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Modelling electrical conduction in nanostructure assemblies through complex networks

Abstract

Carrier transport processes in assemblies of nanostructures rely on morphology-dependent and hierarchical conduction mechanisms, whose complexity cannot be captured by current modelling approaches. Here we apply the concept of complex networks to modelling carrier conduction in such systems. The approach permits assignment of arbitrary connectivity and connection strength between assembly constituents and is thus ideal for nanostructured films, composites and other geometries. Modelling of simplified rod-like nanostructures is consistent with analytical solutions, whereas results for more realistic nanostructure assemblies agree with experimental data and reveal conduction behaviour not captured by previous models. Fitting of ensemble measurements also allows the conduction properties of individual constituents to be extracted, which are subsequently used to guide the realization of transparent electrodes with improved performance. A global optimization process was employed to identify geometries and properties with high potential for transparent conductors. Our intuitive discretization approach, combined with a simple solver tool, allows researchers with little computational experience to carry out realistic simulations.

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Fig. 1: Complex network modelling approach.
Fig. 2: Modelling rod-like constituents.
Fig. 3: Complex assemblies and comparison with experiments.
Fig. 4: Prediction of high-performance TCF materials.

Data availability

The datasets generated during and/or analysed during the current study are available online at https://doi.org/10.6084/m9.figshare.c.4879863.

Code availability

Sample code employed in this work is available in the Supplementary Information and additional examples are available at https://doi.org/10.6084/m9.figshare.c.4879863.

References

  1. 1.

    Hempel, M., Nezich, D., Kong, J. & Hofmann, M. A novel class of strain gauges based on layered percolative films of 2D materials. Nano Lett. 12, 5714–5718 (2012).

    CAS  Google Scholar 

  2. 2.

    Zhang, L. L. & Zhao, X. S. Carbon-based materials as supercapacitor electrodes. Chem. Soc. Rev. 38, 2520–2531 (2009).

    CAS  Google Scholar 

  3. 3.

    Law, M., Greene, L. E., Johnson, J. C., Saykally, R. & Yang, P. Nanowire dye-sensitized solar cells. Nat. Mater. 4, 455–459 (2005).

    CAS  Google Scholar 

  4. 4.

    Park, S., Vosguerichian, M. & Bao, Z. A review of fabrication and applications of carbon nanotube film-based flexible electronics. Nanoscale 5, 1727–1752 (2013).

    CAS  Google Scholar 

  5. 5.

    Blom, P. W. M., Mihailetchi, V. D., Koster, L. J. A. & Markov, D. E. Device physics of polymer:fullerene bulk heterojunction solar cells. Adv. Mater. 19, 1551–1566 (2007).

    CAS  Google Scholar 

  6. 6.

    Manning, H. G. et al. Emergence of winner-takes-all connectivity paths in random nanowire networks. Nat. Commun. 9, 3219 (2018).

    Google Scholar 

  7. 7.

    Yao, H., Hempel, M., Hsieh, Y. P., Kong, J. & Hofmann, M. Characterizing percolative materials by straining. Nanoscale 11, 1074–1079 (2018).

    Google Scholar 

  8. 8.

    Ni, X., Hui, C., Su, N., Jiang, W. & Liu, F. Monte Carlo simulations of electrical percolation in multicomponent thin films with nanofillers. Nanotechnology 29, 075401 (2018).

    Google Scholar 

  9. 9.

    White, S. I. et al. Electrical percolation behavior in silver nanowire–polystyrene composites: simulation and experiment. Adv. Funct. Mater. 20, 2709–2716 (2010).

    CAS  Google Scholar 

  10. 10.

    Li, J., Ray, B., Alam, M. A. & Ostling, M. Threshold of hierarchical percolating systems. Phys. Rev. E 85, 021109 (2012).

    Google Scholar 

  11. 11.

    Sillin, H. O. et al. A theoretical and experimental study of neuromorphic atomic switch networks for reservoir computing. Nanotechnology 24, 384004 (2013).

    Google Scholar 

  12. 12.

    da Rocha, C. G. et al. Ultimate conductivity performance in metallic nanowire networks. Nanoscale 7, 13011–13016 (2015).

    Google Scholar 

  13. 13.

    Chen, T. W., Hsieh, Y. P. & Hofmann, M. Ad-layers enhance graphene’s performance. RSC Adv. 5, 93684–93688 (2015).

    CAS  Google Scholar 

  14. 14.

    Huang, J. C. Carbon black filled conducting polymers and polymer blends. Adv. Polym. Technol. 21, 299–313 (2002).

    CAS  Google Scholar 

  15. 15.

    Zhu, C. et al. Strain engineering in perovskite solar cells and its impacts on carrier dynamics. Nat. Commun. 10, 815 (2019).

    CAS  Google Scholar 

  16. 16.

    Qazilbash, M. M. et al. Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging. Science 318, 1750–1753 (2007).

    CAS  Google Scholar 

  17. 17.

    Zhang, Y. B., Brar, V. W., Girit, C., Zettl, A. & Crommie, M. F. Origin of spatial charge inhomogeneity in graphene. Nat. Phys. 5, 722–726 (2009).

    CAS  Google Scholar 

  18. 18.

    Demis, E. C. et al. Atomic switch networks—nanoarchitectonic design of a complex system for natural computing. Nanotechnology 26, 204003 (2015).

    CAS  Google Scholar 

  19. 19.

    Bauhofer, W. & Kovacs, J. Z. A review and analysis of electrical percolation in carbon nanotube polymer composites. Compos. Sci. Technol. 69, 1486–1498 (2009).

    CAS  Google Scholar 

  20. 20.

    Albert, R. & Barabasi, A. L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).

    Google Scholar 

  21. 21.

    Papo, D., Buldú, J. M., Boccaletti, S. & Bullmore, E. T. Complex network theory and the brain. Phil. Trans. R. Soc. Lond. B 369, 20130520 (2014).

    Google Scholar 

  22. 22.

    Lee, D., Kahng, B., Cho, Y. S., Goh, K. I. & Lee, D. S. Recent advances of percolation theory in complex networks. J. Korean Phys. Soc. 73, 152–164 (2018).

    Google Scholar 

  23. 23.

    Kryven, I. Bond percolation in coloured and multiplex networks. Nat. Commun. 10, 404 (2019).

    CAS  Google Scholar 

  24. 24.

    McRae, B. H., Dickson, B. G., Keitt, T. H. & Shah, V. B. Using circuit theory to model connectivity in ecology and conservation. Ecology 10, 2712–2724 (2008).

    Google Scholar 

  25. 25.

    Anantharaman, R., Hall, K., Shah, V. & Edelman, A. Circuitscape in Julia: high performance connectivity modelling to support conservation decisions. Preprint at https://arXiv.org/abs/1906.03542 (2019).

  26. 26.

    Balberg, I., Binenbaum, N. & Anderson, C. H. Critical-behavior of the two-dimensional sticks system. Phys. Rev. Lett. 51, 1605–1608 (1983).

    Google Scholar 

  27. 27.

    Li, J. & Zhang, S. L. Conductivity exponents in stick percolation. Phys. Rev. E 81, 021120 (2010).

    Google Scholar 

  28. 28.

    Mutiso, R. M. & Winey, K. I. Electrical percolation in quasi-two-dimensional metal nanowire networks for transparent conductors. Phys. Rev. E 88, 032134 (2013).

    Google Scholar 

  29. 29.

    Khanarian, G. et al. The optical and electrical properties of silver nanowire mesh films. J. Appl. Phys. 114, 749–755 (2013).

    Google Scholar 

  30. 30.

    Goh, G. L., Saengchairat, N., Agarwala, S., Yeong, W. Y. & Tran, T. Sessile droplets containing carbon nanotubes: a study of evaporation dynamics and CNT alignment for printed electronics. Nanoscale 11, 10603–10614 (2019).

    CAS  Google Scholar 

  31. 31.

    Frey, E., Tauber, U. C. & Schwabl, F. Crossover from isotropic to directed percolation. Phys. Rev. E 49, 5058–5072 (1994).

    CAS  Google Scholar 

  32. 32.

    Wu, B., Geng, D. & Liu, Y. Evaluation of metallic and semiconducting single-walled carbon nanotube characteristics. Nanoscale 3, 2074–2085 (2011).

    CAS  Google Scholar 

  33. 33.

    Rylkov, V. V. et al. in Novel Magnetic Nanostructures 427–464 (Elsevier, 2018).

  34. 34.

    Rintoul, M. D. & Torquato, S. Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model. J. Phys. A 30, L585–L592 (1997).

    CAS  Google Scholar 

  35. 35.

    Avizienis, A. V. et al. Morphological transitions from dendrites to nanowires in the electroless deposition of silver. Cryst. Growth Des. 13, 465–469 (2013).

    CAS  Google Scholar 

  36. 36.

    Tersoff, J. Contact resistance of carbon nanotubes. Appl. Phys. Lett. 74, 2122–2124 (1999).

    CAS  Google Scholar 

  37. 37.

    Estrada, D. & Pop, E. Imaging dissipation and hot spots in carbon nanotube network transistors. Appl. Phys. Lett. 98, 073102 (2011).

    Google Scholar 

  38. 38.

    Hsieh, Y. P. et al. Direct deposition of single-walled carbon nanotube thin films via electrostatic spray assisted chemical vapor deposition. Nanotechnology 20, 065601 (2009).

    Google Scholar 

  39. 39.

    De, S., King, P. J., Lyons, P. E., Khan, U. & Coleman, J. N. Size effects and the problem with percolation in nanostructured transparent conductors. ACS Nano 4, 7064–7072 (2010).

    CAS  Google Scholar 

  40. 40.

    Mutiso, R. M., Sherrott, M. C., Rathmell, A. R., Wiley, B. J. & Winey, K. I. Integrating simulations and experiments to predict sheet resistance and optical transmittance in nanowire films for transparent conductors. ACS Nano 7, 7654–7663 (2013).

    CAS  Google Scholar 

  41. 41.

    Znidarsic, A. et al. Spatially resolved transport properties of pristine and doped single-walled carbon nanotube networks. J. Phys. Chem. C. 117, 13324–13330 (2013).

    CAS  Google Scholar 

  42. 42.

    Nair, R. R. et al. Fine structure constant defines visual transparency of graphene. Science 320, 1308 (2008).

    CAS  Google Scholar 

  43. 43.

    Turner, P., Hodnett, M., Dorey, R. & Carey, J. D. Controlled sonication as a route to in-situ graphene flake size control. Sci. Rep. 9, 8710 (2019).

    Google Scholar 

  44. 44.

    Wang, S., Yi, M. & Shen, Z. G. The effect of surfactants and their concentration on the liquid exfoliation of graphene. RSC Adv. 6, 56705–56710 (2016).

    CAS  Google Scholar 

  45. 45.

    Ellmer, K. Past achievements and future challenges in the development of optically transparent electrodes. Nat. Photon. 6, 808–816 (2012).

    Google Scholar 

  46. 46.

    Mutiso, R. M. & Winey, K. I. Electrical percolation in quasi-two-dimensional metal nanowire networks for transparent conductors. Phys. Rev. E 88, 032134 (2013).

    Google Scholar 

  47. 47.

    De, S. et al. Flexible, transparent, conducting films of randomly stacked graphene from surfactant-stabilized, oxide-free graphene dispersions. Small 6, 458–464 (2010).

    CAS  Google Scholar 

  48. 48.

    Doherty, E. M. et al. The spatial uniformity and electromechanical stability of transparent, conductive films of single walled nanotubes. Carbon 47, 2466–2473 (2009).

    CAS  Google Scholar 

  49. 49.

    De, S. et al. Silver nanowire networks as flexible, transparent, conducting films: extremely high DC to optical conductivity ratios. ACS Nano 3, 1767–1774 (2009).

    CAS  Google Scholar 

  50. 50.

    Scardaci, V., Coull, R. & Coleman, J. N. Very thin transparent, conductive carbon nanotube films on flexible substrates. Appl. Phys. Lett. 97, 023114 (2010).

    Google Scholar 

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Acknowledgements

M.H. acknowledges funding from the Ministry of Science and Technology (107-2112-M-002 -004 -MY3). Y.-P.H. acknowledges funding from Academia Sinica (AS-iMATE-108-32) and the Ministry of Science and Technology.

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Contributions

H.Y. developed methodology, software and analysis and wrote the original draft, Y.-P.H. provided methodology, experimental resources, revisions and supervision, J.K. contributed conceptualization, experimental ressources and critical review, M.H. aided in formal analysis, visualization, project management and revision.

Corresponding authors

Correspondence to Ya-Ping Hsieh or Mario Hofmann.

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Supplementary information

Supplementary Information

Supplementary text, Table 1 and Figs. 1–5.

Supplementary Video 1

Animated visualization of percolation pathways in composite of resistive matrix with conductive fillers corresponding to the inset of Fig. 3a.

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Yao, H., Hsieh, YP., Kong, J. et al. Modelling electrical conduction in nanostructure assemblies through complex networks. Nat. Mater. 19, 745–751 (2020). https://doi.org/10.1038/s41563-020-0664-1

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