Static and dynamic transport properties of multi-terminal, multi-junction microSQUIDs realized with Nb/HfTi/Nb Josephson junctions

We consider the 4JJSQ as schematically depicted in figure 1, with four ports (p = 1,..., 4). Three independent currents I₁, I₂ and I₄ can be injected via ports 1, 2 and 4, respectively. The current I₃ to port 3 is used as a drain, i.e. we have I₃ = I₁ + I₂ + I₄. Each loop segment between neighboring ports contains one JJ; i.e. the SQUID loop is intersected by four JJs (k = 1,...,4). We consider I₁ as a bias current and I₂ and I₄ as control currents, and we will discuss how the critical current I_c1 (from port 1 to 3) and the voltage V_SQ (between port 1 and 3) will change with the control currents and with Φ_ext.

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The Josephson current through the kth junction is I_J,k = I₀ a_k sin(δ_k ), with asymmetry parameters a_k and gauge-invariant phase differences δ_k of the superconducting wave functions across the kth JJ. Here, we define I₀ as the average (I_0,left-min + I_0,right-min)/2 of the (noise-free) critical currents of those two JJs which have the lower critical current in the left and right SQUID arm, respectively. Accordingly, the sum of the asymmetry parameters of those two 'weakest JJs per SQUID arm' is 2. This choice for the definition of I₀ ensures that the (noise-free) critical current I_c1 vs. Φ_ext for I₂ = I₄ = 0 of the 4JJSQ has a maximum I_c1,max = 2I₀.

Normalizing currents to I₀, we obtain in normalized form for the Josephson currents i_J,k = a_k sin(δ_k ). To capture also dynamic properties of the 4JJSQs, we describe each junction within the resistively and capacitively shunted junction (RCSJ) model [31–33], i.e. we also consider quasiparticle currents through the junction resistance R, and displacement currents across the junction capacitance C. For simplicity, we assume that R and C are the same for all four JJs. The SQUID loop shall have a total inductance L = Σ_k=1 ⁴ L_k , where L_k denote the inductances of the loop segments between the four ports (see figure 1).

In normalized form the RCSJ equations read:

$$\begin{equation}{\beta _{\text{C}}}{\partial _t}^2{\delta _k} + {\partial _t}{\delta _k} + {a_k}\sin \left( {{\delta _k}} \right) + {i_{{\text{N}},k}} = {j_k},\;\;{\text{for}}\;\;k = 1,2,\end{equation} \tag{ 1a }$$

$$\begin{equation}{\beta _{\text{C}}}{\partial _t}^2{\delta _k} + {\partial _t}{\delta _k} + {a_k}\sin \left( {{\delta _k}} \right) + {i_{{\text{N}},k}} = - {j_k},\;\;{\text{for}}\;\;k = 3,4.\end{equation} \tag{ 1b }$$

Here, β_C = 2πI₀ R² C/Φ₀ is the Stewart-McCumber parameter with the magnetic flux quantum Φ₀ = h/2e ≈ 2.0678 · 10⁻¹⁵ Vs. In equation (1) time is normalized to Φ₀/(2πI₀ R) and currents are normalized to I₀. ∂_t indicates derivatives with respect to the normalized time. The currents i_N,k denote normalized noise currents with (normalized) spectral density 4Γ, where Γ = 2πk_B T/(I₀Φ₀) is the noise parameter.

The normalized currents j_k can be found from the conditions i₁ = j₁ − j₄, i₂ = j₂ − j₁, i₃ = i₁ + i₂ + i₄ = j₂ − j₃ and i₄ = j₄ − j₃. With the definition of an average circulating current j_SQUID = Σ_k =1⁴ j_k /4 one obtains:

$$\begin{equation}{j_1} = {j_{{\text{SQUID}}}} + {i_1}/2 - {i_2}/4 + {i_4}/4,\end{equation} \tag{ 2a }$$

$$\begin{equation}{j_2} = {j_{{\text{SQUID}}}} + {i_1}/2 + 3{i_2}/4 + {i_4}/4,\end{equation} \tag{ 2b }$$

$$\begin{equation}{j_3} = {j_{{\text{SQUID}}}} - {i_1}/2 - {i_2}/4 - 3{i_4}/4,\end{equation} \tag{ 2c }$$

$$\begin{equation}{j_4} = {j_{{\text{SQUID}}}} - {i_1}/2 - {i_2}/4 + {i_4}/4.\end{equation} \tag{ 2d }$$

Using (in dimensioned units) Σ_{k =1} ⁴ L_kJ_k + Φ_ext = (Φ₀/2π) [δ₃ + δ₄ − δ₁ − δ₂], the current j_SQUID can be related to the phase differences δ_k via:

$$\begin{equation}{j_{{\text{SQUID}}}} = \left[ { - {\delta _1} - {\delta _2} + {\delta _3} + {\delta _4} - 2\pi \left( {{\varphi _{{\text{ext}}}} + {\varphi _{\text{b}}}} \right)} \right]/\left( {\pi {\beta _{\text{L}}}} \right),\end{equation} \tag{ 3 }$$

with the screening parameter β_L = 2I₀ L/Φ₀. Here, ϕ_ext is the magnetic flux (normalized to Φ₀) applied to the SQUID loop and ϕ_b is the normalized flux due to additional fluxes generated by the normalized bias current i₁ and control currents i₂ and i₄ in the case of asymmetric inductances,

$$\begin{align}{\varphi _{\text{b}}} = {\beta _{\text{L}}}/2 \cdot &\left[ \left( {{l_1} \kern-0.4pt +\kern-0.4pt {l_2} \kern-0.4pt-\kern-0.4pt {l_3}\kern-0.4pt -\kern-0.4pt {l_4}} \right) \cdot {i_1}/2 \kern-0.4pt+\kern-0.4pt \left( {3{l_2} \kern-0.4pt-\kern-0.4pt {l_1} \kern-0.4pt- \kern-0.4pt{l_3} \kern-0.4pt-\kern-0.4pt{l_4}} \right) \cdot {i_2}/4 \right.\nonumber\\ &\ \left.+ \left( {{l_1} + {l_2} + {l_4} - 3{l_3}} \right) \cdot {i_4}/4 \right],\end{align} \tag{ 4 }$$

where l_k = L_k /L.

We solve the equation (1) using a 5th order Runge Kutta method, together with equations (2 a)–(4). The time derivatives ∂_t δ_k equal the normalized voltages u_k = U_k /(I₀ R) across the four junctions, and from that we obtain the time averaged normalized voltages v_k . The time averaged voltage across the SQUID (between port 1 and 3) is v_SQ = v₁ + v₂ = v₃ + v₄. To record an IVC we typically start at zero current with initial conditions δ_k = u_k = 0, let the system relax over 10³–10⁴ time units and then perform time averaging over 10³–10⁴ time units to obtain v_k . Then the current is changed by a small step and the procedure is repeated using the final values of δ_k and u_k of the previous step as the new initial conditions. The SQUID critical current is determined from i₁ vs. v_SQ characteristics by applying a voltage criterion v_crit = 0.05, which was chosen to be close to the experimental voltage criterion (see section 3.2).

The simulations discussed in this paper are for β_C = 0. We further use l_k = 0.25 for all k and i₄ = 0.

To connect to the theoretical predictions shown in [28] we briefly address the symmetric 4JJSQ (a_k = 1) at β_L = 1. We first set the noise parameter to a very small value, Γ = 10⁻⁶. For the perfectly symmetric 4JJSQ, Γ = 0 would produce solutions that are instable against small perturbations.

Figure 2(a) shows the normalized critical current i_c1 vs. ϕ_ext, calculated for five different values of i₂ = 0, ±0.6 and ±1.2. This graph agrees very well with figure 2(b) of [28]. The dependence i_c1 vs. i₂ is shown in figure 2(b) for three values of ϕ_ext = 0, 0.25 and 0.5 and both polarities of i₁. This graph can be compared with figure 3 of [28]. While the overall agreement is very good, there is a difference for i₂ < −1.5 (at i_c1 > 0), where i_c1 in our dynamic simulations shows an increase with decreasing i₂, while i_c1 in [28] continues to decrease, as indicated by the dashed blue and green lines in figure 2(b).

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The differences can be understood from the IVC shown in figure 3(a) for i₂ = −2 and ϕ_ext = 0. The current i₁ was swept in steps of 0.02 using the sequence 0 → 6 → −6 → 0. The SQUID IVC v_SQ vs. i₁ shows a clear supercurrent up to i₁ = 1.96 at positive bias. However, when plotting the individual voltages v_k across junctions 1–4 vs. current i₁ (figure 3(b)) one finds that for currents between 0.06 and 1.96 junctions 1 and 2 are in the resistive state with compensating voltages v₁ and v₂, while the time-averaged voltages across junctions 3 and 4 are zero. This dynamic state could not be captured by the static analysis of [28].

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As another remarkable dynamic feature, figure 4(a) shows IVCs (for i₂ = 0) for different ϕ_ext = 0 (black line), 0.25 (red line) and 0.5 (green line). For ϕ_ext = 0 and 0.25 the IVCs are hysteretic, although β_C was set to zero. One further notes in figure 4(a) that in the resistive state at fixed bias current the voltage v_SQ across the SQUID decreases with increasing flux, opposite to the case of a standard (2-JJ) dc SQUID and somewhat reminiscent to the case of a dc SQUID with nonzero β_C, if the LC-resonance-voltage of the SQUID is exceeded [33, 34]. This is more clearly visible in figure 4(b), which shows v_SQ(ϕ_ext) curves for different values of i₁. Here we initialized the phases δ_k and voltages u_k with zero at a fixed value of i₁ and then swept ϕ_ext from either −1 to 1 or from 1 to −1. For i₁ = 1.5, v_SQ(ϕ_ext) has maxima at ϕ_ext = ± 0.5. With further increasing i₁, those are converted into minima. From figure 4(a) we see that there is a resistive (downsweep) branch in the IVC also near integer values of ϕ_ext (with a return current to the zero-voltage state i_r1 = 0.88 at ϕ_ext = 0), which however, could not be stabilized for i₁ < 1.7 in our sweeps of ϕ_ext. For i₁ = 1.7 it appears only on the beginning of the sweep sequence. Also note that for i₁ > 2.6 there is a splitting of the v_SQ(ϕ_ext) curves near half-integer values of ϕ_ext. We did not investigate this feature further at this point.

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While for i₂ = 0 at given bias current i₁ the time-averaged voltages v_k across all junctions are identical (not shown here), a difference shows up when inspecting the time-dependent voltages u_k , as displayed in figure 5 for i₂ = 0 and i₁ = 1.8. Figure 5(a) shows the case ϕ_ext = 0. Junctions 1 and 2, as well as junctions 3 and 4, oscillate in-phase. However, the voltage oscillations of the junctions 1 and 2 in the left and 3 and 4 in the right arm of the SQUID are out-of-phase, corresponding to a large circulating current around the SQUID. Consequently, the effective critical current i_c1 is lowered compared to the static case, causing the hysteresis in the IVC. Note that for a symmetric two-junction SQUID the circulating current would be zero for ϕ_ext = 0.

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In figure 4(a), we show that the hysteresis in the IVC decreases and eventually vanishes with increasing ϕ_ext. More precisely, the hysteresis disappears near ϕ_ext = 0.314. Figures 5(b) and (c) show u_k (t) for ϕ_ext = 0.25 and 0.5, respectively. At ϕ_ext = 0.25 (figure 5(b)) the phase shift between the JJs in the left and right arm becomes smaller. If we increase ϕ_ext further to ϕ_ext = 0.5 (figure 5(c)), the system seems to become chaotic, and synchronization of JJs within the same SQUID arm gets lost. This chaotic regime seems to coincide with the region where the IVCs are no longer hysteretic. We also calculated families of v_SQ (ϕ_ext) curves for i₂ = ± 0.6 and ±1.2 (not shown). In all cases we observed instabilities in the v_SQ (ϕ_ext) patterns indicating that for the symmetric device chaotic regimes appear over a wide range of i₂ values.

Trivially, if the critical currents of one of the junctions in each arm of the SQUID were much bigger than the critical currents of the two other junctions one should return to the behaviour of a 2-junction SQUID. Thus, to investigate the quite unusual hysteresis in the IVCs (in the absence of JJ capacitance or heating effects) in some more detail we also show simulations for a somewhat asymmetric 4JJSQ with a₁ = a₃ = 1.2 and a₂ = a₄ = 1. The parameters are chosen such that the left and right arm of the SQUID remain symmetric.

Figure 6(a) shows by the red line i_c1 vs. ϕ_ext for the asymmetric case in comparison to the symmetric case (black line). In both cases, the critical current has a maximum i_c1,max = 2 at ϕ_ext = 0 and minima i_c1,min at ϕ_ext = ±0.5, like for a symmetric 2-junction SQUID. But one notes that the modulation depth (i_c1,max − i_c1,min)/i_c1,max of the asymmetric 4JJSQ is larger than for the symmetric 4JJSQ and closer to the modulation depth of 1/2 which would be obtained for a symmetric 2-junction SQUID with β_L = 1.

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Figure 7(a) shows IVCs for the asymmetric 4JJSQ for ϕ_ext = 0 (maximum i_c1) and 0.5 (minimum i_c1). For ϕ_ext = 0 the critical current i_c1 = 2, i.e. the two weaker JJs 2 and 4 (with a = 1) are switching to the resistive state. With further increasing i₁, we find a jump to higher v_SQ at i₁ = 2.4, which means that now also JJs 1 and 3 (with a = 1.2) become resistive. We also find for the slightly asymmetric 4JJSQ a hysteretic IVC for ϕ_ext = 0; however, this is smaller than for the symmetric 4JJSQ, and it is also suppressed with increasing ϕ_ext. In the resistive state one observes a crossover between the two IVCs at ϕ_ext = 0 and 0.5.

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Figure 7(b) displays for the slightly asymmetric 4JJSQ a family of v_SQ(ϕ_ext) curves for different bias currents i₁. Like for the symmetric 4JJSQ, at low bias currents the voltage maximum occurs at half-integer values of ϕ_ext. With increasing i₁ one observes the crossover to the state where the maxima of v_SQ vs. ϕ_ext occur at integer values of ϕ_ext. Also note, that near the crossover the v_SQ vs. ϕ_ext curves look noisy. This is because of the small hysteresis visible in the IVCs.

Finally, for the symmetric device, we also address the case of higher thermal noise. We take Γ = 4.4 · 10⁻³, a value, which we find for our experimental 4JJSQ discussed below. Figure 8(a) shows IVCs for ϕ_ext = 0, 0.25 and 0.5. The large hysteresis observed for Γ = 10⁻⁶ has nearly vanished, and we observe a low-bias region where the voltage at ϕ_ext = 0.5 is higher than the voltage for lower values of ϕ_ext. This can also be seen in the v_SQ vs. ϕ_ext curves displayed in figure 8(b), where one observes the crossover in the position of voltage maxima from half-integer to integer values of ϕ_ext at voltages near 1. Also note that the voltage jumps associated with the splitting of the v_SQ(ϕ_ext) curves at high bias currents are no longer observable.

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The 4JJSQs with deep submicron sandwich-type trilayer SNS JJs with Nb electrodes and HfTi barrier were fabricated using an established Nb planar fabrication technology. This is based on magnetron sputtering deposition on 3 inch Si wafers with 300 nm thermally oxidized SiO₂, electron beam lithography to ensure high alignment precision for nanopatterning of the small features of our SQUIDs and a chemical-mechanical polishing step to planarize the insulating SiO₂ layer between the Nb base and the Nb wiring layer and to open contact windows between the Nb top layer and Nb wiring layer. Those technologies are available at the clean room center at PTB Braunschweig and have been applied before to the fabrication of JJ-based circuits [35–38], including complex and advanced nanoSQUID layouts, such as gradiometers [39] and 3-axis vector nanoSQUIDs [40] and for auxiliary components such as pick-up and feedback coils. Since the fabrication technology is not the focus of this paper, we refer to [41] for the detailed fabrication technology.

We fabricated dc SQUIDs with four SNS JJs with nominal lateral JJ size of 300 nm × 300 nm (figure 9) and a normal conducting Hf_wt50%Ti_wt50% (HfTi) barrier with a thickness of 26 nm. The superconducting Nb loop is constructed as two quadrants in the 160 nm thick Nb base layer which are connected via the four JJs to two more quadrants in the Nb wiring layer, which has a thickness of 200 nm. The applied magnetic flux ϕ_ext to the SQUID can be provided by the modulation currents I_mod,i (i = 1–4) through four inductively coupled modulation lines in the Nb base layer, which are designed to be in a close distance (with a nominal 0.9 µm gap) to the four SQUID loop segments. The SQUID loop was designed to have an inner diameter of 8.0 µm with 1.1 µm linewidth for the Nb base and 8.1 µm diameter with 0.9 µm line width in the wiring layer.

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All measurement data shown here were taken at liquid He temperature (T = 4.2 K) and we used the current I₁, flowing from the top across the SQUID loop to the drain (bottom; see figure 9) as the bias current. The SQUID voltage V_SQ was measured between ports 1 and 3. The modulation current I_mod1 was used to inductively couple magnetic flux Φ₁ to the SQUID loop, via the mutual inductance M₁ = Φ₁/I_mod1. The control current I₂, flowing from the left to the drain, was used to shift the interference patterns. We always kept I₄ = 0, and accordingly, I₃ = I₁ + I₂. Also, we limit the results presented below to one exemplary 4JJSQ, since the other investigated 4JJSQs showed similar behaviour.

The IVCs V_SQ(I₁), with I_mod1 adjusted to obtain maximum critical current I_c1,max and minimum critical current I_c1,min are shown in figure 10(a) for the 4JJSQ with I₂ = 0 and 37.5 µA. These IVCs, as well as all others we obtained, are nonhysteretic. For I₂ = 0 the maximum critical current is 80 µA, which corresponds to I₀ ≈ 40 µA and a critical current density j₀ ≈ 44 kA cm⁻². For the noise parameter we find Γ ≈ 4.4 · 10⁻³ at 4.2 K. Further, we find the best overall agreement between the shapes of the experimental and simulated IVCs for i₂ = 0 by using I₀ R = 15 µV.

We show in figure 10(b) the voltage oscillation V_SQ(I_mod1) for the 4JJSQ (with I₂ = 0) at I₁ = 80 µA. From the oscillation period (current I_mod1,0 required to couple one flux quantum Φ₀ to the SQUID) we determine the inverse mutual inductance 1/M₁ = I_mod1,0/Φ₀ = 1.95 mA/Φ₀ (M₁ = 1.06 pH) between the corresponding modulation line and the 4JJSQ. The maximum modulation voltage of our 4JJSQ is 9 µV (cf figure 10(b)) and the derived maximum transfer coefficient is V_Φ ≈ 50 µV/Φ₀. Using the simulation software 3D-MLSI, to calculate the supercurrent density distribution based on the London equations [42] for the given geometry of our device and with a London penetration depth λ_L = 70–80 nm for our Nb thin films, we obtain for the mutual inductance M₁ = 1.12 pH, which is in good agreement with the value M₁ = 1.06 pH determined experimentally from figure 10(b). For the SQUID inductance we calculate with 3D-MLSI L = 17.0 pH, leading to an estimated screening parameter β_L = 2I₀ L/Φ₀ ≈ 0.66. From direct simulations (see below) we obtain β_L = 0.65 and L = 16 pH, which is in very good agreement.

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To work with dimensionless quantities for comparing our measurement results with calculations from [28] and with our simulations based on the mathematical model described in section 2, we normalize the currents I₁ and I₂ to I₀ and the modulation current I_mod1 to I_mod1,0 = Φ₀/M₁. The voltages are normalized to I₀ R = 15 µV. We denote normalized quantities by lower case letters i and v. ϕ₁ = I_mod1/I_mod1,0 = Φ₁/Φ₀ is the normalized applied flux (modulation current).

For the simulations shown below we use asymmetry parameters a₁ = 1.05 and a₄ = 0.95, so that we have a₁ + a₄ = 2 for the weakest JJs in both SQUID arms. We further use a₂ = 1.15 and a₃ = 5 and β_L = 0.65. These values have been obtained by matching i_c1 vs. ϕ₁ curves for i₂ = 0 and i₂ = 1.25, as well as IVCs for i₂ = 0 and ϕ₁ chosen so that the positive critical current was at a maximum or a minimum. Parameters a₁ to a₄, as well as β_L were varied in steps of 0.05. For the asymmetry parameter a₃ we can give only a lower limit of 4 and for convenience set this parameter to 5. We briefly note here that, by inspecting IVCs for i₄ = 0 only, in the absence of measurements of the individual voltages v_k , we could have exchanged the values of a₃ and a₄ without changing the results. However, using the measured individual junction voltages, as well as using measurements with nonzero values of the current i₄, we find JJ3 to be the junction with significantly higher asymmetry parameter. In experiment the threshold voltage to determine the critical current was 1 µV, translating to a normalized voltage criterion of about 0.07 to determine i_c1 in our simulations. Finally, we note that to match experimental i_c1(ϕ₁) or v_SQ(ϕ₁) curves with simulations, the experimental I_mod-axes needed to be adjusted typically by an offset of 0.05–0.1 mA to account for residual fields of ≈1–2 µT in the cryostat.

Figure 11 shows by solid lines the measured quantum interference patterns i_c1(ϕ₁) of our 4JJSQ together with the corresponding simulated curves (dashed lines). Here we measured IVCs for 101 different values of the modulation current, varying between I_mod1 = −1.48 mA and 2.40 mA, at 5 different values of the applied control current i₂, which were chosen to be between −1.25 and 1.25.

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As predicted by the model described in [28], the quantum interference patterns in figure 11 are not only shifted to lower critical currents, but also along the ϕ₁-axis while applying a positive control current (i₂ > 0). For negative control currents (i₂ < 0) the patterns are shifted only along the ϕ₁-axis, which agrees with the theoretical model, too. Compared to [28] and to our simulations for symmetric 4JJSQs (cf figure 2(a)) we observe a shift of all quantum interference patterns to the left, which is due to the asymmetry in the junction critical currents of our device.

Now, we compare our simulation results to IVCs measured at different values of applied flux and i₂ = 0. The measured curves are presented in figure 12(a), whereas the simulated curves are shown in figure 12(b). In general, there is a good agreement between the measurement and the simulation. Particularly, the measured critical current values and the 'bends' (abrupt changes in the slope) observed in the IVCs for different values of applied flux could be reproduced by simulations.

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An explanation for the bending of the IVCs can be found by having a closer look at the voltages v_k across the four individual junctions, shown in figure 13 for i₂ = 0. The left panels (a), (c) show measurements and the right panels (b), (d) show simulations. The upper panels (a), (b) show v_k (i₁) at maximum critical current (at ϕ₁ = −0.17), and the lower panels are for minimum critical current (at ϕ₁ = 0.41). In both cases, JJ3 stays in the zero-voltage state. The voltage v₁ across JJ1 equals v₄ as long as JJ2 is in the zero-voltage state, while v₁ is close to v₂ as soon as v₂ becomes nonzero. This transition marks the bends in the IVCs shown in figure 12, where the voltages are the sum of the JJs in the right and left arm, respectively.

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With respect to the IVCs we briefly mention that also for nonzero values of i₂ we found a very good agreement between experiment and simulation. However, neither in experiment nor in simulation we found hysteretic IVCs for our strongly asymmetric device, as we found in our simulations for the (nearly) symmetric 4JJSQ, cf figures 4–6.

Figure 14 shows measured (figure 14(a)) and simulated (figure 14(b)) families of v_SQ(ϕ₁) curves for i₂ = 0. Here, we varied the bias current i₁ in steps of 0.1 from 1 to 4 for positive bias and from −1 to −4 for negative bias.

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The v_SQ vs. ϕ₁ curves show a point-symmetric behaviour and for |i₁| ≲ 1.5 one observes the voltage maxima at half-integer values of the applied flux (shifted due to the asymmetric JJs), similar to 2-junction SQUID behaviour. For higher values of the bias current |i₁|, the maxima of the v_SQ(ϕ₁) curves are gradually shifting further to the left and right, for positive and negative i₁, respectively. The simulations are in good agreement with the experimental data. As we neglected inductance and resistance asymmetries in the simulations, the above-described shifts in the voltage maxima are attributed to critical current asymmetries only. Note that in contrast to the simulated v_SQ(ϕ_ext) families shown in figures 4, 7 and 8, we do not observe an abrupt crossover of the position of the voltage maxima from half-integer to integer values of the applied normalized flux. Further note that in contrast to the case of the simulated v_SQ(ϕ_ext) families for the symmetric 4JJSQ with comparable noise parameter, the v_SQ vs. ϕ₁ curves look much less noisy close to v_SQ ≈ 1, which is consistent with the missing hysteresis in the IVCs of our asymmetric device.

Figure 15 shows measured (figure 15(a)) and simulated (figure 15(b)) v_SQ(ϕ₁) curves for i₂ = 0 and ±0.95 while the bias current is chosen to obtain maximum voltage modulation amplitude. We observe a shift of the curves along the ϕ₁-axis while applying a positive control current (i₂ = 0.95) as well as negative control currents (i₂ = −0.95) compared to the case of i₂ = 0. In contrast to the symmetric device, no instabilities appear in the curves.

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Figure 16 displays the measured (a) and simulated (b) critical currents i_c1 vs. control current i₂ for both polarities of i_c1 and different values of applied flux (taking into account that for i₂ = 0 the i_c1(ϕ₁) curves have their maximum at ϕ₁ = −0.17 for i_c1 > 0, and at 0.16 for i_c1 < 0. The agreement between the measured and the simulated curves is very good and, like for the symmetric case (cf figure 2(b)), we observe both in experiment and simulation an increase of the critical current for positive bias, for i₂ < −1.5 and ϕ₁ adjusted to be near the i_c1(ϕ₁) maximum for i₂ = 0.

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The upper panel of figure 17 displays experimental (a) and simulated (b) IVCs v_SQ(i₁) for i₂ = −2 and the same three values of ϕ₁ that were chosen for the i_c1(i₂) curves in figure 16 for i_c1 > 0. Again, the agreement between experiment and simulation is very good. The lower panel of figure 17 shows measured (c) and simulated (d) individual junction voltages v_k (i₁) for ϕ₁ = 0.31 and i₂ = −2. Like for the symmetric case, cf figure 3(b), the simulations yield a state where the junctions in the left arm are resistive, with opposite dc voltages, while the junctions in the right arm remain in their zero-voltage state. This state is also realized in the experimental device.

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As explained in [28] the current-phase relation F(δ) of junctions 1 and 2 can be reconstructed from measurements of the critical current i_c1(ϕ₁,i₂). For a given value of i₂ one determines the flux value ϕ₁ where i_c1(ϕ₁) has a maximum. This constitutes the curve ϕ_1,max(i₂) which encodes the CPR of JJ1 for i₂ > a₂ − a₁ and the CPR of JJ2 for i₂ ⩽ a₂ − a₁.

For junction 2 the phase δ₂ is found from:

$$\begin{equation}{\delta _2} = - 2\pi {\varphi _{1,{\text{max}}}} - \pi {\beta _L}{i_2}/4 + \mu ,\end{equation} \tag{ 5 }$$

and the normalized CPR F₂(δ₂)/I₀ is obtained via:

$$\begin{equation}{F_2}\left( {{\delta _2}} \right)\!/\!{I_0} = {j_2} = {i_2} + {a_1}.\end{equation} \tag{ 6 }$$

The phase µ is a constant and can, e.g. be determined by demanding F₂(0) = 0.

From equation (3), using the definition of j_SQUID and further assuming sinusoidal CPR's and under the assumption that junction 4 is at its critical current we can also give an analytic expression for µ,

$$\begin{equation}\mu = {a_4}/{a_3} - \pi {\beta _{\text{L}}}\left( {{a_1} - {a_4}} \right)\!/2.\end{equation} \tag{ 7 }$$

For junction 1 one makes the replacements i₂ → −i₂, a₁ ↔ a₂, δ₂ → δ₁, and for µ one finds:

$$\begin{equation}\mu = {a_4}/{a_3} - \pi {\beta _{\text{L}}}\left( {{a_2} - {a_4}} \right)\!/2.\end{equation} \tag{ 8 }$$

Figure 18(a) shows a measured contour plot i_c1(ϕ₁,i₂). Here, we measured IVCs for 52 different values of ϕ₁ from −0.9 to 1 for 81 different values of i₂ between −2 and 2. ϕ_1,max(i₂) is indicated by the pink symbols for ϕ₁ between −0.25 and 0.3 and for both polarities of i₂. The reconstructed normalized CPRs F_i (δ_i )/(a_iI₀) are shown in figure 19(a) for junction 1 and in figure 19(b) for junction 2. In the graphs we also plotted by lines the sinusoidal dependences. For the phase shift we used µ = −0.22 for junction 1 and µ = −0.12 for junction 2, to fulfil F₁(0) = F₂(0) = 0. Apparently, the data points are compatible with a sinusoidal CPR, although the error bars are large. Using the sinusoidal CPRs we also inverted equations (5) and (6) and show the resulting ϕ_1,max(i₂) curves as solid white lines in figure 18(a).

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Finally, for completeness and in order to further test the evaluation method we applied the same procedure for the simulated i_c1(ϕ₁,i₂) curves, where by ansatz we have sinusoidal CPRs. Figure 18(b) shows the corresponding contour plot i_c1(ϕ₁,i₂). In the calculations both ϕ₁ and i₂ were varied in steps of 0.02.

Figures 19(c) and (d) show the extracted CPRs, where we used µ = 0 for JJ1 and µ = 0.1 for JJ2 to fulfil F₁(0) = F₂(0) = 0. Those values are close to µ = −0.01 and 0.09, found from equations (7) and (8), respectively. We further note that the experimental data yield values for µ that are shifted by Δµ = −0.22 relative to the simulated ones. This may indicate that a₃ of the experimental device is much larger than the value of 5 used for simulation. Indeed, a₃ → ∞ would roughly lead to the observed value of Δµ.