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Interdependent superconducting networks

Nature Physics volume 19, pages 1163–1170 (2023)Cite this article

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Abstract

Cascades are self-amplifying processes triggered by feedback mechanisms that may cause a substantial part of a macroscopic system to change its phase in response to a relatively small local event. The theoretical background for these phenomena is rich and interdisciplinary, with interdependent networks providing a versatile framework to study their multiscale evolution. However, laboratory experiments aimed at validating this ever-growing volume of predictions have not been accomplished, mostly because of the lack of a physical mechanism that realizes interdependent couplings. Here we demonstrate an experimental realization of an interdependent system as a multilayer network of two disordered superconductors separated by an electric insulating film. We show that Joule heating effects due to large driving currents act as dependency links between the superconducting layers, igniting overheating cascades via adaptive and heterogeneous back-and-forth electrothermal feedback. We characterize the phase diagram of mutual superconductive transitions and spontaneous microscopic critical processes that physically realize interdependent percolation and generalize it beyond structural dismantling. This work establishes a laboratory manifestation of the theory of interdependent systems, enabling experimental studies to control and to further develop the multiscale phenomena of complex interdependent materials.

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Fig. 1: Design and experimental setup of thermally interdependent superconducting networks.
Fig. 2: Thermally interdependent networks of RSJJs.
Fig. 3: Microscopic kinetics and their lifetime at the first-order transition thresholds.
Fig. 4: Mutual phase diagram showing theory versus experiments.

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Data availability

Source data are provided with this paper. All data supporting our findings are available from the corresponding author upon reasonable request.

Code availability

Source codes and videos showing the states of resistors, their currents and the power dissipated in both layers during the transition can be freely accessed at the GitHub repository: https://github.com/BnayaGross/Interdependent-SC-networks.

References

  1. Yang, Y., Nishikawa, T. & Motter, A. E. Small vulnerable sets determine large network cascades in power grids. Science 358, eaan3184 (2017).

    Google Scholar 

  2. Schäfer, B., Witthaut, D., Timme, M. & Latora, V. Dynamically induced cascading failures in power grids. Nat. Commun. 9, 1975 (2018).

    ADS  Google Scholar 

  3. Rinaldi, S. M., Peerenboom, J. P. & Kelly, T. K. Identifying, understanding, and analyzing critical infrastructure interdependencies. IEEE Control Syst. 21, 11–25 (2001).

    Google Scholar 

  4. Rosato, V. et al. Modelling interdependent infrastructures using interacting dynamical models. Int. J. Crit. Infrastruct. 4, 63–79 (2008).

    Google Scholar 

  5. Hokstad, P., Utne, I. B. & Vatn, J. Risk and Interdependencies in Critical Infrastructures (Springer, 2012).

  6. Haldane, A. G. & May, R. M. Systemic risk in banking ecosystems. Nature 469, 351–355 (2011).

    ADS  Google Scholar 

  7. Pocock, M. J., Evans, D. M. & Memmott, J. The robustness and restoration of a network of ecological networks. Science 335, 973–977 (2012).

    ADS  Google Scholar 

  8. Helbing, D. Globally networked risks and how to respond. Nature 497, 51–59 (2013).

    ADS  Google Scholar 

  9. Rocha, J. C., Peterson, G., Bodin, Ö. & Levin, S. Cascading regime shifts within and across scales. Science 362, 1379–1383 (2018).

    ADS  Google Scholar 

  10. Scheffer, M. Critical Transitions in Nature and Society, Vol. 16 (Princeton Univ. Press, 2020).

  11. Borge-Holthoefer, J., Banos, R. A., González-Bailón, S. & Moreno, Y. Cascading behaviour in complex socio-technical networks. J. Complex Netw. 1, 3–24 (2013).

    Google Scholar 

  12. Morone, F. & Makse, H. A. Influence maximization in complex networks through optimal percolation. Nature 524, 65–68 (2015).

    ADS  Google Scholar 

  13. Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).

    ADS  Google Scholar 

  14. Bianconi, G. Multilayer Networks: Structure and Function (Oxford Univ. Press, 2018).

  15. Barabási, A.-L. Network Science (Cambridge Univ. Press, 2016).

  16. Baxter, G. J., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Avalanche collapse of interdependent networks. Phys. Rev. Lett. 109, 248701 (2012).

    ADS  Google Scholar 

  17. Bashan, A., Berezin, Y., Buldyrev, S. V. & Havlin, S. The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys. 9, 667–672 (2013).

    Google Scholar 

  18. Radicchi, F. Percolation in real interdependent networks. Nat. Phys. 11, 597–602 (2015).

    Google Scholar 

  19. Klosik, D. F., Grimbs, A., Bornholdt, S. & Hütt, M.-T. The interdependent network of gene regulation and metabolism is robust where it needs to be. Nat. Commun. 8, 534 (2017).

    ADS  Google Scholar 

  20. Nicosia, V., Skardal, P. S., Arenas, A. & Latora, V. Collective phenomena emerging from the interactions between dynamical processes in multiplex networks. Phys. Rev. Lett. 118, 138302 (2017).

    ADS  Google Scholar 

  21. Danziger, M. M., Bonamassa, I., Boccaletti, S. & Havlin, S. Dynamic interdependence and competition in multilayer networks. Nat. Phys. 15, 178–185 (2019).

    Google Scholar 

  22. Morris, R. G. & Barthelemy, M. Transport on coupled spatial networks. Phys. Rev. Lett. 109, 128703 (2012).

    ADS  Google Scholar 

  23. Saito, Y., Nojima, T. & Iwasa, Y. Highly crystalline 2d superconductors. Nat. Rev. Mater. 2, 16094 (2017).

    ADS  Google Scholar 

  24. Sacépé, B. et al. Localization of preformed cooper pairs in disordered superconductors. Nat. Phys. 7, 239–244 (2011).

    Google Scholar 

  25. Doron, A., Levinson, T., Gorniaczyk, F., Tamir, I. & Shahar, D. The critical current of disordered superconductors near 0 K. Nat. Commun. 11, 2667 (2020).

    ADS  Google Scholar 

  26. Parshani, R., Buldyrev, S. V. & Havlin, S. Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105, 048701 (2010).

    ADS  Google Scholar 

  27. Havlin, S. & Ben-Avraham, D. Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987).

    ADS  Google Scholar 

  28. Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 45, 574–588 (1973).

    ADS  Google Scholar 

  29. Coniglio, A. Cluster structure near the percolation threshold. J. Phys. A 15, 3829–3844 (1982).

    MathSciNet  ADS  Google Scholar 

  30. Skvortsov, M. A. & Feigel’man, M. V. Superconductivity in disordered thin films: giant mesoscopic fluctuations. Phys. Rev. Lett. 95, 057002 (2005).

    ADS  Google Scholar 

  31. Gurevich, A. V. I. & Mints, R. G. Self-heating in normal metals and superconductors. Rev. Mod. Phys. 59, 941–999 (1987).

    ADS  Google Scholar 

  32. Danziger, M. M., Bashan, A. & Havlin, S. Interdependent resistor networks with process-based dependency. New J. Phys. 17, 043046 (2015).

    ADS  Google Scholar 

  33. Bonamassa, I., Gross, B. & Havlin, S. Interdependent couplings map to thermal, higher-order interactions. Preprint at https://doi.org/10.48550/arXiv.2110.08907 (2021).

  34. Cho, Y. S., Hwang, S., Herrmann, H. J. & Kahng, B. Avoiding a spanning cluster in percolation models. Science 339, 1185–1187 (2013).

    ADS  Google Scholar 

  35. De Gennes, P.-G. On a relation between percolation theory and the elasticity of gels. J. Physique Lett. 37, 1–2 (1976).

    ADS  Google Scholar 

  36. Ponta, L., Carbone, A., Gilli, M. & Mazzetti, P. Resistive transition in granular disordered high tc superconductors: a numerical study. Phys. Rev. B 79, 134513 (2009).

    ADS  Google Scholar 

  37. Berman, R. The thermal conductivities of some dielectric solids at low temperatures (experimental). Proc. R. Soc. Lond. Ser. A 208, 90–108 (1951).

    ADS  Google Scholar 

  38. Moore, A. L. & Shi, L. Emerging challenges and materials for thermal management of electronics. Mater. Today 17, 163–174 (2014).

    Google Scholar 

  39. Binder, K. Theory of first-order phase transitions. Rep. Prog. Phys. 50, 783–859 (1987).

    ADS  Google Scholar 

  40. Krzakala, F. & Zdeborová, L. On melting dynamics and the glass transition. i. glassy aspects of melting dynamics. J. Chem. Phys. 134, 034512 (2011).

    ADS  Google Scholar 

  41. Zhou, D. et al. Simultaneous first-and second-order percolation transitions in interdependent networks. Phys. Rev. E 90, 012803 (2014).

    ADS  Google Scholar 

  42. Zapperi, S., Lauritsen, K. B. & Stanley, H. E. Self-organized branching processes: mean-field theory for avalanches. Phys. Rev. Lett. 75, 4071–4074 (1995).

    ADS  Google Scholar 

  43. Halperin, B. I. & Nelson, D. R. Theory of two-dimensional melting. Phys. Rev. Lett. 41, 121–124 (1978).

    MathSciNet  ADS  Google Scholar 

  44. Huang, B. et al. Layer-dependent ferromagnetism in a van der waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).

    ADS  Google Scholar 

  45. Gibertini, M., Koperski, M., Morpurgo, A. F. & Novoselov, K. S. Magnetic 2D materials and heterostructures. Nat. Nanotechnol. 14, 408–419 (2019).

    ADS  Google Scholar 

  46. Shurakov, A., Lobanov, Y. & Goltsman, G. Superconducting hot-electron bolometer: from the discovery of hot-electron phenomena to practical applications. Supercond. Sci. Technol. 29, 023001 (2015).

    ADS  Google Scholar 

  47. Meijer, G. I. Who wins the nonvolatile memory race? Science 319, 1625–1626 (2008).

    Google Scholar 

  48. Orr, B. G., Jaeger, H. M., Goldman, A. M. & Kuper, C. G. Global phase coherence in two-dimensional granular superconductors. Phys. Rev. Lett. 56, 378–381 (1986).

    ADS  Google Scholar 

  49. Chakravarty, S., Ingold, G.-L., Kivelson, S. & Luther, A. Onset of global phase coherence in josephson-junction arrays: a dissipative phase transition. Phys. Rev. Lett. 56, 2303–2306 (1986).

    ADS  Google Scholar 

  50. Chakravarty, S., Ingold, G.-L., Kivelson, S. & Zimanyi, G. Quantum statistical mechanics of an array of resistively shunted josephson junctions. Phys. Rev. B 37, 3283–3294 (1988).

    ADS  Google Scholar 

  51. Abraham, D. W., Lobb, C. J., Tinkham, M. & Klapwijk, T. M. Resistive transition in two-dimensional arrays of superconducting weak links. Phys. Rev. B 26, 5268–5271 (1982).

    ADS  Google Scholar 

  52. Lobb, C. J., Abraham, D. W. & Tinkham, M. Theoretical interpretation of resistive transition data from arrays of superconducting weak links. Phys. Rev. B 27, 150–157 (1983).

    ADS  Google Scholar 

  53. Josephson, B. D. Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251–253 (1962).

    MATH  ADS  Google Scholar 

  54. Ambegaokar, V. & Baratoff, A. Tunnelling between superconductors. Phys. Rev. Lett. 10, 486–489 (1963).

    ADS  Google Scholar 

  55. Dubi, Y., Meir, Y. & Avishai, Y. Nature of the superconductor-insulator transition in disordered superconductors. Nature 449, 876–880 (2007).

    ADS  Google Scholar 

  56. Baturina, T. I., Mironov, A. Y., Vinokur, V. M., Baklanov, M. R. & Strunk, C. Localized superconductivity in the quantum-critical region of the disorder-driven superconductor-insulator transition in tin thin films. Phys. Rev. Lett. 99, 257003 (2007).

    ADS  Google Scholar 

  57. Sacépé, B. et al. Disorder-induced inhomogeneities of the superconducting state close to the superconductor-insulator transition. Phys. Rev. Lett. 101, 157006 (2008).

    ADS  Google Scholar 

  58. Ponta, L., Andreoli, V. & Carbone, A. Superconducting-insulator transition in disordered josephson junctions networks. Eur. Phys. J. B 86, 1–5 (2013).

    MathSciNet  MATH  Google Scholar 

  59. Aslamazov, L. G. & Larkin, A. I. in 30 Years Of The Landau Institute Vol. 11 (ed. Khalatnikov, I. M.) 23–28 (World Scientific Series in 20th Century Physics, 1996).

  60. Baturina, T. I. et al. Superconductivity on the localization threshold and magnetic-field-tuned superconductor-insulator transition in tin films. J. Exp. Theor. Phys. Lett. 79, 337–341 (2004).

    Google Scholar 

  61. Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).

    ADS  Google Scholar 

  62. Gao, J., Buldyrev, S. V., Stanley, H. E. & Havlin, S. Networks formed from interdependent networks. Nat. Phys. 8, 40–48 (2012).

    Google Scholar 

  63. Rowe, D. M. Thermoelectrics Handbook: Macro to Nano (CRC Press, 2018).

  64. Motter, A. E. & Yang, Y. The unfolding and control of network cascades. Phys. Today 70, 32–39 (2017).

    ADS  Google Scholar 

  65. Li, W., Bashan, A., Buldyrev, S. V., Stanley, H. E. & Havlin, S. Cascading failures in interdependent lattice networks: the critical role of the length of dependency links. Phys. Rev. Lett. 108, 228702 (2012).

    ADS  Google Scholar 

  66. Danziger, M. M., Shekhtman, L. M., Berezin, Y. & Havlin, S. The effect of spatiality on multiplex networks. Europhys. Lett. 115, 36002 (2016).

    ADS  Google Scholar 

  67. Gross, B., Bonamassa, I. & Havlin, S. Interdependent transport via percolation backbones in spatial networks. Physica A 567, 125644 (2021).

    Google Scholar 

  68. Mezard, M. & Montanari, A. Information, Physics, and Computation (Oxford Univ. Press, 2009).

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Acknowledgements

S.H. acknowledges financial support from the Israel Science Foundation (ISF), the China–Israel Science Foundation, the Office of Naval Research (ONR), the Bar-Ilan University Center for Research in Applied Cryptography and Cyber Security, the EU project RISE, the US-Israel Binational Science Foundation (NSF-BSF) grant no. 2019740 and the Defense Threat Reduction Agency (DTRA) grant no. HDTRA-1-19-1-0016. I.B., A.F. and S.H. acknowledge partial support from the Italy-Israel grant ‘EXPLICS’. A.F. acknowledges partial support from the ISF Israel–China grant no. 3192/19. B.G. acknowledges the support of the Mordecai and Monique Katz Graduate Fellowship Program. We thank A. Bashan, M. M. Danziger and S. Boccaletti for stimulating discussions.

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Author notes

  1. These authors contributed equally: I. Bonamassa, B. Gross, M. Laav.

Authors and Affiliations

  1. Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

    I. Bonamassa, B. Gross, M. Laav, I. Volotsenko, A. Frydman & S. Havlin

  2. Department of Network and Data Science, CEU, Vienna, Austria

    I. Bonamassa

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Contributions

I.B., A.F. and S.H. initiated and designed the research. M.L., I.V. and A.F. fabricated the samples, carried on the experiments and collected the data. I.B. and B.G. developed the modelling and the adaptive algorithm for solving the thermally coupled Kirchhoff equations. B.G. designed the codes. B.G. and I.B. carried out the numerical simulations. I.B. designed and created the figures. I.B. developed the mean-field theory. I.B. was the leading writer of the paper with contributions from B.G., S.H. and A.F. A.F. and S.H. supervised the research. All authors critically reviewed and approved the paper.

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Correspondence to I. Bonamassa.

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Nature Physics thanks Juan Rocha, Simon Levin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–8.

Supplementary Video 1

Evolution of the current flow during the metastable plateau stage both for the heating and cooling directions.

Supplementary Video 2

Evolution of the power dissipated during the metastable plateau stage both for the heating and cooling directions.

Source data

Source Data Fig. 1

Data plotted in Fig. 1c,f.

Source Data Fig. 2

Data plotted in Fig. 2c,d.

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Bonamassa, I., Gross, B., Laav, M. et al. Interdependent superconducting networks. Nat. Phys. 19, 1163–1170 (2023). https://doi.org/10.1038/s41567-023-02029-z

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