Evidence of Universal Temperature Scaling in Self-Heated Percolating

Subscriber access provided by LAURENTIAN UNIV

Communication

Evidence of Universal Temperature Scaling in Self-heated Percolating Networks Suprem R. Das, Amr M.S. Mohammed, Kerry Maize, Sajia Sadeque, Ali Shakouri, David B. Janes, and Muhammad Ashraf Alam Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b00428 • Publication Date (Web): 12 Apr 2016 Downloaded from http://pubs.acs.org on April 18, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Evidence of Universal Temperature Scaling in Selfheated Percolating Networks Suprem R. Das┴, ‡,| |, #, Amr M.S. Mohammed ┴, ‡,| |, Kerry Maize ┴, ‡, Sajia Sadeque ┴, ‡, Ali Shakouri ┴, ‡, David B. Janes ┴, ‡, Muhammad A. Alam ┴, ‡, * ┴

School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA ‡

||

Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA

These authors contributed equally to this work.

KEYWORDS: Universality, Scaling law, Co-Percolation network, Nanowires, Self-heating, Thermoreflectance imaging, Weibull distribution, Percolation Theory.

ABSTRACT

During routine operation, electrically percolating nanocomposites are subjected to high voltages, leading to spatially heterogeneous current distribution. The heterogeneity implies localized selfheating that may (self-consistently) reroute the percolation pathways and even irreversibly damage the material. In the absence of experiments that can spatially resolve the current distribution and a nonlinear percolation model suitable to interpret them, one relies on empirical rules and safety factors to engineer these materials. In this paper, we use ultra-high resolution

ACS Paragon Plus Environment

1

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 25

thermo-reflectance imaging, coupled with a new imaging processing technique, to map the spatial distribution Δ(, ; ) and histogram (Δ) of temperature rise due to self-heating in two types of 2D networks (percolating and co-percolating). Remarkably, we find that the selfheating can be described by a simple two-parameter Weibull distribution, even under voltages high enough to reconfigure the percolation pathways. Given the generality of the phenomenological argument supporting the distribution, other percolating networks are likely to show similar stress distribution in response to sufficiently large stimuli. Furthermore, the spatial evolution of the self-heating of network was investigated by analyzing the spatial distribution and spatial correlation respectively. An estimation of degree of hotspot clustering reveals a mechanism analogous to crystallization physics. The results should encourage nonlinear generalization of percolation models necessary for predictive engineering of nanocomposite materials. Main Manuscript Since the 1970s, percolation theory has interpreted counterintuitive electrical1 , mechanical2 and optical3 responses of two and three dimensional composite materials. For example, the theory explains the average electrical conduction ( ), the nonuniform current distribution, (Δ ), and even the magnitude of maximum voltage drop (Δ ) of a network with simple scaling relationships defined by only a few parameters, such as percolation threshold and transport exponents, etc4,5. Despite its elegance and sophistication (constants known to 5 decimal places!6), the theory/interpretation is valid only for small driving forces. In practice, complex nanostructured material must operate in nonlinear mode, with large forces and correlated response (e.g., electro-thermal). The statistical distribution is equally important: if the force at a point exceeds a threshold, the material may fracture. It is perceived that nonlinear statistical


2

Page 3 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

response of complex material must be irreducibly complicated, beyond the simple scaling relationships proffered by percolation theory. The inability to experimentally record the evolution of space- and time-resolved nonlinear responses in these materials has perpetuated this perception. Over the last two decades, a new class of electrical percolation networks, composed of 1D Nanowires (NW) and nanotubes (NT), have found applications as high-performance flexible thin-film transistors

7,8

, nanobio9 and chemical sensors10,11. Similarly, copercolating networks

consisting of a randomly dispersed NT/NW network coupled to a 2D conductor have demonstrated record performance as transparent conductors

12–14

. Unlike natural systems, the

parameters of the network (e.g., NT/NW density and dimensions, operating conditions) can be tuned continuously. A broad range of techniques can be used to characterize their electrical15, thermal16, chemical10,11 and optical17 response. As a result, these networks have emerged as powerful test beds for linear and nonlinear percolation models. Today, the network-averaged nonlinear response of these networks is well-understood in terms of a generalized multi-terminal percolation theory 18, that includes the inter-tube transfer resistances and nonlinear voltage drop across the network. In addition, these networks show remarkably rich spatial distributions of current and selfheating temperature rise within the network Δ(, ; ), especially at high bias conditions. High-spatial resolution measurements of Δ over significant areas are of interest in terms of understanding both conduction pathways within the device and physical effects which limit reliability, particularly related to electro-thermal fracture. The existence of Δ reflects a confluence of geometry and physics, as follows. By construction, a percolating network is spatially inhomogeneous, therefore, at any cross-section perpendicular to the current flow, the


3

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 25

current density and self-heating are spatially nonuniform, i.e. Δ(, ; → 0) ≠ 0. Apart from this geometrical effect, at higher , self-heating correlates the electrical and thermal resistances in a way19 that may further localize Δ(, ; ). Such localization has global implications for the network, leading to dynamic hot-spot formation that eventually leads to electro-thermal fracture of the network20,21 and variability of performance from one device to the next22 . Theoretically, one expects the linear-response distribution at Δ(, ; → 0) to be multi-fractal and described by simple scaling function23. One wonders, if the nonlinear, emergent statistical distribution associated with hot spot distribution, could likewise be described by a simple scaling relationship. Unfortunately, the existing imaging techniques (optical or infra-red24) cannot resolve Δ(, ; ), in sufficient detail to see if the temperature histogram (Δ , ) satisfies a simple formula. In this paper, we measure the temperature distribution within a network using a high resolution themoreflectance (TR) imaging technique19. A careful statistical analysis of these TR images shows that – despite the spatio-temporal complexity – the temperature distribution is indeed described by a simple universal scaling formula, characterized fully by only two scalefactors! The paper is arranged as follows. First, we describe the samples, the imaging techniques, and the image analysis methodology. The results are interpreted in terms of scaling function later, where we also offer a simple scaling relationship, and its analogy to crystallization kinetics. We conclude the paper explaining how our work may inspire statistical generalization of the classical percolation to nonlinear regime, and inspire search for similar scaling functions for other (nonelectrical) linear and nonlinear networks. Moreover, the results can be used to predict the probability of electrostatic fracture in such networks.


4

Page 5 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

In this paper, we will examine the nonlinear response on two network topologies: (1) a percolating random network of Ag nanowires (NW network), and (2) a co-percolating network of Ag-NWs covered by a single layer of 2D polycrystalline graphene (Hybrid network). Samples have comparable Ag-NW density, and the Ag NWs have average diameters of 90 nm and average lengths of 40 microns. Both networks operate well above the percolation threshold for the corresponding network type. The networks are well known -- many groups have studied them as transparent conductors12,25–27 (as an alternative to Indium Tin Oxide (ITO)) for displays and solar cells. Apart from simple fabrication, three other features make the networks well-suited for research objectives: First, the two networks rely on fundamentally different types of percolating transport12, with very different percolation thresholds and orders of magnitude difference in network resistance at the same NW density; thus, each sample represents a different regime within the broad spectrum of percolating networks reported in the literature26,28–31. Second, despite the differences, the network resistance is dominated by the transfer resistance between the NWs12, therefore, the hot spots are localized at the junctions. This would simplify image processing. Third, hot spots change the temperature of the metallic NW. The corresponding change in the reflectance can be captured by high-resolution thermo-reflectance imaging system, to be discussed below. For electrical tests, a pair of concentric circular electrodes defines the device region (see Figure 1(a)). The electrodes [Ti(1nm)/ Pd(30nm)/ Au(20nm) stack] were fabricated using standard photolithography, followed by e-beam evaporation of metals, and metal liftoff. Details of the synthesis process for NW and Hybrid networks and device fabrication have been described previously12. The channel resistance has been measured for several devices of both network


5

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 25

types before and after TR imaging. No hysteresis was observed during the forward and reverse sweep of the I-V characteristics. As discussed previously, optical or IR images cannot sufficiently resolve the temperature distribution of the samples. One pixel may average over a point with high temperature, surrounded by several points of much lower temperature, while another pixel may average over several points of intermediate temperatures. The inability to distinguish these pixel erases the characteristics features (if any) of a probability distribution. Second, a probability distribution is distinguished by its tails, containing pixels with temperature barely above the background. The poor SNR of optical/IR images cannot differentiate signal vs. noise in these pixels. Therefore, classical imaging can be used to identify dynamics of hot spot formation, but cannot be used to analyze them quantitatively. The TR imaging (TRI) used in this work addresses the issues of spatial resolution and SNR simultaneously. Figure 1 (a) shows the simplified schematic arrangement/operational principle of the TRI measurement technique. First, to achieve submicron resolution, a narrow band light emitting diode (LED) centered at 530nm illuminates the network surface and a CCD camera, with broad dynamic range, captures the image. Second, the SNR ratio is improved by averaging over more than 10,000 images taken over 20 minutes, as follows. The samples are electrically excited by a sequence of boxcar pulses, namely, current pulses of 1ms duration at 150 Hz repetition rate (see the method section for details). Each pulse heats the NW network and perturbs its surface reflectance. The current pulses, LED, and CCD camera are synchronized, so that the sample is illuminated (for 100 µsec) and imaged only once per boxcar excitation (See the Supporting Information for more details). The differential TR is obtained by alternatively imaging (every 2 sec) the reflectance during active (boxcar on) and passive (boxcar off) phases.


6

Page 7 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Signal averaging over the differential measurements dramatically improves the SNR, see SI for quantitative error estimates. Later, we will use a custom MATLAB image processing tool to extract the histogram of temperature distribution from these images, and consider whether the data is described by simple scaling relation. The Δ(, ; ) was measured for devices consisting of i) NW network and ii) Hybrid network, with comparable NW densities in each sample.

Figure 1 (b) shows the evolution of

Δ(, , ; ) for three current values (I=15mA, 17 mA and 29 mA) for the NW network at low magnification (20X); the sequence of video of images as a function of current is posted as a video in the Supporting Information (SI). A nonlinear response of the network is observed. At low currents, the network is linear and percolating current paths (indicated by constellation of hotspots) are stationary. Increasing current brightens the image, but does not change its spatial distribution. At intermediate and high-currents, however, significant self-heating modifies the percolation network itself: some of the existing hotspots dim or disappear, while new hotspots come into view, etc. The current redistribution does not imply network damage: the process is reversible, that is, the spatial pattern observed at a given current is restored when the current is reduced to that level. The nonlinear response can be quantified by the terminal I-V characteristics, and by observing the evolution of the network by monitoring the spatially averaged temperatures at Regions 1 and 2(each of size 30µmX30µm, indicated in Figure 1(c) inset) shown in Figure 1(c), Δ and Δ, respectively. Figure 1(c) shows that the voltage developed (left axis) begins to deviate from linear Ohm’s law for > 10 mA, suggesting a change in percolation pathways. Indeed, Δ and Δ begin to deviate from classical quadratic dependence (〈Δ〉 ∝ , where is the thermal resistance) around the same threshold current. Specifically, at > 15 mA, Δ begins to


7

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 25

drop, along with a correlated increase in Δ. By ∼ 17 mA, Δ ∼ 0, region 2 effectively disappears from the percolating pathways, as if the network is apparently damaged. Yet, the region is restored to ‘life’ at > 25mA, again with correlated decrease in Δ . This complex, non-ohmic response of percolating network is expected7. This reversible darkening (and subsequent brightening) of the segment of the network must necessarily arise from currentdependent dynamical opening and closing of the electronic paths associated the nanowirenanowire junction. Furthermore, the channel resistance has been measured for several devices of both network types before and after TR imaging to ensure that no irreversible change (permanent damage) happened to the network structures. The real question however is this: Despite the complexity, could the network be still described by relatively simple temperature distribution? A detailed analysis of the current-specific images at higher magnification (100X), that allows field view of 100µmX100µm, to ensure that the resolved temperature is not mostly dominated by inactive region within the single pixel would answer this question, as follows. The probability distribution of (Δ, ) is obtained by a MATLAB script as follows. For each current level, a binary mask is obtained from the optical image of the network by selecting all pixels with reflectance values above an arbitrary threshold. Since the nanowires are relatively more reflective than the surrounding substrate, this procedure identifies nanowire pixels, and generally excludes pixels corresponding to regions without nanowires. This binary mask is used to select pixels from the TR image to be included in the analysis, so as to extract Δ(, ; ) of the network (Figure 2(a)), without considering pixels corresponding to the substrate. The use of the binary mask allows the analysis to focus on NWs rather than on the surrounding areas. The (Δ, ) is obtained by plotting the histogram of the temperature distribution. Histograms were


8

Page 9 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

obtained for both types of samples, and at all current levels. For illustration, we plot a subset of these histograms in Figure 2 (b) for Ag-NW and Hybrid networks, respectively. In each case, several current levels are presented, along with Weibull distributions at the corresponding current levels (discussed in next section). The histograms are smooth, reflecting the high spatial resolution, large dataset, and improved SNR of TR images. The histograms of other current levels are shown in SI. To see if the experimental distribution corresponds to a known analytical distribution, we plot ()

the cumulative probability distribution function (CDF), #(Δ, ) = %* (Δ& , )'& , in a double-log plot, + = ,-(−,-(1 − #)). Figure 3 (a, b) show W versus ln(∆T) for the NW network and Hybrid network devices, respectively. For comparison, various analytical distributions, i.e., Normal, Weibull, Extreme Value and Rayleigh, are also shown in the plots. A double-log plot accentuates the tail of the distribution, thereby reducing the chances of its misidentification. A simple two-parameter Weibull distribution, namely, ()

#(Δ, ) = 1 – 0& 1 − () (3)4

5(3)

2

(1)

fits the experimental distributions very well (except for the minor deviation at Δ → 0 related to the thermal noise of the measurement, and quantified in detail in SI). This background noise broadens the histogram slightly (see supporting information), making it deviate slightly with respect to the ideal (noise-free) Weibull distribution. This deviation is no more than a few percent close to 6 = 0, and decreases rapidly at higher temperature. All other distributions are positively excluded. While a double-log CDF plot identifies the best-fit distribution to the experimental data, it is important to check if the fit is equally effective at high-temperature (Δ ≫ * ), especially


9

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 25

because these hot spots will eventually lead to electro-thermal fracture of the network. For 8 → 1, and Δ ≫ *, Weibull distribution assumes the form '# Δ 5 ,-() = ,- 9 : → − 9 : (2) ' * Figure 3(c, d) show that the Weibull distribution captures the high-temperature tail of the distribution down to the noise limit as well. In both types of networks, the experimental temperature distributions obtained at all measured current levels can be fit quantitatively using a Weibull distribution. We find that (i) the scale parameter (Δ*) scales quadratically with current (Δ* = ; ) (Figure 4), where the constant ; depends on the network topology through its electrical and thermal resistances. The Hybrid network exhibits a lower sheet resistance, and therefore lower overall power dissipation at a given current. In addition, the graphene layer provides an additional pathway for lateral heat spreading, which also reduces the average temperature rise. In contrast, the shape parameter (8, also known as the Weibull slope) drops linearly with the current (i.e. 8 ≅ 8* – = ), where 8* defines the NW-density-dependent geometrical inhomogeneity of the network at low-currents, and = indicates current-induced self-localization of Joule heating in the network12. For the specific networks under consideration, the following parameterization captures the essence of the experimental data fully, #(Δ, ; ;, 8) = 1 −

?@ D2 EFB

1> 4 0 ABC

(3)

where ; = .0244 I/K for the NW network and ; = .006 I/K for the hybrid network. This difference in ; reflects the reduced Joule heating (due to smaller resistance) and more effective heat dissipation of the Hybrid network. The shape parameter follows the relation: 8 = 8* − = where (8* , =)= (1.257,0.0063) for the NW device, (1.281,0.0072) for the Hybrid device.


10

Page 11 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

One might ask how the above Weibull is related with the percolation threshold. To elaborate this, we recall that the two network types considered in this study (NW and Hybrid) are representative of percolation and co-percolation networks. Also, the current study, to start with, focused on devices with NW densities sufficiently large to provide operation well above the percolation threshold. In the case of the Hybrid sample, the co-percolation mechanism corresponds to a percolation threshold at a significantly lower NW density (compared to NW network), so the Hybrid sample is farther from the percolation limit. The observation that data from both singlecomponent (NW) and two-component (Hybrid) networks follow the same distribution function provides evidence that behavior is universal within the percolation regime considered in this study. Although it is possible that a comparable universality exists in 2D networks operated below the percolation threshold and in 3D percolation networks, the qualitatively different conductance behavior in these networks makes it difficult to directly extrapolate the current results into those regimes. Interestingly, the Weibull distributions integrate, as limiting cases, phenomenological observations previously reported in the literature19,32. For example, previously reported superJoule heating (Δ ∼ M , with - > 2) of the hotspots can be understood as a consequence of current-dependent shape factor, 8( ), so that the tail of the distribution scales superlinearly with the network current .

At more macroscopic dimensions, the behavior is consistent with

conventional Joule heating, which would correspond to an ensemble average of the distribution. The percolation theory must be generalized to the nonlinear regime to explain why selfheating is Weibull distributed. While this generalization is beyond the scope of the paper, a phenomenological argument – analogous to the physics of crystallization – can be helpful. When a network is heated by a fixed total current , the density of pixels with temperature between Δ


11

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 25

and Δ + O, namely (Δ, ), depends on two factors: the number of pixels still available (1 − #), multiplied by the ability of the network to sustain the local temperature above the characteristic average temperature, * , namely, P(/* ), so that QR Q)

)

≡ () = P 1) 4 [1 − #] (4) 2

Now if we assume that P(/* ) = 8(⁄* )5> , then the integral has the form given by the Weibull distribution, Eq. (1). The heterogeneity of a percolating network (see Figure 1) is essential to generate the Weibull form: A homogeneous Ohmic network cannot sustain a temperature difference (it tries to erase any temperature differential), so that 8 → ∞. Integrating, we find () = (* ), as expected. Therefore, the factor 8(≡ 8* − = ) captures two different types of inhomogeneity inherent to nonlinear percolating networks: specifically, 8* defines the degree of geometrical heterogeneity characterized by the density of the network and its thermal environment, while = is related to the additional degree of localization due to self-heating. The exact function dependence of B(I) remains to be verified through a nonlinear generalization of the percolation model. In addition to the temperature distribution, it is interesting to analyze the spatial distribution of the TR image in order to develop a comprehensive picture of the underlying physics. A probability distribution function (PDF) of the hot spots, a key concept to investigate short, medium or long-range electrothermal activity of materials, was calculated. For this, we focus on nanowire pixels with ∆T well above noise level. At each bias point, an arbitrary threshold of 15 % of the maximum ∆T for that bias point was chosen, and only NW pixels with ∆T values above this threshold were included. The spatial distribution has been calculated by considering the nearest-neighbor inter-pixel distance between all such pixels33, and generating corresponding histograms at various current values. Figure 5 (a) and (b) show the spatial distribution for the


12

Page 13 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

NW network and the Hybrid network samples, respectively, with representative current levels shown in each case. The insets show corresponding binary TR images in which pixels above the threshold are represented as white spots. Several observations arise from such analysis: First, the spatial distribution for both cases is relatively independent of current. Second, the hybrid sample shows a narrower main peak than that of the NW sample, along with a broad shoulder (80-130 µm). This reflects a more scattered nature of the hot spots in the NW network with short range correlation and a relatively stronger binding and correlation in the Hybrid network. This is analogous to measuring structural correlation between atoms/molecules in a gas vs. liquid. Therefore, although the transport in the percolating and co-percolating networks is often presumed random, they are distinguished by the degree of spatial correlation present.

For a more quantitative measure for hot spot clustering phenomena, we calculate Global Moran’s Index ( ) for the spatial distributions of both cases. Moran’s index, which measures spatial autocorrelation based on both feature locations and feature values (e.g. temperature) simultaneously, determines whether the pattern is clustered (0 < < 1), dispersed (−1 < < 0), or random ( = 0)34. Moran’s Index has been used extensively for geographical data analysis and spatial demographics35,36. Figure 5(c) shows the Moran’s index as function of current for the Hybrid and NW devices. The NW sample exhibits a low degree of correlation. The Hybrid sample exhibits a larger Moran’s Index, although still less than one, which indicates moderate correlation. These observations that the Hybrid sample shows a narrower main peak in the spatial distribution and a higher degree of spatial correlation can be explained in terms of an increased number of conductive pathways in the Hybrid sample, which may occur in addition to higher average conductance in pathways within the NW network. Figure 5(d) shows a schematic


13

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 25

diagram comparing a partial conduction pathway in a NW network (top) and two possible mechanisms for completing that pathway in the Hybrid network (bottom): A. Two NWs may come into contact via mechanical force due to graphene layer; B. Current can flow from a NW through graphene to another NW, even if the NWs are not in physical contact. The increased spatial correlation in the Hybrid sample can be qualitatively explained using this model. If a new pathway (A or B) forms, local heating at that location will only occur if this new pathway completes a current path, i.e. if both of the NWs are connected to other NWs within a conductive path. Therefore, new hot spots are more likely to arise in the vicinity of existing (within NW network) hotspots.

We have used a thermo-reflectance imaging technique and image analysis methodology to conclude that the self-heating in representative percolating and co-percolating networks are described by a Weibull distribution, a simple two-parameter distribution function. Such elegant simplicity was unexpected, given the apparent complexity of the structure and activity of the nonlinear network.

The observation has important practical implications: since the high-

temperature tail of the Weibull distribution decays more slowly than that of other distributions, a larger number of hot-spots (exceeding a specified threshold) will form at a given stress current. The network will have to be operated at a reduced voltage to avoid electro-thermal fracture of the network. Furthermore, the degree of hotspot clustering at nonlinear transport regime reveals an evidence of mechanism analogous to crystallization physics. The phenomenological arguments for the emergence of a Weibull-like distribution and spatial correlation are general. Therefore, if all percolation networks demonstrate Weibull distributed self-heating and hotspot clustering and correlation, as ours do so conclusively, the phenomena of


14

Page 15 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

self-heating may be far less complicated and a nonlinear generalization of percolation theory may be simpler than previously presumed. Therefore, our network analysis framework may pave the way toward better designing of increasing number of complex nanoscale networks for wide spread applications such as electrodes for transparent conducting electrodes, energy harvesting, and biochemical sensing. ASSOCIATED CONTENT Thermoreflectance Imaging method, details for generating temperature histograms from thermal Image, and data Analysis. Movie showing the evolution of self-heating in the nanowires network is also provided. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author *[email protected] Present Address # Iowa State University, Ames, IA 50011, USA; Ames Laboratory, Ames, IA 50011, USA Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. | | These authors contributed equally. ACKNOWLEDGMENT This work is supported as part of the Center for Re-Defining Photovoltaic Efficiency through Molecule Scale Control, an Energy Frontier Research Center (EFRC) funded by the U.S.


15

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001085, and by National Science Foundation under grant ECCS 1408346.

REFERENCES

(1)

Zallen, R. The Physics of Amorphous Solids; Zallen, R., Ed.; Wiley-VCH Verlag GmbH: Weinheim, Germany, 1998.

(2)

Phillips, J. C. Reports Prog. Phys. 1996, 59 (9), 1133.

(3)

Shalaev, V. M. Optical Properties of Nanostructured Random Media; Shalaev, V. M., Ed.; Topics in Applied Physics; Springer Berlin Heidelberg: Berlin, Heidelberg, 2002; Vol. 82.

(4)

Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor & Francis, 1994.

(5)

Stanley, H. E. Rev. Mod. Phys. 1999, 71 (2), S358–S366.

(6)

Li, J.; Ray, B.; Alam, M. A.; Östling, M. Phys. Rev. E 2012, 85 (2), 021109.

(7)

Kumar, S.; Murthy, J. Y.; Alam, M. A. Phys. Rev. Lett. 2005, 95 (6), 066802.

(8)

Cao, Q.; Kim, H.; Pimparkar, N.; Kulkarni, J. P.; Wang, C.; Shim, M.; Roy, K.; Alam, M. A.; Rogers, J. A. Nature 2008, 454 (7203), 495–500.

(9)

Nair, P. R.; Alam, M. A. Phys. Rev. Lett. 2007, 99 (25), 256101.

(10)

Sysoev, V. V.; Goschnick, J.; Schneider, T.; Strelcov, E.; and Andrei Kolmakov. Nano Lett. 2007, 7 (10), 3182–3188.

(11)

Go, J.; Sysoev, V. V.; Kolmakov, A.; Pimparkar, N.; Alam, M. A. 2009 IEEE Int. Electron Devices Meet. 2009, 1–4.


16

Page 17 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(12)

Chen, R.; Das, S. R.; Jeong, C.; Khan, M. R.; Janes, D. B.; Alam, M. A. Adv. Funct. Mater. 2013, 23 (41), 5150–5158.

(13)

Jeong, C.; Nair, P.; Khan, M.; Lundstrom, M.; Alam, M. A. Nano Lett. 2011, 11 (11), 5020–5025.

(14)

Kholmanov, I. N.; Magnuson, C. W.; Aliev, A. E.; Li, H.; Zhang, B.; Suk, J. W.; Zhang, L. L.; Peng, E.; Mousavi, S. H.; Khanikaev, A. B.; Piner, R.; Shvets, G.; Ruoff, R. S. Nano Lett. 2012, 12 (11), 5679–5683.

(15)

Clerc, J. P.; Giraud, G.; Laugier, J. M.; Luck, J. M. Adv. Phys. 1990, 39 (3), 191–309.

(16)

Yu, A.; Ramesh, P.; Sun, X.; Bekyarova, E.; Itkis, M. E.; Haddon, R. C. Adv. Mater. 2008, 20 (24), 4740–4744.

(17)

Simien, D.; Fagan, J. A.; Luo, W.; Douglas, J. F.; Migler, K.; Obrzut, J. ACS Nano 2008, 2 (9), 1879–1884.

(18)

Pimparkar, N.; Kocabas, C.; Seong Jun Kang; Rogers, J.; Alam, M. A. IEEE Electron Device Lett. 2007, 28 (7), 593–595.

(19)

Maize, K.; Das, S. R.; Sadeque, S.; Mohammed, A. M. S.; Shakouri, A.; Janes, D. B.; Alam, M. A. Appl. Phys. Lett. 2015, 106 (14), 143104.

(20)

Wahab, M. A.; Alam, M. A. IEEE Trans. Electron Devices 2014, 61 (12), 4273–4281.

(21)

Liao, A.; Alizadegan, R.; Ong, Z.-Y.; Dutta, S.; Xiong, F.; Hsia, K. J.; Pop, E. Phys. Rev. B 2010, 82 (20), 205406.

(22)

Gupta, M. P.; Behnam, A.; Lian, F.; Estrada, D.; Pop, E.; Kumar, S. Nanotechnology 2013, 24 (40), 405204.

(23)

Redner, S. Encyclopedia of Complexity and Systems Science; Meyers, R. A., Ed.; Springer New York: New York, NY, 2009; pp 3737–3754.

(24)

Bae, M.-H.; Ong, Z.-Y.; Estrada, D.; Pop, E. Nano Lett. 2010, 10 (12), 4787–4793.


17

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 25

(25)

Lee, M.-S.; Lee, K.; Kim, S.-Y.; Lee, H.; Park, J.; Choi, K.-H.; Kim, H.-K.; Kim, D.-G.; Lee, D.-Y.; Nam, S.; Park, J.-U. Nano Lett. 2013, 13 (6), 2814–2821.

(26)

Hu, L.; Kim, H. S.; Lee, J.-Y.; Peumans, P.; Cui, Y. ACS Nano 2010, 4 (5), 2955–2963.

(27)

Wu, Z. .; Wu, Z.; Chen, Z.; Du, X.; Logan, J. M.; Sippel, J.; Nikolou, M.; Kamaras, K.; Reynolds, J. R.; Tanner, D. B.; Hebard, A. F.; Rinzler, A. G. Science (80-. ). 2004, 305 (5688), 1273–1276.

(28)

Liang, J.; Li, L.; Tong, K.; Ren, Z.; Hu, W.; Niu, X.; Chen, Y.; Pei, Q. ACS Nano 2014, 8 (2), 1590–1600.

(29)

De, S.; Higgins, T. M.; Lyons, P. E.; Doherty, E. M.; Nirmalraj, P. N.; Blau, W. J.; Boland, J. J.; Coleman, J. N. ACS Nano 2009, 3 (7), 1767–1774.

(30)

Coleman, J. N.; Curran, S.; Dalton, A. B.; Davey, A. P.; McCarthy, B.; Blau, W.; Barklie, R. C. Phys. Rev. B 1998, 58 (12), R7492–R7495.

(31)

Hu, L.; Hecht, D. S.; Grüner, G. Nano Lett. 2004, 4 (12), 2513–2517.

(32)

Grosse, K. L.; Dorgan, V. E.; Estrada, D.; Wood, J. D.; Vlassiouk, I.; Eres, G.; Lyding, J. W.; King, W. P.; Pop, E. Appl. Phys. Lett. 2014, 105 (14), 143109.

(33)

Alam, M. A.; Varghese, D.; Kaczer, B. IEEE Trans. Electron Devices 2008, 55 (11), 3150–3158.

(34)

Moran, P. A. P. Biometrika 1950, 37 (1/2), 17–23.

(35)

Anselin, L. Geogr. Anal. 1995, 27 (2), 93–115.

(36)

Bjørnstad, O. N.; Ims, R. A.; Lambin, X. Trends Ecol. Evol. 1999, 14 (11), 427–432.


18

Page 19 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

FIGURES

Figure 1: (a) Simplified schematic arrangement/ operational principles of the TR imaging technique applied to the network devices. A pulsed boxcar averaging technique utilizing a short duration LED pulse (~ tens of nanosecond) is synchronized with a device excitation. The change in device temperature and hence the reflectivity is shown in the schematic. (b) TR images at the indicated bias currents for the representative NW network device described in the text. (c) Measured voltage (V) and temperature change (∆T) versus I for the two regions (each of size 30 µm X 30 µm) indicated in the CCD image (inset). The corresponding I vs. V data is analyzed to support the correlation of heating in both regions. The light brown-shaded region marks the linearity of I-V characteristics while the green-shaded region marks the restoration of the linearity of I-V characteristics.


19

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 25

Figure 2: (a) Binary mask (generated from CCD image showing the nanowire pixels) is overlaid with the corresponding thermal image to select pixels used to produce the histogram. The histogram shows the temperature distribution of nanowire pixels, expressed as a percentage of the overall number of nanowire pixels. (b) Histograms generated at biasing currents applied to both type of networks (NW & Hybrid). The corresponding Weibull distributions (solid lines) fit the data well.


20

Page 21 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 3: Statistical analysis of the thermal images at each current level for both NW &Hybrid devices (a-b) Comparing the cumulative distribution function (F) of temperature of nanowire pixels (NW at I=31mA, Hybrid at I=37mA, respectively) to different statistical distributions, Weibull stands clearly to be the best one describing the temperature distribution in both the NW and the Hybrid network. (c-d) The logarithm of Probability density function (log(f)) versus change in temperature, showing that the logarithm of the pdf drops almost linearly at high temperature, as anticipated for Weibull distribution.


21

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 25

Figure 4: Weibull Distribution parameters (scale parameter (A) and shape parameter (B)) plotted as function of the current for both the NW and Hybrid networks. The scale parameter (A) goes quadratically with the current (α=0.024 for NW, 0.006 for Hybrid). The shape parameter drops linearly with current, corresponding to a signature of increased localization of self-heating in nanowires. The straight line constants are as follows (Bo,m):(1.257,0.0063) for NW network, and (1.281,0.0072) for Hybrid network.


22

Page 23 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 5: (a) – (b) Spatial distribution of hot spots for NW and Hybrid network, respectively. The insets show corresponding binary TR images in which pixels above the threshold are represented as white spots. (c) Global Moran’s Index as function of the current for the two networks, showing a larger degree of correlation for the Hybrid case. (d) Schematics showing conductive NW-NW junctions for NW network (top) and Hybrid network (bottom), with red circles indicating conductive NW-NW junctions. The comparison of spatial distributions indicates that new conductive paths emerge in the Hybrid network, through either: A. additional NWNW junctions, or B. local shunt path through the graphene. These additional pathways are consistent with the increased Moran’s Index in the Hybrid sample.


23

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 25

TOC FIGURE


24

Page 25 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

75x39mm (300 x 300 DPI)


Evidence of Universal Temperature Scaling in Self-Heated Percolating

Recommend Documents