PA1
Influence of external magnetic fields on the inductive properties of rapid single-flux-quantum digital circuits Romain Collot, Pascal Febvre
Coenrad Fourie
University of Savoie, IMEP-LAHC, CNRS UMR5130 73376 Le Bourget du Lac Cedex, France
[email protected]
Department of Electrical and Electronic Engineering Stellenbosch University Stellenbosch, 7600, South Africa
Jean-Luc Issler, Thierry Robert
Juergen Kunert, Ronny Stolz, Hans-Georg Meyer
Centre National d'Etudes Spatiales 18 avenue Edouard Belin 31401 Toulouse Cedex 9, France Abstract— A method has been developped to take into account the presence of a local external magnetic field in Superconducting Quantum Interference Devices (SQUIDs) loops composing Rapid Single-Flux-Quantum (RSFQ) circuits, through modifications of circuit inductances. A good agreement is observed between analytical formulas and simulations performed by time-domain softwares. The agreement with the first experiments is still only qualitative at this stage. Keywords—Josephson junction; rapid-single-flux-quantum; RSFQ; SFQ; SQUID; superconducting electronics
I.
INTRODUCTION
RSFQ circuits are based on Josephson junctions embedded with inductive lines that lead to specific digital functionalities [1]. Magnetic disturbances can influence their operation because magnetic flux quanta, used as the carriers of the logical bits for the RSFQ logic, correspond to faint magnetic fields, lower than 20µT for a typical RSFQ loop with an area of 100 µm2. In practice RSFQ circuits are optimum biased for large margins of stable operation and increase the stability. A slight change of bias currents (margins are typically in the 1030% range) can result in malfunctions of the logical circuit, since some overbiased Josephson junctions can switch without any incoming RSFQ pulse, or, on the contrary, underbiased junctions will not switch upon arrival of an RSFQ pulse. The difference between these two extreme cases usually corresponds to a change of a fraction of magnetic flux quantum in the RSFQ loops, if one sees RSFQ circuits as a smart organization of non-hysteretic SQUIDs. As an example Josephson Transmission Lines (JTLs) that are widely used in RSFQ circuits and are known to be tolerant to deviations of parameters from their nominal values are usually biased at about 75% of the junction critical currents. One fourth of a magnetic flux quantum produced in the loop by an external magnetic field is sufficient to make it switch. The situation is
Institute of Photonic Technology Albert-Einstein-Str. 9 D-07745 Jena, Germany
worse for many other cells, especially those with storing loops such as SFQ-to-DC converters used to monitor output signals, Toggle-Flip-Flops (TFFs), Delay-Flip-Flops (DFFs), etc… Therefore RSFQ circuits are usually embedded in a magnetically shielded environment to screen the ambient magnetic field of about 50 µT (the average value of the geomagnetic field). Nevertheless, some local magnetic fields are created on the chip by the bias currents of the circuit [2-5]. The associated magnetic field cannot be screened and some techniques have to be used to minimize its impact on the operation of the RSFQ circuit [6-7]. Besides, for the readout of superconducting sensors, it is often necessary to operate the digital circuits without magnetic shield or with a weak shielding, as is the case for digital SQUID magnetometers [89]. Since the basic element of RSFQ circuits is the SQUID, we studied experimentally and theoretically the behavior of nonhysteretic SQUIDs in presence of an externally applied magnetic field. We chose parameters (critical currents, shunt resistors, connecting inductors) typically used for RSFQ circuits. Figure 1 shows the schematics and picture of a SQUID fabricated at the European FLUXONICS Foundry [10, 11] used for our analysis. The goal is to determine the parameters that can be modified locally in the schematics, through the netlist of the circuit used by the time-domain RSFQ simulator (JSIM in our case [12]) in order to take into account the local magnetic field. The technique can take non uniform external fields into account, as long as one can have access to the experimental information. In all cases the analysis allows us to evaluate with simulations the behaviour of circuits in presence of magnetic field, and corroborate and complement in particular the experimental analysis performed earlier [13]. We started with loops with only two Josephson junctions but the work can be extended to more complex loops and circuits.
PA1 need to keep the total loop inductance constant to simulate the behaviour of the same physical circuit. One can do so by modifying each inductance of the two SQUID branches by the quantity ΔL: one inductance is increased by ΔL, while the other one is decreased by the same quantity. From the two equations mentioned above, it can be shown that: Fig. 1. The Nb/Al-AlOx/Nb non-hysteretic SQUID used for our analysis. Both Josephson junctions have a 250 µA critical current. The loop inductance extracted with Inductex [14] is La+Lb=10.35 pH (La =Lb).
II.
DESCRIPTION OF THE METHOD
The magnetic interference pattern of two-junction DC SQUIDs is well known [15]. It can be calculated theoretically by using the first Josephson equation I=I0 sinϕ, where I0 is the maximum critical current of the Josephson junction (in absence of magnetic field) and ϕ is the phase across the junction. Also the relation corresponding to the quantization of the fluxoid in the SQUID loop is: 2πn = φa – φb+2π Фext + βa sin φa + βb sin φb
ΔL= Фext / Ic(Фext)
(2)
where Ic is the critical current of one junction of the SQUID in presence of an external magnetic flux Фext. Equation (2) is non-linear through the expression of Ic(Фext) shown in Fig. 2. Hence ΔL can be derived from (2) for each magnetic flux. Normalized results are shown in Fig. 3. Then the behaviour of the SQUID loop with the modified inductances can be calculated with any time-domain simulator to mimic the presence of a local external flux.
(1)
where Фext is the external magnetic flux , βa = 2πLaIa / Ф0 and βb = 2πLbIb/Ф0 are the screening parameters, Ia and Ib (resp. ϕa and ϕb) are the critical currents (resp. the phases) of the two Josephson junctions. Ф0 is the magnetic flux quantum. Applying these relations to the SQUID of Fig. 1, one can calculate theoretically the interference pattern, shown in Fig. 2. In practice a circulating current is induced in the SQUID loop to screen the external magnetic field. The consequence is that the total current that can flow through the SQUID is lower than the maximum one, since the phase of both Josephson junctions has departed a priori from its π/2 value due to the screening current.
Fig. 3. Theoretical curve showing ΔL as a function of Фext
III.
SIMULATIONS OF THE MODIFIED CIRCUIT
For several values of ΔL, we extracted the corresponding critical current of the SQUID from JSIM simulations. The initial configuration for the inductances in absence of external magnetic flux was La = Lb = 5 pH. On the other hand one can relate analytically the critical current Ic of the SQUID to ΔL from Fig. 2 and Fig. 3.
Fig. 2. Theoretical interference pattern for the SQUID of Fig. 1.
Since the RSFQ simulator that we use does not take into account the presence of a magnetic field in the loops, we propose to modify some parameters of the SQUID loops in the circuit netlist, e.g. inductors, to mimic the presence of the magnetic field. Nevertheless, since the SQUID loop inductances are fixed at the design and fabrication levels, we
Fig. 4. Comparison, for different inductances ΔL, of the normalized critical current calculated from analytical formulas and with JSIM simulator.
PA1 The comparison between the analytical curve and JSIM simulation results in Fig. 4 shows an excellent agreement for the method. IV.
COMPARISON WITH EXPERIMENTS
We measured the interference pattern of the SQUID of Fig. 1 by using an external magnetic field applied through the SQUID loop. This magnetic field is generated in situ by a set of superconducting coils surrounding the chip under test. We biased the SQUID above its critical current for several bias currents and we measured the voltage for several values of the external magnetic field. Results are presented in Fig.5.
The discrepancy in periodicity is believed to be due to the Meissner effect by the ground plane of the chip that deviates some magnetic field lines. Also the different shapes of the curves are likely due to some flux trapped in the SQUID, that leads to a higher, less strongly modulated voltage. V.
CONCLUSION
We have proposed a method to take into account the influence of an external magnetic field in time-domain simulators like JSIM. The method has been theoretically validated and will be helpful to calculate the performance of RSFQ circuits in presence of magnetic field. The comparison with experiment s, based on a 2-junction SQUID circuit gives only a qualitative agreement at this stage: more measurements are needed to find and fit the same parameters in the simulator and the circuit. REFERENCES [1]
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[4] Fig. 5. Experimental interference patterns, for different bias current of the SQUID, obtained with an external parallel magnetic field
The same SQUID with a bias current of 648 µA was simulated with JSIM. For each ΔL, the average voltage of the SQUID was extracted. From Fig. 3 used as a calibration curve, we can plot the simulated average voltage as a function of the external magnetic field by taking into account the effective area of the SQUID. The result is shown in Fig. 6. The experimental periodicity is about 21 µT, assuming that all the flux created by the coils goes through the SQUID loop [14]. Fig. 6 shows that the periodicity simulated by JSIM is rather close to 33 µT.
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[14] Fig. 6. Average SQUID voltage simulated by JSIM with a magnetic field taken into account through the variation of inductance DL. The SQUID bias current is 648 µA.
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