A Voltage Monitoring Approach for Eliminating Flux Trapping in Josephson Junction Arrays

A Voltage Monitoring Approach for Eliminating Flux Trapping in Josephson Junction Arrays | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Voltage Monitoring Approach for Eliminating Flux Trapping in Josephson Junction Arrays Huan Qiao, Wenhui Cao, Jinjin Li, Jun Wan, Guanhua She, Qing Zhong, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3890830/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Aug, 2024 Read the published version in Journal of Low Temperature Physics → Version 1 posted 8 You are reading this latest preprint version Abstract Based on the characteristic change in resistance of superconducting devices during the superconducting transition, a quantitative method to determine the temperature of superconducting devices has been proposed. This method can detect in real time whether the temperature of the superconducting device exceeds the superconducting transition temperature, thereby enabling quick elimination of the trapped magnetic flux in devices. We apply this method in our experiment, which uses a superconducting 22 subarrays programmable Josephson voltage standard (PJVS) device and obtains satisfactory results. We conclude that this method can efficiently eliminate the tapped magnetic flux in superconducting device testing systems that do not possess thermometers and facilitate testing. Programmable Josephson junction array flux trapping superconductivity voltage standard Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction Programmable Josephson junction arrays [ 1 – 7 ] are superconducting integrated circuits that operate at a low temperature of 4.2 K. External electromagnetic and thermal disturbances can lead to trapped magnetic flux in these kinds of devices [ 8 – 11 ]. This results in the localized trapping of quantized magnetic flux [ 12 – 18 ], which can suppress the critical current of the Josephson junction, thereby affecting its normal operation [ 22 – 25 ]. For superconducting quantum devices, a common method to prevent trapped magnetic flux is to add moat structures to the device design [ 8 , 9 , 10 , 11 , 13 , 15 , 19 , 20 , 21 ], allowing the magnetic flux to be confined in designated areas. Another way is to cool rapidly so that superconducting materials can quickly become superconducting, reducing the trapped flux. Once the device experiences trapped magnetic flux, the usual method for eliminating this flux is to warm the device, which changes the material state from a superconducting state to a resistive state; hence, the trapped flux can be eliminated. Conventional probe-inserted programmable Josephson voltage standards have advantages, such as faster temperature raising or lowering than cryocooler systems, good temperature stability, and low system noise. However, these devices typically do not offer temperature measurements, which makes it difficult to determine whether the device's temperature exceeds the superconducting transition temperature during the warming process [ 26 – 30 ]. When a sample experiences trapped magnetic flux, the sample probe needs to be pulled a sufficient distance out of the dewar, increasing the device temperature well above the superconducting transition temperature to remove the flux. Without knowing the temperature in real time, it usually takes a long time to increase the temperature of the sample to room temperature and then decrease it again to 4.2 K, which greatly reduces the testing efficiency and delays the testing process. To address the aforementioned issues, a method was proposed that is based on the resistance change of superconductive devices during the superconducting transition. By applying a small current to the device and monitoring the voltage change across the programmable Josephson junction device, one can quickly determine the temperature of the device and rapidly eliminate trapped magnetic flux. This method allows for quantitative temperature monitoring of the device, improving the efficiency of magnetic flux removal, reducing the time required for the elimination of trapped flux, and facilitating the testing process. 2. Introduction to the Experiment and Josephson junction array sample The experimental setup included a liquid helium dewar for providing low temperature, a sample probe, a multichannel current source, a digital voltmeter, a programmable Josephson junction array, a computer for collecting and analyzing the data, etc., as shown in Fig. 1 . The whole programmable Josephson junction array is partitioned into 8 branches employing three stages of Wilkinson microwave power dividers. Interconnecting adjacent branches involves the use of resistances shunted with inductance low-pass filters, and the effective junctions are systematically arranged in series. Capacitances are strategically positioned at the initiation and termination points of each branch to maintain the overall floating-to-ground configuration of the series array. The two large bilateral subarrays were subsequently subdivided into eight triple-fold subarrays. Therefore, there are 22 subarrays in this sample. The sample is fabricated by a microfabrication process on a silicon wafer. We simply introduce this here. In an ultrahigh-vacuum chamber, Nb/(NbxSi1-x/Nb)2 layers are sputtered in situ. The Nb base and top layers were deposited at a sputtering power of 500 W and a deposition rate of 0.75 nm/s. Simultaneously, the NbxSi1-x film is co-sputtered with Nb and Si at 40 W and 200 W, respectively, resulting in a deposition rate of approximately 0.16 nm/s. The barrier and middle Nb layers have thicknesses of 31 nm and 67 nm, respectively, with the barrier containing approximately 15.3% Nb. To define the junction area, the top Nb layer, alternating Nb middle layer, and NbxSi1-x barrier layer were subjected to reactive ion etching (RIE) coupled with inductively coupled plasma. SF6 serves as the etching gas, while C4F8 acts as a passivating gas, safely guiding the junction profile and maintaining sharp, clean sidewalls to preserve the uniformity of the stacked junctions. For the Nb base electrode, CF4 and O2 were used as etching gases. An insulating layer of SiO2 is deposited through plasma-enhanced chemical vapor deposition. Subsequently, CHF3 and O2 are employed to etch SiO2. Prior to depositing the 500-nm-thick Nb wiring electrode, the connection area underwent DC-biased RF cleaning. The wiring layer is etched using SF6 in RIE. Finally, the microwave-absorbing resistor patterns are defined through photolithography and deposited using the magnetron sputtering method. The sample was fixed to the probe and placed inside a shielded tube with Nb superconducting magnetic shielding, which was subsequently inserted into a liquid helium dewar. A multichannel current source is used for biasing and performing DC and AC I-V characteristic scans on the 22 subarrays of the device. After analyzing the acquired data, whether the 22 subarrays were in a flux-trapping state could be determined. If some subarrays are in a flux-trapping state, then a process for eliminating them will be executed as follows. First, a small electric current is applied to the device, and the voltage across the sample is continuously measured. Then, the sample probe is pulled directly to a height slightly above the level of liquid helium (which can be estimated by the helium liquid level gauge of the dewar), waiting for the liquid helium in the probe to completely flow through the small hole in the magnetic shielding cylinder. After approximately 160 seconds, the sample rod was pulled out for another two to three centimeters until a sudden increase in voltage was observed across the device. This means that the device temperature exceeds the superconducting transition temperature and that the trapped magnetic flux in the device dissipates at the same time. At this point, the rod was stopped, and the sample probe was immediately placed back into the liquid helium dewar. Then, DC and AC scans are performed again on the I-V characteristic curves of the sample's 22 channels to verify whether the trapped magnetic flux phenomenon has been eliminated. 3. Analysis of the experimental data Based on the experimental steps mentioned in the previous section, the DC and AC I-V characteristics of the 22 subarrays of the programmable Josephson junction array with flux trapping were first measured. The critical current of each subarray was obtained through analysis, as shown in Table 1. Due to the varying degrees of suppression by the flux trapping, there is a significant difference in the critical current of each subarray. Table.1 Critical current of each sub junction array of the programmable array in the flux-trapping state No. Number Ic (mA) No. Number Ic (mA) 0 2 4.33 11 8734 6.76 1 2916 2.08 12 8734 5.89 2 108 3.47 13 8734 3.99 3 36 4.68 14 324 2.95 4 972 2.77 15 4 5.32 5 12 7.97 16 12 5.55 6 4 7.97 17 972 2.95 7 324 7.63 18 36 4.68 8 8734 2.43 19 103 5.03 9 8734 5.63 20 2916 2.95 10 8734 6.93 21 2 6.41 Due to space limitations, the I-V curves for the other subarrays are not shown here one by one, and only the I-V characteristic curve of the third subarray, which has very serious flux trapping, is presented. The black and red curves in Fig. 2 display the DC and AC I-V characteristic curves, respectively, of the third subarray under the flux trapping state. The data were measured at 4.2 K with the device biased at 18.2 GHz and 4 dbm. The DC I-V characteristic curve of the third subarray shows a significantly reduced critical current due to the suppression of flux trapping. Since the amount of flux trapping varies among the individual Josephson junctions within the same subarray, their critical currents differ from each other. This results in a "rounding" phenomenon in the overall I-V characteristic curve of the third subarray. After applying microwaves, the width of the voltage step is also suppressed by flux trapping, decreasing. Following the experimental steps from the previous section, a small current is applied to one of the subarrays in the programmable Josephson junction array device, and the voltage across it is observed as the temperature increases with increasing probe temperature from the Dewar. Currently, we choose to apply a tiny current of approximately 30 µA to the third subarray and monitor the voltage changes across it in real time. As shown in Fig. 3 , the horizontal axis represents the voltage across the sample, while the vertical axis represents the heating time. As the sample is heated by raising the probe, the voltage across the array initially remains constant. As the temperature continues to increase, the sample transitions from a superconducting state to a normal resistive state. The voltage jumps from approximately 0 mV to approximately 1.4 mV, which is directly proportional to the number of junctions in the subarray. At this point, we determine that the sample has fully transitioned to a resistive state, and the flux trapping is consequently eliminated. We measured the voltages of different subarrays with various numbers of junctions after the superconducting transition, as shown in Table 2. Subarrays with more junctions correspond to longer superconducting circuits. When the sample transitions out of superconductivity, the corresponding resistance is much greater, leading to a more significant voltage jump. For arrays with junction counts on the order of hundreds or less, the voltage difference after the transition is not very noticeable. This is mainly because the current source for biasing the device is designed to output a current of mA. When some channels are set to output a current of 0 mA for the left subarrays, leakage currents still occur at the µA level. This leads to the presence of a background voltage of approximately 1 mV after the device transitions out of superconductivity. Therefore, small subarrays correspond to nearly 1 mV voltage transitions. Table.2 Voltage of the subarrays with different junction numbers after eliminating the flux trapping Quantities of junctions in each sub arrays Normal state voltage (V) 8734 0.02 324 0.00195 4 0.0013 12 0.0013 972 0.0077 36 0.0014 108 0.0015 2916 0.006 2 0.0015 In actual applications, we need to observe voltage changes only in one of the subarrays in the device. When the voltage at which the flux trapping occurs is terminated, the sample probe is quickly returned to the Dewar to reach a temperature of 4.2 K. As the temperature decreases, the sample transitions back from the resistive state to the superconducting state. Table 3 shows the critical currents of the 22 subarrays in the programmable array after elimination of the flux trapping. Compared to Table 1, it is evident that the critical current of each subarray, now free from the suppression of flux trapping, significantly increases and becomes more uniform. Table.3 The critical current of the subarrays of the programmable array after flux trapping is eliminated No. Number Ic (mA) No. Number Ic (mA) 0 2 8.36 11 8734 7.61 1 2916 7.80 12 8734 7.61 2 108 8.17 13 8734 7.61 3 36 8.17 14 324 7.99 4 972 7.80 15 4 8.17 5 12 8.36 16 12 7.99 6 4 8.54 17 972 7.61 7 324 8.17 18 36 7.99 8 8734 7.80 19 103 7.99 9 8734 7.61 20 2916 7.61 10 8734 7.61 21 2 8.73 Figure 4 displays the DC and AC I-V characteristic curves after the elimination of the magnetic flux. After the flux trapping is eliminated, the DC I-V characteristic curve shows a significantly larger critical current value. When the current applied to the device reaches the critical current, the rounding phenomenon of the transition from a zero-voltage state to a voltage state disappears, and the transition process becomes very clear. Similarly, the rounding phenomenon on the AC characteristic curve also disappears, and the voltage steps become very distinct, with a dramatically increased width. After adopting these magnetic flux elimination steps, the time required for magnetic flux elimination was significantly reduced from the original 30 minutes to approximately 3 minutes. 4. Conclusions Most researchers are aware that Josephson junctions are relatively fragile structures that are prone to electrostatic breakdown or damage by test circuits. Therefore, during the process of temperature change, it is common to maintain a short-circuit state at both ends of the junction, avoiding the application of current to the device. This experiment introduces the innovative approach of applying a very small electric current to the device and detecting whether the device undergoes a superconducting transition. Experiments were conducted on a programmable Josephson junction containing 22 subarrays. The results indicate that this approach can avoid device damage while efficiently and rapidly eliminating flux trapping in programmable junction array devices. This approach enables the restoration of the critical current and voltage steps for 22 subarrays to normal levels. This method addresses the issue of flux trapping in testing without temperature measurement devices, significantly enhancing array testing efficiency and expediting the development of programmable Josephson junction array devices for various applications. Declarations Author Contribution H.Q.: device preparation and measure, wrote the main manuscript text;W.H.C.: design experiments, conceptualization, project administration, resources, and supervision; J.J.L., J.W., G.H.S, Q.Z., Y.Z.andS.J.W.: revised draft preparation;All authors reviewed the manuscript. References R. Knipper, S. Anders, M. Schubert et al., A low power 10 V programmable array based on NbxSi1 – x Josephson Junctions for metrology applications[J]. Supercond. Sci. Technol. 29 (9), 095015 (2016) R. Behr, O. Kieler, J. Kohlmann et al., Development and metrological applications of Josephson arrays at PTB[J]. 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Maruyama et al., Investigation of the flux state in single-flux-quantum circuits with moats by scanning SQUID microscope[J]. Supercond. Sci. Technol. 19 (5), S316 (2006) A.A. Golubov, M.Y. Kupriyanov, Effect of solitary Abrikosov vortices on the properties of Josephson tunnel junctions[J]. Zh Eksp. Teor Fiz. 92 , 1512–1523 (1987) S.L. Miller, K.R. Biagi, J.R. Clem et al., Critical currents of cross-type superconducting-normal-superconducting junctions in perpendicular magnetic fields[J]. Phys. Rev. B 31 (5), 2684 (1985) A.A. Golubov, M.Y. Kupriyanov, Theoretical investigation of Josephson tunnel junctions with spatially inhomogeneous superconducting electrodes[J]. J. Low Temp. Phys. 70 , 83–130 (1988) A. Golubov, M. Kupriyanov, Influence of single trapped Abricosov's vortices on the properties of Josephson tunnel junctions[J]. IEEE Trans. Magn. 23 (2), 785–787 (1987) C.J. Burroughs, S.P. Benz, T.E. Harvey et al., 1 volt DC programmable Josephson voltage standard[J]. IEEE Trans. Appl. 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Cite Share Download PDF Status: Published Journal Publication published 22 Aug, 2024 Read the published version in Journal of Low Temperature Physics → Version 1 posted Editorial decision: Revision requested 16 May, 2024 Reviews received at journal 09 Feb, 2024 Reviewers agreed at journal 29 Jan, 2024 Reviewers agreed at journal 29 Jan, 2024 Reviewers invited by journal 29 Jan, 2024 Submission checks completed at journal 27 Jan, 2024 Editor assigned by journal 27 Jan, 2024 First submitted to journal 23 Jan, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3890830","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":269933056,"identity":"82683d6d-9d6d-42d7-91fc-0e987221b080","order_by":0,"name":"Huan Qiao","email":"","orcid":"","institution":"College of Metrology and Measurement Engineering，China Jiliang University","correspondingAuthor":false,"prefix":"","firstName":"Huan","middleName":"","lastName":"Qiao","suffix":""},{"id":269933057,"identity":"c2920e6f-9698-44e3-8ca1-12e597ba0f61","order_by":1,"name":"Wenhui 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1","display":"","copyAsset":false,"role":"figure","size":261663,"visible":true,"origin":"","legend":"\u003cp\u003eMeasurement and analysis system\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3890830/v1/e36c85e8841ed4b9bbeae1d1.png"},{"id":50352352,"identity":"79817920-392d-4b89-b1dc-0e94fa6b2214","added_by":"auto","created_at":"2024-01-30 08:14:14","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":15028,"visible":true,"origin":"","legend":"\u003cp\u003eDC and AC I-V characteristic curves in the flux trapping state of the third subarray\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3890830/v1/565381463553db3d412347f9.png"},{"id":50352350,"identity":"15441da9-59a1-4c16-be17-e9123271f67c","added_by":"auto","created_at":"2024-01-30 08:14:14","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":10789,"visible":true,"origin":"","legend":"\u003cp\u003eVoltage variation in the third subarray with time for increasing temperature\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3890830/v1/8e6841259e715f7643a9e2b2.png"},{"id":50352667,"identity":"0e6d5bc7-9310-45d3-9af7-8544de8b85e3","added_by":"auto","created_at":"2024-01-30 08:22:14","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":16058,"visible":true,"origin":"","legend":"\u003cp\u003eThe DC and AC I-V characteristic curves of the third subarray after eliminating the flux trapping\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3890830/v1/65650fbb0a16655de99ae386.png"},{"id":63300052,"identity":"a9fc6401-b313-46d1-8f8b-ff38750490d3","added_by":"auto","created_at":"2024-08-26 16:10:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":693754,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3890830/v1/930fc11a-b55d-473d-ad4e-861d20400842.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Voltage Monitoring Approach for Eliminating Flux Trapping in Josephson Junction Arrays","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eProgrammable Josephson junction arrays [\u003cspan additionalcitationids=\"CR2 CR3 CR4 CR5 CR6\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] are superconducting integrated circuits that operate at a low temperature of 4.2 K. External electromagnetic and thermal disturbances can lead to trapped magnetic flux in these kinds of devices [\u003cspan additionalcitationids=\"CR9 CR10\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. This results in the localized trapping of quantized magnetic flux [\u003cspan additionalcitationids=\"CR13 CR14 CR15 CR16 CR17\" citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], which can suppress the critical current of the Josephson junction, thereby affecting its normal operation [\u003cspan additionalcitationids=\"CR23 CR24\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. For superconducting quantum devices, a common method to prevent trapped magnetic flux is to add moat structures to the device design [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], allowing the magnetic flux to be confined in designated areas. Another way is to cool rapidly so that superconducting materials can quickly become superconducting, reducing the trapped flux. Once the device experiences trapped magnetic flux, the usual method for eliminating this flux is to warm the device, which changes the material state from a superconducting state to a resistive state; hence, the trapped flux can be eliminated. Conventional probe-inserted programmable Josephson voltage standards have advantages, such as faster temperature raising or lowering than cryocooler systems, good temperature stability, and low system noise. However, these devices typically do not offer temperature measurements, which makes it difficult to determine whether the device's temperature exceeds the superconducting transition temperature during the warming process [\u003cspan additionalcitationids=\"CR27 CR28 CR29\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. When a sample experiences trapped magnetic flux, the sample probe needs to be pulled a sufficient distance out of the dewar, increasing the device temperature well above the superconducting transition temperature to remove the flux. Without knowing the temperature in real time, it usually takes a long time to increase the temperature of the sample to room temperature and then decrease it again to 4.2 K, which greatly reduces the testing efficiency and delays the testing process. To address the aforementioned issues, a method was proposed that is based on the resistance change of superconductive devices during the superconducting transition. By applying a small current to the device and monitoring the voltage change across the programmable Josephson junction device, one can quickly determine the temperature of the device and rapidly eliminate trapped magnetic flux. This method allows for quantitative temperature monitoring of the device, improving the efficiency of magnetic flux removal, reducing the time required for the elimination of trapped flux, and facilitating the testing process.\u003c/p\u003e"},{"header":"2. Introduction to the Experiment and Josephson junction array sample","content":"\u003cp\u003eThe experimental setup included a liquid helium dewar for providing low temperature, a sample probe, a multichannel current source, a digital voltmeter, a programmable Josephson junction array, a computer for collecting and analyzing the data, etc., as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThe whole programmable Josephson junction array is partitioned into 8 branches employing three stages of Wilkinson microwave power dividers. Interconnecting adjacent branches involves the use of resistances shunted with inductance low-pass filters, and the effective junctions are systematically arranged in series. Capacitances are strategically positioned at the initiation and termination points of each branch to maintain the overall floating-to-ground configuration of the series array. The two large bilateral subarrays were subsequently subdivided into eight triple-fold subarrays. Therefore, there are 22 subarrays in this sample.\u003c/p\u003e \u003cp\u003eThe sample is fabricated by a microfabrication process on a silicon wafer. We simply introduce this here. In an ultrahigh-vacuum chamber, Nb/(NbxSi1-x/Nb)2 layers are sputtered in situ. The Nb base and top layers were deposited at a sputtering power of 500 W and a deposition rate of 0.75 nm/s. Simultaneously, the NbxSi1-x film is co-sputtered with Nb and Si at 40 W and 200 W, respectively, resulting in a deposition rate of approximately 0.16 nm/s. The barrier and middle Nb layers have thicknesses of 31 nm and 67 nm, respectively, with the barrier containing approximately 15.3% Nb.\u003c/p\u003e \u003cp\u003eTo define the junction area, the top Nb layer, alternating Nb middle layer, and NbxSi1-x barrier layer were subjected to reactive ion etching (RIE) coupled with inductively coupled plasma. SF6 serves as the etching gas, while C4F8 acts as a passivating gas, safely guiding the junction profile and maintaining sharp, clean sidewalls to preserve the uniformity of the stacked junctions.\u003c/p\u003e \u003cp\u003eFor the Nb base electrode, CF4 and O2 were used as etching gases. An insulating layer of SiO2 is deposited through plasma-enhanced chemical vapor deposition. Subsequently, CHF3 and O2 are employed to etch SiO2. Prior to depositing the 500-nm-thick Nb wiring electrode, the connection area underwent DC-biased RF cleaning. The wiring layer is etched using SF6 in RIE. Finally, the microwave-absorbing resistor patterns are defined through photolithography and deposited using the magnetron sputtering method.\u003c/p\u003e \u003cp\u003eThe sample was fixed to the probe and placed inside a shielded tube with Nb superconducting magnetic shielding, which was subsequently inserted into a liquid helium dewar. A multichannel current source is used for biasing and performing DC and AC I-V characteristic scans on the 22 subarrays of the device. After analyzing the acquired data, whether the 22 subarrays were in a flux-trapping state could be determined. If some subarrays are in a flux-trapping state, then a process for eliminating them will be executed as follows. First, a small electric current is applied to the device, and the voltage across the sample is continuously measured. Then, the sample probe is pulled directly to a height slightly above the level of liquid helium (which can be estimated by the helium liquid level gauge of the dewar), waiting for the liquid helium in the probe to completely flow through the small hole in the magnetic shielding cylinder. After approximately 160 seconds, the sample rod was pulled out for another two to three centimeters until a sudden increase in voltage was observed across the device. This means that the device temperature exceeds the superconducting transition temperature and that the trapped magnetic flux in the device dissipates at the same time. At this point, the rod was stopped, and the sample probe was immediately placed back into the liquid helium dewar. Then, DC and AC scans are performed again on the I-V characteristic curves of the sample's 22 channels to verify whether the trapped magnetic flux phenomenon has been eliminated.\u003c/p\u003e"},{"header":"3. Analysis of the experimental data","content":"\u003cp\u003eBased on the experimental steps mentioned in the previous section, the DC and AC I-V characteristics of the 22 subarrays of the programmable Josephson junction array with flux trapping were first measured. The critical current of each subarray was obtained through analysis, as shown in Table\u0026nbsp;1. Due to the varying degrees of suppression by the flux trapping, there is a significant difference in the critical current of each subarray.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTable.1\u003c/b\u003e Critical current of each sub junction array of the programmable array in the flux-trapping state\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIc (mA)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eIc (mA)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.76\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.08\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.89\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e108\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e324\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.95\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.32\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.55\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.95\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e324\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.68\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e103\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.03\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.95\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e6.93\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e6.41\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eDue to space limitations, the I-V curves for the other subarrays are not shown here one by one, and only the I-V characteristic curve of the third subarray, which has very serious flux trapping, is presented. The black and red curves in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e display the DC and AC I-V characteristic curves, respectively, of the third subarray under the flux trapping state. The data were measured at 4.2 K with the device biased at 18.2 GHz and 4 dbm.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe DC I-V characteristic curve of the third subarray shows a significantly reduced critical current due to the suppression of flux trapping. Since the amount of flux trapping varies among the individual Josephson junctions within the same subarray, their critical currents differ from each other. This results in a \"rounding\" phenomenon in the overall I-V characteristic curve of the third subarray. After applying microwaves, the width of the voltage step is also suppressed by flux trapping, decreasing.\u003c/p\u003e \u003cp\u003eFollowing the experimental steps from the previous section, a small current is applied to one of the subarrays in the programmable Josephson junction array device, and the voltage across it is observed as the temperature increases with increasing probe temperature from the Dewar. Currently, we choose to apply a tiny current of approximately 30 \u0026micro;A to the third subarray and monitor the voltage changes across it in real time. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the horizontal axis represents the voltage across the sample, while the vertical axis represents the heating time. As the sample is heated by raising the probe, the voltage across the array initially remains constant. As the temperature continues to increase, the sample transitions from a superconducting state to a normal resistive state. The voltage jumps from approximately 0 mV to approximately 1.4 mV, which is directly proportional to the number of junctions in the subarray. At this point, we determine that the sample has fully transitioned to a resistive state, and the flux trapping is consequently eliminated.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe measured the voltages of different subarrays with various numbers of junctions after the superconducting transition, as shown in Table\u0026nbsp;2. Subarrays with more junctions correspond to longer superconducting circuits. When the sample transitions out of superconductivity, the corresponding resistance is much greater, leading to a more significant voltage jump. For arrays with junction counts on the order of hundreds or less, the voltage difference after the transition is not very noticeable. This is mainly because the current source for biasing the device is designed to output a current of mA. When some channels are set to output a current of 0 mA for the left subarrays, leakage currents still occur at the \u0026micro;A level. This leads to the presence of a background voltage of approximately 1 mV after the device transitions out of superconductivity. Therefore, small subarrays correspond to nearly 1 mV voltage transitions.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTable.2\u003c/b\u003e Voltage of the subarrays with different junction numbers after eliminating the flux trapping\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabb\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQuantities of junctions in each sub arrays\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNormal state voltage (V)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e324\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.00195\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0013\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0013\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0077\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0014\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e108\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0015\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.006\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0015\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn actual applications, we need to observe voltage changes only in one of the subarrays in the device. When the voltage at which the flux trapping occurs is terminated, the sample probe is quickly returned to the Dewar to reach a temperature of 4.2 K. As the temperature decreases, the sample transitions back from the resistive state to the superconducting state. Table\u0026nbsp;3 shows the critical currents of the 22 subarrays in the programmable array after elimination of the flux trapping. Compared to Table\u0026nbsp;1, it is evident that the critical current of each subarray, now free from the suppression of flux trapping, significantly increases and becomes more uniform.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTable.3\u003c/b\u003e The critical current of the subarrays of the programmable array after flux trapping is eliminated\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabc\" border=\"1\"\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eIc (mA)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNo.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eNumber\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eIc (mA)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e108\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e324\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.17\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e324\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e8.17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e103\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2916\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e8734\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e7.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e8.73\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e displays the DC and AC I-V characteristic curves after the elimination of the magnetic flux. After the flux trapping is eliminated, the DC I-V characteristic curve shows a significantly larger critical current value. When the current applied to the device reaches the critical current, the rounding phenomenon of the transition from a zero-voltage state to a voltage state disappears, and the transition process becomes very clear. Similarly, the rounding phenomenon on the AC characteristic curve also disappears, and the voltage steps become very distinct, with a dramatically increased width.\u003c/p\u003e \u003cp\u003eAfter adopting these magnetic flux elimination steps, the time required for magnetic flux elimination was significantly reduced from the original 30 minutes to approximately 3 minutes.\u003c/p\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eMost researchers are aware that Josephson junctions are relatively fragile structures that are prone to electrostatic breakdown or damage by test circuits. Therefore, during the process of temperature change, it is common to maintain a short-circuit state at both ends of the junction, avoiding the application of current to the device. This experiment introduces the innovative approach of applying a very small electric current to the device and detecting whether the device undergoes a superconducting transition. Experiments were conducted on a programmable Josephson junction containing 22 subarrays. The results indicate that this approach can avoid device damage while efficiently and rapidly eliminating flux trapping in programmable junction array devices. This approach enables the restoration of the critical current and voltage steps for 22 subarrays to normal levels. This method addresses the issue of flux trapping in testing without temperature measurement devices, significantly enhancing array testing efficiency and expediting the development of programmable Josephson junction array devices for various applications.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eH.Q.: device preparation and measure, wrote the main manuscript text;W.H.C.: design experiments, conceptualization, project administration, resources, and supervision; J.J.L., J.W., G.H.S, Q.Z., Y.Z.andS.J.W.: revised draft preparation;All authors reviewed the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eR. Knipper, S. Anders, M. Schubert et al., A low power 10 V programmable array based on NbxSi1\u0026thinsp;\u0026ndash;\u0026thinsp;x Josephson Junctions for metrology applications[J]. Supercond. Sci. 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Technol. 2023,\u003cb\u003e67\u003c/b\u003e(03):35\u0026ndash;42\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-low-temperature-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jltp","sideBox":"Learn more about [Journal of Low Temperature Physics](http://link.springer.com/journal/10909)","snPcode":"10909","submissionUrl":"https://submission.nature.com/new-submission/10909/3","title":"Journal of Low Temperature Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Programmable Josephson junction array, flux trapping, superconductivity, voltage standard","lastPublishedDoi":"10.21203/rs.3.rs-3890830/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3890830/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBased on the characteristic change in resistance of superconducting devices during the superconducting transition, a quantitative method to determine the temperature of superconducting devices has been proposed. This method can detect in real time whether the temperature of the superconducting device exceeds the superconducting transition temperature, thereby enabling quick elimination of the trapped magnetic flux in devices. We apply this method in our experiment, which uses a superconducting 22 subarrays programmable Josephson voltage standard (PJVS) device and obtains satisfactory results. We conclude that this method can efficiently eliminate the tapped magnetic flux in superconducting device testing systems that do not possess thermometers and facilitate testing.\u003c/p\u003e","manuscriptTitle":"A Voltage Monitoring Approach for Eliminating Flux Trapping in Josephson Junction Arrays","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-30 08:14:10","doi":"10.21203/rs.3.rs-3890830/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-05-16T16:30:03+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-02-09T13:25:01+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"00f11401-765b-49fa-bc6e-51649ae5707b","date":"2024-01-29T18:47:47+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"99f41445-dfd6-4c89-98fc-f0c378ab4f80","date":"2024-01-29T16:28:28+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-01-29T16:24:14+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-01-27T08:03:02+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-01-27T08:03:02+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Low Temperature Physics","date":"2024-01-23T10:37:35+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-low-temperature-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jltp","sideBox":"Learn more about [Journal of Low Temperature Physics](http://link.springer.com/journal/10909)","snPcode":"10909","submissionUrl":"https://submission.nature.com/new-submission/10909/3","title":"Journal of Low Temperature Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"e8d5f8c0-49be-4836-ab03-51d9de1f18fd","owner":[],"postedDate":"January 30th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-08-26T15:59:38+00:00","versionOfRecord":{"articleIdentity":"rs-3890830","link":"https://doi.org/10.1007/s10909-024-03196-w","journal":{"identity":"journal-of-low-temperature-physics","isVorOnly":false,"title":"Journal of Low Temperature Physics"},"publishedOn":"2024-08-22 15:57:04","publishedOnDateReadable":"August 22nd, 2024"},"versionCreatedAt":"2024-01-30 08:14:10","video":"","vorDoi":"10.1007/s10909-024-03196-w","vorDoiUrl":"https://doi.org/10.1007/s10909-024-03196-w","workflowStages":[]},"version":"v1","identity":"rs-3890830","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3890830","identity":"rs-3890830","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}