# Charging of Superconducting Layers in Arrays of Coupled Josephson Junctions for Overcritical Currents

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## Abstract

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## 1. Introduction

## 2. Model and Method

## 3. Main Results

#### 3.1. Appearance of Branching for Overcritical Currents

#### 3.2. Charging of the Superconducting Layers

#### 3.3. The Effect of Number o Junctions in the Stack

#### 3.4. Effects of Boundary Conditions

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

JJ | Josephson junctions |

TW | Traveling wave |

IV-characteristic | Current voltage characteristic |

FFT | Fast Fourier Transform |

## References

- McCumber, D. Effect of ac impedance on dc voltage-current characteristics of superconductor weak-link junctions. J. Appl. Phys.
**1968**, 39, 3113–3118. [Google Scholar] [CrossRef] - Weinstock, H.; Nisenoff, M. Superconducting Electronics; Springer Science & Business Media: Berlin, Germany, 2013; Volume 59. [Google Scholar]
- Holmes, D.S.; Ripple, A.L.; Manheimer, M.A. Energy-efficient superconducting computing—Power budgets and requirements. IEEE Trans. Appl. Superconduct.
**2013**, 23, 1701610. [Google Scholar] [CrossRef] - Cassidy, M.; Bruno, A.; Rubbert, S.; Irfan, M.; Kammhuber, J.; Schouten, R.; Akhmerov, A.; Kouwenhoven, L. Demonstration of an ac Josephson junction laser. Science
**2017**, 355, 939–942. [Google Scholar] [CrossRef] [PubMed][Green Version] - Clark, T. Experiments on coupled Josephson junctions. Phys. Lett. A
**1968**, 27, 585–586. [Google Scholar] [CrossRef] - Tilley, D. Superradiance in arrays of superconducting weak links. Phys. Lett. A
**1970**, 33, 205–206. [Google Scholar] [CrossRef] - Welp, U.; Kadowaki, K.; Kleiner, R. Superconducting emitters of THz radiation. Nat. Photonics
**2013**, 7, 702. [Google Scholar] [CrossRef] - Ioffe, L.; Feigel’man, M. Possible realization of an ideal quantum computer in Josephson junction array. Phys. Rev. B
**2002**, 66, 224503. [Google Scholar] [CrossRef][Green Version] - Kadigrobov, A.; Shekhter, R.; Jonson, M. Quantum spin fluctuations as a source of long-range proximity effects in diffusive ferromagnet-super conductor structures. EPL Europhys. Lett.
**2001**, 54, 394. [Google Scholar] [CrossRef] - Stoutimore, M.J.A. Non-Sinusoidal Current-Phase Relations in Superconductor-Ferromagnet-Superconductor Josephson Junctions; University of Illinois at Urbana-Champaign: Champaign, IL, USA, 2009. [Google Scholar]
- Macklin, C.; O’Brien, K.; Hover, D.; Schwartz, M.; Bolkhovsky, V.; Zhang, X.; Oliver, W.; Siddiqi, I. A near-quantum-limited Josephson traveling-wave parametric amplifier. Science
**2015**, 350, 307–310. [Google Scholar] [CrossRef] - O’Brien, K.; Macklin, C.; Siddiqi, I.; Zhang, X. Resonant phase matching of Josephson junction traveling wave parametric amplifiers. Phys. Rev. Lett.
**2014**, 113, 157001. [Google Scholar] [CrossRef] - Davidson, A. Distributed Array of Josephson Devices with Coherence. U.S. Patent 4,344,052, 10 August 1982. [Google Scholar]
- Kleiner, R.; Müller, P. Intrinsic Josephson effects in high-T c superconductors. Phys. Rev. B
**1994**, 49, 1327. [Google Scholar] [CrossRef] [PubMed] - Shimakage, H.; Tamura, Y. Chaotic oscillations in Josephson junctions for random number generation. IEEE Trans. Appl. Superconduct.
**2015**, 25, 1–4. [Google Scholar] [CrossRef] - Stewart, W. Current-voltage characteristics of Josephson junctions. Appl. Phys. Lett.
**1968**, 12, 277. [Google Scholar] [CrossRef] - Irie, A.; Shukrinov, Y.M.; Oya, G. Experimental manifestation of the breakpoint region in the current-voltage characteristics of intrinsic Josephson junctions. Appl. Phys. Lett.
**2008**, 93, 152510. [Google Scholar] [CrossRef][Green Version] - Shukrinov, Y.M.; Mahfouzi, F.; Suzuki, M. Structure of the breakpoint region on current-voltage characteristics of intrinsic Josephson junctions. Phys. Rev. B
**2008**, 78, 134521. [Google Scholar] [CrossRef][Green Version] - Shukrinov, Y.M.; Mahfouzi, F. Branching in current–voltage characteristics of intrinsic Josephson junctions. Superconduct. Sci. Technol.
**2006**, 20, S38. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Mahfouzi, F.; Pedersen, N.F. Investigation of the breakpoint region in stacks with a finite number of intrinsic Josephson junctions. Phys. Rev. B
**2007**, 75, 104508. [Google Scholar] [CrossRef][Green Version] - Shukrinov, Y.M.; Hamdipour, M.; Kolahchi, M. Effect of interjunction coupling on superconducting current and charge correlations in intrinsic Josephson junctions. Phys. Rev. B
**2009**, 80, 014512. [Google Scholar] [CrossRef] - Du, J.; Hellicar, A.; Leslie, K.; Nikolic, N.; Hanham, S.; Macfarlane, J.; Foley, C. Towards large scale HTS Josephson detector arrays for THz imaging. Superconduct. Sci. Technol.
**2013**, 26, 115012. [Google Scholar] [CrossRef] - Holdengreber, E.; Moshe, A.; Mizrahi, M.; Khavkin, V.; Schacham, S.; Farber, E. High sensitivity high Tc superconducting Josephson junction antenna for 200 GHz detection. J. Electromagn. Waves Appl.
**2019**, 33, 193–203. [Google Scholar] [CrossRef] - Hamdipour, M. Detailed investigation of the bifurcation diagram of capacitively coupled Josephson junctions in high-Tc superconductors and its self similarity. Phys. C Superconduct. Appl.
**2018**, 547, 66–68. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Rahmonov, I.; Kulikov, K.; Seidel, P. Effects of LC shunting on the Shapiro steps features of Josephson junction. EPL Europhys. Lett.
**2015**, 110, 47001. [Google Scholar] [CrossRef] - Machida, M.; Koyama, T.; Tachiki, M. Dynamical breaking of charge neutrality in intrinsic Josephson junctions: Common origin for microwave resonant absorptions and multiple-branch structures in the I-V characteristics. Phys. Rev. Lett.
**1999**, 83, 4618. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Gaafar, M. Charging of superconducting layers and resonance-related hysteresis in the current-voltage characteristics of coupled Josephson junctions. Phys. Rev. B
**2011**, 84, 094514. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Hamdipour, M. Charge creation and nucleation of the longitudinal plasma wave in coupled Josephson junctions. EPL Europhys. Lett.
**2010**, 92, 37010. [Google Scholar] [CrossRef] - Machida, M.; Sakai, S. Unified theory for magnetic and electric field coupling in multistacked Josephson junctions. Phys. Rev. B
**2004**, 70, 144520. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Mahfouzi, F. Influence of coupling between junctions on breakpoint current in intrinsic Josephson junctions. Phys. Rev. Lett.
**2007**, 98, 157001. [Google Scholar] [CrossRef] [PubMed] - Shukrinov, Y.M.; Hamdipour, M. The c-axis charge traveling wave in a coupled system of Josephson junctions. JETP Lett.
**2012**, 95, 307–313. [Google Scholar] [CrossRef][Green Version] - Lyatti, M.; Wolff, M.; Savenko, A.; Kruth, M.; Ferrari, S.; Poppe, U.; Pernice, W.; Dunin-Borkowski, R.; Schuck, C. Experimental evidence for hotspot and phase-slip mechanisms of voltage switching in ultrathin YBa
_{2}Cu_{3}O_{7−x}nanowires. Phys. Rev. B**2018**, 98, 054505. [Google Scholar] [CrossRef] - Lyatti, M.; Savenko, A.; Poppe, U.; Dunin-Borkowski, R. High-quality YBa
_{2}Cu_{3}O_{7−x}nanobridges fabricated by FIB etching. arXiv**2016**, arXiv:1603.03459. [Google Scholar] - Shukrinov, Y.M.; Rahmonov, I. Diffusion current in a system of coupled Josephson junctions. J. Exp. Theor. Phys.
**2012**, 115, 289–302. [Google Scholar] [CrossRef] - Koyama, T.; Tachiki, M. I-V characteristics of Josephson-coupled layered superconductors with longitudinal plasma excitations. Phys. Rev. B
**1996**, 54, 16183. [Google Scholar] [CrossRef] [PubMed] - Matsumoto, H.; Sakamoto, S.; Wajima, F.; Koyama, T.; Machida, M. Simulation of I-V hysteresis branches in an intrinsic stack of Josephson junctions in high-T c superconductors. Phys. Rev. B
**1999**, 60, 3666. [Google Scholar] [CrossRef] - Shukrinov, Y.M.; Mahfouzi, F.; Seidel, P. Equidistance of branch structure in capacitively coupled Josephson junctions model with diffusion current. Phys. C Superconduct.
**2006**, 449, 62–66. [Google Scholar] [CrossRef] - Shukrinov Yu, M.; Rahmonov, I.R.K.K.V. Application of Numerical Methods for the Study of the Josephson Effect: Textbook; JINR: Dubna, Russia, 2016. [Google Scholar]
- Kondepudi, D.; Prigogine, I. Modern Thermodynamics: From Heat Engines to Dissipative Structures; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]

**Figure 1.**IV-characteristics for a junction for $\beta =0.9$ with the return current ${I}_{r}$ is shown with a dark blue line and for the first junction in the stack of 3 junctions for $\beta =0.9$ and $\alpha =1$ with the return current $I{\u2019}_{r}$, shown with an orange line. The directions of current are shown with arrows next to the IV-characteristics. Simulations have been done for periodic boundary conditions.

**Figure 2.**Dependences on current of voltage (dark color lines), together with minimal and maximal values of charge (light color lines), are shown for the increase (

**a**) and for the decrease (

**b**) of current. The direction of current is shown with arrows. Labels for outermost switching and the wave switching currents are shown next to the corresponding current values.

**Figure 3.**Charge traveling wave for the first, second and third superconducting layers are presented for current value of $I/{I}_{c}=1.3$ for the increase (

**a**) and the decrease (

**c**) of current. Corresponding results of Fast Fourier Transform (FFT) analysis are presented in (

**b**,

**d**).

**Figure 4.**The I-V characteristics for the JJs arrays with different number of junctions in the stack N and the same dissipation of $\beta =0.8$ and the coupling of $\alpha =1$. The branching (for arrays with $N\ge 2$ junctions) is put in evidence by dashed ellipses.

**Figure 5.**(

**a**) IV-characteristics for a system of 3 junctions in a stack with coupling parameter $\alpha =1$, $\beta =0.9$ and periodic boundary conditions (shown with a green line) and $\gamma =0.5$ (shown with an orange line). (

**b**) Zoom at the IV-characteristic for the region marked by an elipse (currents from $1.36$ ${I}_{c}$ to $1.48$ ${I}_{c}$) for junctions 1, 2, and 3.

**Figure 6.**Maximal and minimal values of charge (light color lines) on the superconducting layers of the stack for $\alpha =1$, $\beta =0.9$ and $\gamma =0.5$. For the increase of current (

**a**) on the 1st superconducting layer (

**c**) on the 2nd superconducting layer. For the decrease of current (

**b**) on the 1st superconducting layer ($l=1$), and (

**d**) on the 2nd superconducting layer ($l=2$). The IV-characteristics for the corresponding direction of currents are presented with dark lines.

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**MDPI and ACS Style**

Cuzminschi, M.; Zubarev, A. Charging of Superconducting Layers in Arrays of Coupled Josephson Junctions for Overcritical Currents. *Crystals* **2019**, *9*, 327.
https://doi.org/10.3390/cryst9070327

**AMA Style**

Cuzminschi M, Zubarev A. Charging of Superconducting Layers in Arrays of Coupled Josephson Junctions for Overcritical Currents. *Crystals*. 2019; 9(7):327.
https://doi.org/10.3390/cryst9070327

**Chicago/Turabian Style**

Cuzminschi, Marina, and Alexei Zubarev. 2019. "Charging of Superconducting Layers in Arrays of Coupled Josephson Junctions for Overcritical Currents" *Crystals* 9, no. 7: 327.
https://doi.org/10.3390/cryst9070327