic content of its CPR,³⁸ and depending on the ratio $L/ξ0$⁠, where $ξ0=ℏD/Δw,0$ is the coherence length in the Cu nanowires, one can sit in the multi-harmonic or the sinusoidal CPR regime. For Cu proximitized by Al banks, we set $D≈$ 80 cm²/s as the diffusion coefficient of Cu and $Δw,0=1.764kBTc≈$ 200 μeV as the superconducting gap of Al, leading to a coherence length of $ξ0∼$ 50 nm. Consequently, the ratio $L/ξ0$ spans from $≈1.8$ to $≈2.2$⁠, thereby setting the frame of the intermediate-length junction regime. Our theoretical model based on the KO-1 relation [see Eq. (1)] is valid strictly in the short junction limit, i.e., $L/ξ0≪1$⁠, and, therefore, does not take into account the variation of the harmonic content, which happens when the junction is longer. A reduction in the harmonic content of the weak links, which takes place in the long regime, reflects in a reduced interference among the branches of the DL-SQUID. This said, and considering the unknown asymmetries among the branches, we can appreciate that the experimental data are qualitatively reproduced by our theoretical model if one considers the overall behavior of the curves.

The experimental rectification $η(Φ1,Φ2)$ as a function of the fluxes in the two loops is shown in Fig. 5. The contributions of the single fluxes were de-embedded from the experimental parameters by inverting Eqs. (6) and (7). This allows us to compare the experiment with the simulated value reported in Fig. 1. The data extend mainly around the $Φ1=Φ2$ line since this is in the parameter region explored by sweeping the current in the superconducting coil. Here, $η$ ranges from $+8%$ to $−8%$ in the explored fluxes with a clear periodic feature in the flux quantum. Its behavior is not compatible with the symmetric configurations reported in Fig. 1, suggesting an asymmetry between the three junctions, as reported in Fig. 1(d). Again, the reduced rectification with respect to the KO-1 model is consistent with the reduced harmonic content of our weak links caused by the intermediate-length regime. In this sense, the harmonic content results to be a key ingredient to enhance the JDE.

In conclusion, we have demonstrated a double-loop interferometer based on Al/Cu/Al superconductor-normal metal-superconductor weak links. The KO-1 formula describes these weak links, which provide a multi-harmonic CPR yielding sizable JDE when inserted into a SQUID-like structure. In our research, we used rectification $η$ as a tool to examine the anharmonicity of the weak links within the interferometer. We explored the tunability of the device with respect to magnetic fluxes in the double-loop geometry. We hope this work will encourage the use of JDE to investigate nonlinear superconducting building blocks, leading to further developments in superconducting electronics.

We acknowledge the EU's Horizon 2020 Research and Innovation Framework Programme under Grant Nos. 964398 (SUPERGATE) and 101057977 (SPECTRUM), and the PNRR MUR project PE0000023-NQSTI for partial financial support.

The authors have no conflicts to disclose.

Angelo Greco: Conceptualization (lead); Investigation (lead). Quentin Pichard: Data curation (equal); Formal analysis (equal). Elia Strambini: Investigation (supporting); Methodology (supporting). Francesco Giazotto: Conceptualization (supporting); Funding acquisition (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.