An Empirical Method for Predicting Unshunted Josephson Junction Critical Current in Shielded Room from Measurement in Unshielded Room

1. Introduction

Superconducting electronic circuit technology has the characteristics of high clock frequency [1] and the lowest bit manipulation energy dissipation [2,3], making it an important candidate for future high-performance superconducting computers MIT Lincoln Laboratory (MIT LL) has made a series of advancements in the manufacturing technology of superconducting rapid single flux quantum (RSFQ) circuits [4,5,6,7], designed for high-performance superconducting multi bit processors with clock frequencies ranging from 10 GHz to 20 GHz for high-performance computing [8]. The conventional process often uses shunted JJ (implemented in parallel with unshunted JJ resistors) as the basic unit to form the circuit. Although these JJs can achieve good noise resistance and thus circuit stability, it occupies a large area, which limited the JJ process integration density to be generally lower than 10⁶ JJs/cm² [9]. To increase the integrated scale of SFQ circuits to over 10⁶ JJs/cm², it is necessary to reduce the area occupied by JJs and shunt resistors. The unshunted JJ is beneficial for significantly reducing the unit area. Therefore, unshunted JJ has become a potential candidate for high integration density, high current density, and high clock frequency superconducting Josephson circuits [9].

Compared to conventional unshunted JJ, shunted JJ is more sensitive to electromagnetic and magnetic noise from circuits, spaces, and other sources. As one of the important physical parameters of JJ, critical current (I_c) is crucial for the later device, circuit design, process preparation [10]. The accuracy of Ic measurement is directly related to the margin of various electrical parameters of subsequent circuits. In order to obtain accurate measurement values, the test should be carried out in a suitable shielding room [11]. However, due to the limitations of laboratory conditions, the actual test is usually carried out in unshielded environment. This inevitably introduces various environmental noises to JJ, and brings great uncertainty to the measurement results of JJ current. Also, the future superconductor computer will work under noisy environment, comparing to the environment of traditional computer, to obtain the application competitive with semiconductor computer. Therefore, it is necessary to carry out the accurate measurement research of unshunted JJ current under the unshielded environment, and evaluate the filtering technics applicable to the unshielded environment.

According to Ambegaokar-Baratoff theory [12], the theoretical value I_c0 can be calculated by formula: I_c0=I_g * π/4. I_g is defined as the current at which the junction goes into a normal state and V_g as gap voltage of Josephson junction. The actually measured I_c value is usually smaller than the theoretical value I_c0 due to the effect of the complex noise effect, which seriously affects the circuit performance of superconducting Josephson devices. Yasushi Ishikawa et al. analyzed the influence of magnetic noise on the characteristics of superconductor/normal-metal/superconductor (SNS) JJ I_c , and found that the dependence of Josephson currents on the magnetic field are sensitive to the width of the normal metal [13]. Ohkubo et al. [14] have studied the standard measurement method of I_c in the over-damped regime, but such a study for under-damped Nb/Al-AlO_x/Nb junctions under unshielded conditions is lacking.

In this paper, the influence of electromagnetic, magnetic and flux noises on the accurate measurement of JJ I_c is experimentally studied based on the unshunted JJ under unshielded environment. This study provides reference for avoiding or reducing the impact of test environment and magnetic flux noise on I_c under actual experimental conditions, and has guiding significance and application value for the subsequent research and application of RSFQ circuits and superconducting large-scale digital integrated circuits.

2. Results and Discussion

2.1. JJ Fabrication Process and I_c Test Equipment

Comprehensive and efficient process control monitors (PCMs) are essential to keep the physical parameter of JJ stable. Here, we carry out the measurement based on the unshunted Nb/Al-AlOx/Nb JJ sample fabricated following the standard fabrication process [15,16], which is named as SIMIT Nb03, for RSFQ integrated circuits. Considering that the quality of junctions is influenced by the stress of Nb film [16,17], junctions on a SiO₂ flat field and on a via to a Mo resistor are both designed. In our PCM, two kinds of JJs are included, the unshunted junctions on SiO₂ fields (labeled as juf) and those on Mo resistors (labeled as jur). The thickness of Mo film is 40 nm, and the thickness of SiO₂ is 100 nm. Nb is patterned and etched by inductively coupled plasma reactive ion etching (ICP-RIE) equipment with endpoint detector. SiO₂ is deposited by plasma enhanced chemical vapor deposition (PECVD) equipment and etched by reactive ion etching (RIE) equipment [18].

The JJ device test circuit diagram is shown in Figure 1. The JJ is put 10 cm below the liquid level of liquid Helium to assure the JJ in superconducting state during the process of I-V curve test. Four π-type lower-pass filter (LPF) units are integrated in the circuit to filter the environmental EM noise out of the test circuit. In the actual test environment, the test space is often full of electromagnetic noise at various frequencies, including power frequency (50 Hz) and RF signals (30 kHz-30 GHz). These spatial noises will couple into the circuit to form conductive noise, which will affect the JJ current test results. Therefore, it is necessary to filter these noises by filtering circuit.

The setup of I_c measurement is shown in Figure 2. Keithley6221 current source and Keithley2182A voltmeter are used for testing I-V curve, and I_c value can be obtained automatically from the curve. The tested PCM cell is connected to the PCB by wire bonding. The Al foils (10 layers) are wrapped around the Nb cylinder for shielding magnetic noise. Just after the sample temperature is cooled down to ~4.3 K, the amplitude of the supply current has to be adjusted to twice the estimated maximum value of I_c and the I-V curve can be scanned and recorded in computer. The maximum value of I_c is usually known from the preceding measurement of a current–voltage characteristic. The whole experiment is controlled by a PC computer. During testing, the sample must be located below 10cm of the liquid helium level. The de-fluxing mentioned in this paper refers to the following process: 1) Raising the test rod to increase the sample temperature; 2) Rotate the test rod 90^o, the rotation may help to increase the I_c value, which may be caused by the difference of magnetic field in certain direction; 3) Keep the sample above 15 K for 5 minutes. Based on our experimental knowledge, in the different maximum temperature (e.g, heating the sample to 10 K, 11 K, 12 K, 15 K, each for 10 s, respectively) , the 15 K is the suitable temperature to get the highest I_c value; 4) then lower the test rod to the original position for testing.

The relationship of measured and simulated normalized output amplitude from filter with the frequency in Figure 3a and the photo of the filter PCB board is shown in Figure 3b. The equivalent circuit diagram of the low-pass filter is shown in the illustration of Figure 3b. The simulated curve is the transfer function of a first order Butterworth type filter. Determination of the cut-off frequency is illustrated in Figure 3a, where the value of this function is 0.707. Comparing Gfilter (f_c=50 Hz) experiment and simulation data, it is found that when the frequency is less than 90 Hz, the experiment and simulation data are in good agreement. However, when the frequency is greater than 90 Hz, the experimental data is gradually higher than the simulation value. Also in the Ffilter (f_c=2 kHz) test, it is found that when the frequency is higher than 4 kHz, the experimental data is gradually higher than the simulation data. This may be due to frequency related noise interference above the cut-off frequency. To assure the filtering effect, a filter with f_c (R=5.1 kHz, C=10 F) of 5 Hz is used in the paper.

2.2. The Definition of Physics Parameters Related to the Measurement of I_c

The sketch map of I-V curve is shown in Figure 4. The reciprocal of slope of L1, L2 and L3 correspond to normal state resistance R_N, semi-superconducting state resistance R_s, and sub-gap resistance R_sg. I_g is determined as vertical coordinate value of intersection (point N) of L1 and L2. I_M (=V_M/R_s) is determined as vertical coordinate value of intersection (point M) of L2 and L3. Vg of JJ is defined as 2Δ/e. Here, e is the electron charge, and Δ is the superconducting energy gap [19]. L2 has the same slope with the I-V curve (R_sI_M<V<R_NI_g). R_N is the differential value of V to I when V >I_gR_N. In this paper, R_sg is the differential value of V to I when V<2 mV. V_g is calculated by the formula V_g=0.5*(V_N+V_S) in the line L2, where I equals to about 0.5 I_g.

In an ideal environment, the measured I_c value inclines to the ideal critical current I_c0 and the zero-voltage hysteresis current Ir inclines to be zero when temperature is 0 K. When I_c increase from 0 to I_c0, the voltage is zero, when I >=I_c0, V increases rapidly to V_g. when I >I_g (quasi-particle current), JJ enters normal state, with resistance R_N. In a practical test, the magnetic and EM noise may move some fluxes into the JJ sample, and thus decrease the I_c value to a lower value and increase I_r to a higher value.

3. Discussion

3.1. The Effect of Low-Pass Filter with Cut-Off Frequency f_c on I_c

The I_c measurements, repeated 30 times for sample s0506-juf3.1 for each filtering configuration, outside the magnetic shield room, are shown Figure 5. It is obvious that average I_c without filter (upward triangle) is much lower than that with filters, which means that filter is a necessary tool to filter conductive noise along with the cables to assure higher I_c value. The bulge in I_c curve of no filter may be caused by sudden disappearance of surrounding noise, which induces I_c to increase. The I_c values without filter after 22 times measurement seems almost constant compared to the values before 21 times measurement, which may be due to flux trapping under unshielded environment.

The histograms of I_c for sample s0506-juf3.1 using different filters with different cut-off frequencies are shown in Figure 6, it is not difficult to find that the average I_c is almost the same (~520 μA) by comparing the two curves with different filter cut frequency f_c of 50 Hz (Gfilter Figure 6b and 2 kHz (Ffilter, Figure 6a. I_c obtained through filter with lower cut-off frequency (f_c=50 Hz) shows better I_c uncertainty than that with higher f_c (2 kHz), which indicates that in order to obtain good uncertainty and higher average I_c, filter with lower cut-off frequency is better than that with higher one. Electromagnetic (such as power frequency) noise can be filtered out by filter with a cut-off frequency lower than 50 Hz, thus improve the measured value of I_c and reducing the uncertainty.

3.2. The Effect of Nb Cylinder and Multi-Layer Al Foils on the Measured I_c Value

Magnetic field, flux vortex and electromagnetic field can penetrate superconducting films and devices in many ways, resulting in I_c measurement uncertainty and ultimately affecting the operation of circuits. Therefore, it is necessary to prevent or reduce these effects through shielding in measurements.

The I_c measurements, repeated 41 times for sample s0506-juf3.1 with or without Al cover are shown Figure 7a. The probability of I_c with and without Al foils are shown in Figure 7b. The shielding structure consists of an Nb cylinder and 10 layers of Al foils surrounding. The average I_c and uncertainty with cover is 349 μA and ±11% , respectively，and those without cover is 299 μA and ±11% , respectively.

3.3. Influence of Multiple de-Fluxing on I_c with and Without Spatial EM and Magnetic Noise

The I_c repeatability for different de-flux times is shown in Figure 8a. Histogram of the same data is shown in Figure 8b. It is not hard to find that different de-fluxing time corresponds to different average I_c. The uncertainty and average value of I_c of no de-flux, de-flux 1st , 2nd , 3rd , 4th outside magnetic shield room, and in magnetic shield room(full shield) are reported in table I. The average I_c after the 2nd de-fluxing cycle is the lowest (102 μA), corresponding to largest uncertainty. The deviation of I_c from the average I_c of corresponding de-fluxing time (U=10.27-76.47%) outside magnetic shield room is caused by environmental EM noise, which may introduce some fluxes into sample and reduce I_c. In contrast to the measurement outside magnetic room, the highest average I_c (560.13 μA) and lower uncertainty (0.71%) are obtained in magnetic shield room. The theoretical I_c (=I_g π/4) is calculated to be 668.82 μA.

Table I. The uncertainty and average value of I_c with no de-fluxing, after 1st, 2nd, 3rd, 4th de-fluxing outside magnetic shielding room, in shielding room and the theoretical I_c.

item	No de-flux	de-flux1	de-flux2	de-flux3	de-flux4	full shield	ideal
U(%)	15.39	10.27	76.47	35.82	28.43	0.71	0
Ic(μA)	523.07	526.05	102	469.02	495.95	560.1	668.8

Table I. The uncertainty and average value of I_c with no de-fluxing, after 1st, 2nd, 3rd, 4th de-fluxing outside magnetic shielding room, in shielding room and the theoretical I_c.

item	No de-flux	de-flux1	de-flux2	de-flux3	de-flux4	full shield	ideal
U(%)	15.39	10.27	76.47	35.82	28.43	0.71	0
Ic(μA)	523.07	526.05	102	469.02	495.95	560.1	668.8

3.4. Flux Noise Impact and Its Removal or Reduction

The I_c distribution for no-de-flux, de-flux1, de-flux2, de-flux3 and de-flux4 are shown in Figure 8b. The peak position of I_c stands for the most often measured value of I_c. Different times of de-fluxing shows different I_c peak value and distribution, and the 1st de-fluxing shows maximum I_c in Figure 8a, which means that I_c value is not positively correlated with the de-fluxing number. Based on this method, the obtained test I_c value under EM and magnetic noise environment can reach 526.05 μA, which reaches 93.9% of the value (560.13 μA) tested in magnetic shielding room.

It is easy to find that the uncertainty U of the sample varies with the de-fluxing times in the Table I, and the uncertainty in fully shielded environment is smaller than all of the other tested data under the unshielded environment. The data tested under the unshielded environment also varies greatly with the de-fluxing times, but the uncertainty does not show a positive or negative correlation with the number of de-fluxing times. This may be explained as that the spatial magnetic field and EM noise may randomly affect the flux trapping mechanism in the tested sample under unshielded environment, thus cause the uncertainty of I_c to change. Therefore, it is necessary to de-flux the sample under magnetic shielded environment to reduce the uncertainty value if the test condition allow.

3.5. Magnetic Noise Impact and Its Removal or Reduction

This study shows that magnetic shielding can reduce the influence of magnetic noise on the measurement of junction critical current I_c by wrapping 10 layers of Al foils on the Nb cylinder outside the sample. General principles of magnetic and EM shielding are treated by textbooks [20,21,22], which include design and calculations of shielding enclosures and their effectiveness. Y. Iwashita et al. proposed a passive magnetic shielding method based on niobium/aluminum structure. Theoretically, it is proved that this structure(with high permeability material and superconducting material) can not only break through the limit of the shielding ability by the critical field (about 170 mT of niobium), and thus improve the shielding effect, but also optimize the temperature gradient of the radial thermal conductive layer, thus further enhancing magnetic flux exclusion effect in sample [23]. But no experimental report has been found in the author’s paper. Based on the test data of Nb/Al-AlO_x/Nb samples, it is found that the average I_c of sample in the Nb cylinder wrapped with Al foils is 16.7% higher than that without Al foils (Figure 7). It is experimentally confirmed that wrapping Al foils outside Nb cylinder can help reduce magnetic leakage and improve the probability of magnetic flux exclusion out of the superconducting layer, which is consistent with Iwashita’s theoretical prediction.

3.6. An Empirical Formula for Predicting I_c Values in Shielded Environments from Unshielded Environment Data

Figure 9 shows the relationship of average JJ current I_c-av with uncertainty U of I_c data of sample s0506-juf3.1 from repeatability study. Based on an test procedure, we propose a formula for predicting measured I_c value under magnetic shielded environment.

$${\mathrm{I}}_{\mathrm{c}\mathrm{i}-\mathrm{a}\mathrm{v}}=\mathrm{A}-\mathrm{B}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{U}/\mathrm{C}\right)$$

(1)

The procedure is as following: 1) Place the sample at the bottom of the testing rack; 2) Insert the test rod into the liquid helium Dewar and keep the sample 10 cm below the liquid helium level during the testing process; 3) Connect all devices; 4) Wait for the temperature of sample under test to be below 4.3 K; 5) Measure I_cij (i=1:n; j=1:m). Firstly, measure I_cij m times, if j<m, then, j=j+1 and measure I_c; if j=m, obtain the average value I_ci-av and uncertainty U_i, then heat the sample to de-flux; 6) carry out N times of measurements of I_ci-av, if i<n, then i=i+1; if i=n, end, output I_ci-av and U_i; 7) setup the relationship between I_ci-av and Ui,in formula (1);

By fitting the data in Figure 9, it is not difficult to give out: A=749 μA, B=-193 μA, and C=20. Based on the I_ci-av and U_i relationship, the average I_ci can be extrapolated to the point where U=0 and I_c = 556 μA, which is the empirical value of I_c under the condition of no shielded room. The experimentally measured I_c under shielding room was 560 μA. The deviation is less than 1%. which verifies the effectiveness of the empirical formula for predicting I_c values of sample s0506-juf3.1 in shielded environments using data from unshielded environments.

4. Conclusions

The influence of magnetic and electromagnetic noise on the accurate measurement of junction critical current I_c is investigated with unshunted Nb/Al-AlO_x/Nb JJ samples. The accumulation of magnetic flux inside the junction is the main reason for the decrease of JJ I_c ; EM noise as an external cause is the main reason for the increase of junction I_c uncertainty and the decrease of I_c value. The results is as following:

We have experimentally demonstrated that magnetic flux pinning in the SIS junction is one of the reasons for the decrease in measured JJ current. Multiple de-fluxing (changing the direction of rotation of the test rod before each de-fluxing ) is beneficial for obtaining higher JJ measurement I_c. The measured I_c value can be as high as possible by repeatedly de-fluxing (more than 5 times) and selecting the optimum I_c value (526.05 μA) reaches, in a batch of 41 cycles, 93.9% of that (560.13 μA) tested in shielding room;

The structure of Nb cylinder wrapped with multi-layer aluminum foil can reduce magnetic interference. The structure can increase JJ I_c value by 16.7%.

We setup an empirical formula for predicting I_c values in shielded environments from un-shielded environment data.The deviation of empirical value of I_c and the experimentally measured value is less than 1%, verifying the effectiveness of the empirical formula.

This paper provides a relatively accurate and low cost test idea of JJ I_c under unshielded conditions, which provides guidance for the fine test of I_c under conventional laboratory conditions without expensive magnetic shielding equipment, and has important practical value for speeding up the design and manufacturing process of RSFQ devices and the development of large-scale superconducting computer integrated circuit chips.

5. Patents

A low temperature superconducting Josephson junction current testing method without magnetic shielding environment and parallel resistance, by Zhou Jian, Wang Juan, Wang Yongliang, and Peng Wei. Invention patent, application number: 202311206205.0, application date: 2023.9.19 Accepted. Authorization number: ZL202311206205.0, authorization date: August 4, 2026.