Research ArticleAPPLIED SCIENCES AND ENGINEERING

Avalanches and criticality in self-organized nanoscale networks

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Science Advances  01 Nov 2019:
Vol. 5, no. 11, eaaw8438
DOI: 10.1126/sciadv.aaw8438
  • Fig. 1 Device schematic, atomic switches, and percolating networks.

    (A) Top: Schematic diagram illustrating our simple two-terminal contact geometry and the percolating network of nanoparticles. The different colors represent groups of particles that are in contact with one another. The zoomed region shows a schematic of the growth of an atomic filament within a tunnel gap (switching event) when a voltage is applied. Bottom: The same network schematic presented so as to show the conducting pathways (black) that result from atomic filament formation within the gaps between groups. (B) Schematic showing the conductance of a device during deposition of conducting nanoparticles follows (28) a power law ~(ppc)1.3 (30) (the critical surface coverage is pc ~68%). arb, arbitrary units. (C) A scanning electron microscope image of a percolating device. Scale bar, 200 nm. (D) Schematic diagrams showing the following: left—in the subcritical, insulating phase at low coverage, groups of particles are small and well separated, so that if an atomic switch connects two groups, then there are few possibilities that this will trigger another switching event; right—in the supercritical, conducting phase at higher coverages, highly connected pathways across the network mean that when an atomic filament bridges a tunnel gap, an avalanche can propagate only to a few nearby tunnel gaps; center—in the critical phase, avalanches propagate on multiple length and time scales. See also fig. S2 for further details.

  • Fig. 2 Spatial and temporal self-similarity and correlation in switching activity.

    (A) Percolating devices produce complex patterns of switching events that are self-similar in nature. The top panel contains 2400 s of data, with the bottom panels showing segments of the data with 10, 100, and 1000 times greater temporal magnification and with 3, 9, and 27 times greater magnification on the vertical scale (units of G0 = 2e2/h, the quantum of conductance, are used for convenience). The activity patterns appear qualitatively similar on multiple different time scales. (B and E) The probability density function (PDF) for changes in total network conductance, PG), resulting from switching activity exhibits heavy-tailed probability distributions. (C and F) IEIs follow power law distributions, suggestive of correlations between events. (D and G) Further evidence of temporal correlation between events is given by the autocorrelation function (ACF) of the switching activity (red), which decays as a power law over several decades. When the IEI sequence is shuffled (gray), the correlations between events are destroyed, resulting in a significant increase in slope in the ACF. The data shown in (B) to (D) (sample I) were obtained with our standard (slow) sampling rate, and the data shown in (E) to (G) (sample II) were measured 1000 times faster (see Materials and Methods), providing further evidence for self-similarity.

  • Fig. 3 Statistical distributions of avalanche properties and shape collapse.

    The distributions shown in (A) to (E) (sample I, slow sampling rate) and (F) to (J) (sample II, fast sampling rate) demonstrate the existence of self-similar avalanches across multiple time scales. (A and F) PDFs of avalanche size and (B and G) PDFs of avalanche duration both follow power laws with exponents τ ≈ 2 and α ≈ 2.7, respectively, providing strong evidence for temporal correlations. Red, distribution presented using standard (linear) bin sizes; blue, distribution presented using logarithmic bin sizes to allow visualization of the heavy tail; gray, distribution after shuffling of the sequence of IEIs in the original data to destroy correlations. (C and H) The mean avalanche sizes are a power law function of their duration with exponent 1/σνz ≈ 1.6. (D and I) The mean avalanche shapes for each unique duration are shown to collapse (E and J, respectively) onto scaling functions (black lines). This shape collapse yields an independent measure of the critical exponent 1/σνz ≈ 1.6. As discussed in the main text and shown explicitly in Fig. 4, the values of the critical exponents τ, α, and 1/σνz satisfy the crackling relationship (Eq. 4) within the measurement uncertainties.

  • Fig. 4 Demonstration of criticality.

    Estimates of three independent measures of 1/σνz are obtained from the crackling relationship (green), plots of mean avalanche size given duration (blue), and avalanche shape collapse (cyan). The blue, cyan, and green symbols agree within the measurement uncertainty in almost every case. Each panel also shows the power law exponents of avalanche size (τ; red) and avalanche duration (α; amber). (A) Critical exponents for sample I over a range of low voltages and for a combined dataset (low sampling rate; see Materials and Methods), showing that the critical exponents are substantially independent of voltage. (B) Comparison of critical exponents measured for sample II for repeated, independent 6 V DC measurements (fast sampling rate; see Materials and Methods). (C) Critical exponents for sample III as a function of voltage (slow sampling rate). (D) Data from a second sequence of measurements on sample III identical to that in (C), showing that while the exponents α and τ vary because of internal reconfigurations of the percolating device, the three estimates of 1/σνz remain in good agreement at every voltage. (E and F) Voltage-dependent data from sample IV (E and F, fast and slow sampling rates, respectively), showing that criticality and self-similarity are observed on vastly different time scales. The mean values of 1/σνz were found to be 1.46 ± 0.05, 1.40 ± 0.03, and 1.40 ± 0.04 from the crackling relationship, 〈S〉(T) and shape collapse, respectively (uncertainties are 1 SD), indicating that there is no significant difference between the estimates from the three independent methods. This is confirmed by a single-factor analysis of variance (ANOVA) test (P = 0.47). The (τ, α) data are replotted in fig. S6 to allow a different comparison with the three calculations of 1/σνz (27).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/11/eaaw8438/DC1

    Fig. S1. Illustration of different types of two-dimensional (2D) percolation model.

    Fig. S2. Illustration of 2D percolation with tunneling and atomic switches.

    Fig. S3. Choice of threshold.

    Fig. S4. Choice of time bin size.

    Fig. S5. Comparison of power law and other fits to the avalanche size and duration distributions.

    Fig. S6. Scatter plot of α(τ).

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Illustration of different types of two-dimensional (2D) percolation model.
    • Fig. S2. Illustration of 2D percolation with tunneling and atomic switches.
    • Fig. S3. Choice of threshold.
    • Fig. S4. Choice of time bin size.
    • Fig. S5. Comparison of power law and other fits to the avalanche size and duration distributions.
    • Fig. S6. Scatter plot of α(τ).

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