Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Quantitative Visualization of Topology and Morphing of Percolation Path in Nanoparticle Network Array Exhibiting Coulomb Blockade at Room Temperature Peter Wilson, Jason K. Y. Ong, Abhijeet Prasad, and Ravi F. Saraf* Department of Chemical and Biomolecular Engineering, Nebraska Center for Materials and Nanosciences, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, United States Downloaded via UNIV OF LOUISIANA AT LAFAYETTE on August 5, 2019 at 16:05:26 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
S Supporting Information *
ABSTRACT: A network of one-dimensional (1D) necklaces of 10 nm Au nanoparticles was fabricated by a directed self-assembly to synthesize 1D necklaces followed by self-limiting monolayer deposition to form a two-dimensional (2D) network array. Scanning electron microscope (SEM) image analysis revealed a percolation threshold lower than random 2D arrays signifying the local 1D structure. The topology of (shortest) percolation paths (tortuosity) and the fraction of clusters isolated from the percolating array were quantified to relate the network morphology to the observed non-Ohmic (Coulomb blockade effect) behavior. Leveraging charge contrast in SEM, the morphing of the percolation path as a function of the kinetic energy of the conduction electron was visualized and quantified to understand the dynamic nature of the percolation behavior. The morphology can be systematically tailored by tuning the two self-assembly processes to obtain the same coverage of the array with significantly diverse non-Ohmic behavior. It was concluded that tortuosity and void fraction unify the Coulomb blockade behavior for a range of fabrication conditions leading to varying network morphologies with a threshold blockade bias ranging from 0.5 to 5.5 V at room temperature. This self-assembly avenue will allow the development of highly sensitive, allmetal electrochemical field effect transistors for applications in biology.
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INTRODUCTION Metallic nanoparticle arrays are an important class of nanomaterials due to their electronic properties, such as freeelectron plasmon oscillations and low capacitance. The plasmon oscillation resonance in the optical range provides a convenient near-field optical arrangement1 for developing highly sensitive optical devices to probe chemical environments, such as by reflectivity (surface plasmon resonance)2−4 and spectroscopy (surface-enhanced Raman scattering).5−7 Owing to low charging energy, typically ∼60 meV for a 10 nm metal particle capped with organics allows for the possibility of fabricating a transistor where the current can be gated by a Coulomb blockade caused by single-electron charging.8,9 In a nanoparticle array, the effect is enhanced due to multiple Coulomb blockades along the percolation path to obtain nonOhmic behavior where the current (I) and applied bias (V) exhibit a threshold, VT, given as I ∼ (V−VT)ζ, where ζ is a critical exponent.10−12 The conduction gap, VT, can be gated to form a single-electron transistor.13 The nonlinear (i.e., non-Ohmic) electrical behavior of a metallic nanoparticle array at cryogenic temperatures is well understood in terms of variable range hopping, i.e., an Efros− Shklovskii-like law, mediated by the cotunneling phenomenon14 and stationary offset charge from the substrate (referred to as quenched charge distribution).11,15 At room temperature, the hopping length should vanish to a single particle,16 and the Coulomb blockade may be overcome due to high-energy © XXXX American Chemical Society
conduction electrons, i.e., the tail of the Fermi−Dirac distribution. However, in a monolayer network of onedimensional (1D) necklaces of 10 nm Au particles, a nonOhmic I−V behavior at room temperature is observed17,18 that is functionally similar to the Coulomb blockade effect observed in dense, two-dimensional (2D) nanoparticle arrays at cryogenic temperatures.12,19−21 The high-temperature nonOhmic behavior in (rather) large Au nanoparticles is attributed to multitunnel junctions (MTJs) along the percolation path.18 In water, the conduction can be gated to obtain a gain of 1−2 orders of magnitude larger than graphene and carbonnanotube-based electrochemical transistors, respectively.22 Cellular activity in the cell can gate the necklace array to quantitatively measure the photosynthesis and single viral infection processes.23,24 It is reasonable to expect that the barriers due to (random) MTJs will affect the topology of the percolation path, and the locus will morph as the barriers are sequentially overcome as bias increases. This effect of topology of the percolation path on the I−V behavior in a 2D nanoparticle array is well recognized by the wide range of ζ from 5/3 to ∼4.3 obtained both experimentally and by simulation.10 The theoretical ζ ∼ 5/3 for a random structure11 can be enhanced to ∼4.3 for a Received: June 10, 2019 Revised: July 15, 2019 Published: July 17, 2019 A
DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C more tortuous percolation path constrained by voids.20 Simulation and experimental studies have shown a change in ζ with bias, as the percolation path morphs from being more linear to branched.12,25 At higher biases, simulations demonstrate that the tortuosity again decreases forming a linear, “smectic-like” percolation path causing the ζ to decrease.26 Here, we experimentally studied the topology of the percolation path and its effect on electron transport. Using a monolayer network of 1D nanoparticle necklaces, we quantitatively studied the tortuosity of the percolation and its possible effect on bias. The 1D network was especially convenient to quantitatively map the percolation path by analyzing scanning electron microscope (SEM) images. Using charge contrast in SEM images, we observed and quantified the effect of morphing of the percolation path as the MTJ barriers were overcome. The topology of the network was systematically tuned by a two-step self-assembly process to regulate the overall density (i.e., coverage) and the junction density (i.e., void content). The first step was a directed self-assembly (dSA) process to synthesize 1D necklaces in solution. The second step was the self-limiting monolayer deposition to form a 2D network.
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Figure 1. Fabrication of a nanoparticle necklace network array. (a) UV−vis before (black) and after (blue) exposure to CaCl2 salt. Inset: change in color of the solution due to necklace formation after td = 18 h. (b) Centrifuge process to deposit the necklaces on the chip. The rotation velocity is converted to acceleration as Ng. (c) SEM images of necklaces made at td = 18 h deposited at different centrifuge speeds.
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EXPERIMENTAL SECTION
RESULTS AND DISCUSSION The observation we wished to understand was as follows: typical I−V characteristics of a nanoparticle array deposited between two electrodes with a gap of 10 μm exhibit highly non-Ohmic behavior at room temperature (T ∼ 295 K) (Figure 2a). The I−V behavior follows I ∼ (V−VT)ζ behavior,
The assembly of individual Au nanoparticles into necklace arrays was a two-step process. Briefly, the first step was the directed self-assembly (dSA) of nanoparticles to form 1D necklaces by slowly adding calcium chloride (CaCl2) solution to a colloidal suspension of 10 nm citrate-terminated Au nanoparticles. The solution was stirred in a shaker for dSA time, td, of 18 and 27 h. The dSA process was similar to previous studies.18,22−24 The process was conveniently monitored as a red shift in the surface plasmon resonance peak due to delocalization of electrons (Figure 1a). (The details on experimental conditions are in the Supporting Information (SI), Section S.1.) The second step was the self-limiting monolayer deposition of negatively charged necklaces on the positively charged surface of poly(dimethylsiloxane) treated with NH3 plasma. The NH3 plasma modified the surface with amine groups. The deposition was performed by centrifugation for 10 min with necklace suspension in contact with the chip with six devices (Figure 1b). (The details of the experimental conditions and the design of the chip are discussed in the SI, Section S.2 and Figure S1.) The deposition process resulted in a self-assembled monolayer that was self-limited to a monolayer due to electrostatic repulsion between the charged necklace clusters on the surface and in the solution. The necklace coverage increased by increasing the centrifugal force, measured as the number of g’s, N, where g is the acceleration due to gravity (Figure 1c). Each chip after deposition had six devices composed of a 15 μm wide array between two Au electrodes at a gap of 10 μm. The electrical characterization of the necklace array was performed on a homemade, two-point probe instrument. The bias, V, at a step size of 100−200 mV was applied across the two pads that are 10 μm apart, and the current, I, was measured via a DAQ card.
Figure 2. I−V characteristics of the necklace array. (a) Typical I−V behavior of two nanoparticle necklace devices. (b) Threshold bias, VT, as a function of area coverage, ϕ. Error bar for VT was based on the six devices on the chip. Error bar for coverage is based on at least six randomly selected SEM images on the chip.
as previously reported..17,18 As the coverage area of the nanoparticle necklace array, ϕ (described later, see Figure 3b), increases, there is a monotonic decrease in VT, indicating that the I−V tends toward more Ohmic behavior (Figure 2b), as expected for a dense array.10,25 The following inference may be drawn from the observation in Figure 2: first, the necklace array is percolating at a coverage as low as ∼34% (Figure 2b), which is well below the percolation threshold, ϕC, of ∼45% for a 2D random array of particles.27 Thus, the system is highly correlated at the local level due to the 1D nature of the necklaces. Second, for the same coverage, ϕ, the nonlinearity changes significantly depending on the td (Figure 2b). The VT is significantly enhanced for smaller td; however, the currents are highly diminished. Thus, beyond the overall coverage, the B
DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C
higher td indicated that the growth process also involved cluster−cluster aggregation. From the B/W image, the fraction of the area covered by nanoparticles, ϕ, was also calculated as the fraction of “white pixels”. The monotonic increase in ϕ as a function of N indicated that the clusters were in an unstable suspension that tended to separate from the solution to deposit on the substrate (Figure 3b). For larger td = 27 h, owing to the larger cluster size, it was reasonable to observe that the particles commenced to deposit at lower N. The ϕ saturated around N ∼ 400, indicating that the deposition was self-limited due to electrostatic repulsion between depositing clusters and clusters on the surface. The deposition of smaller clusters for td = 18 h was slower. The monotonic, nominally linear deposition growth at lower td was attributed to weak electrostatic repulsion between the clusters. Interestingly, for both td, the saturation was at ϕ ∼ 42%. Although N = 500 was the limit of our centrifuge, depositing the necklaces for a longer time (∼30 min) for both values of td did not increase the coverage beyond 42%, confirming saturation. The error bars were based on at least six images for each td and N. As visible precipitation was observed for td ∼7 days, suspension for td below 2 days was reasonably stable. Next (and in the rest of the manuscript), we consider the percolation characteristics of the cluster. It is apparent that at large N, the array became percolating (for example, N = 500 in Figure 1c). As the deposition became denser, we defined a cluster to be isolated if it was disconnected (at all peripheral points) by at least one pixel (i.e., ∼4 nm) apart from the surrounding necklace network. Similar to Figure 3a, we calculated the second-moment average cluster size as a function of total coverage ϕ. At least six images were analyzed per condition. Interestingly, at ϕ ∼ 30%, the average size abruptly increased (Figure 3c). The jump in the average cluster size by 2 orders of magnitude was attributed to the onset of percolation, where adjacent clusters joined to form a continuous path over a distance of 10 μm (i.e., device length). Thus, ϕ = ϕT of ∼32% for both td of 18 and 27 h marked the classical percolation threshold defined by the morphology of the network (rather than transport properties). The ϕT at ∼ 32% was significantly lower than the threshold of ϕC ∼45% for a random array of particles in two dimensions.27 As noted in the first paragraph, the ϕT < ϕC signifies the local 1D nature of the necklace network. It is expected that for ϕ between ϕT (∼32%) and saturation (i.e., ∼42%), there will be some clusters that are isolated from the surrounding (percolating) matrix. This heterogeneity is (beautifully) visual as charge contrast in SEM (Figure 4a). The clusters connected to the grounded electrode on the chip are brighter, while some of the farther clusters appear gray as they are isolated. The grayscale seems inverted from the conventional charge imaging contrast where the charging increases the apparent sample-detector bias to increase brightness.28−30 A possible explanation may be that due to the nanoscale nature of the network with high porosity, the local charging decreased the incidence of (probe) electron beam flux on the sample due to electrostatic repulsion to cause the isolated cluster to appear grayer. On increasing the probe energy of the SEM electron beam, some of the isolated clusters got connected as the current may have tunneled through the barrier of the disconnection point (Figure 4b). A remarkable reversible connection/disconnection cluster in the image can be
Figure 3. Percolation characteristics of the necklace network array. (a) Histogram of necklace size distribution. Inset: typical image of necklace deposition at N = 30. (b) Area coverage of necklaces, ϕ, is a function of centrifuge speed in g’s. The error bar is based on at least six images on the same chip. The second-moment average necklace size as a function of ϕ, indicating the percolation behavior where the size jumps abruptly.
topology of the nanoparticle necklace network is important in understanding the observed non-Ohmic behavior. Third, although, the Coulomb blockade due to local (single-electron) charging of a 10 nm particle is roughly ∼2.5 kT at 295 K (where k is Boltzmann’s constant),9 a strong non-Ohmic behavior is not expected because an alternate percolation path may diminish the effectiveness of the electrostatic barrier at room temperature. For the study, we quantitatively considered the topology of the necklace network array to attempt to explain the above observation. As the time scale required for the necklace to form, i.e., td is several hours and centrifuge deposition time is 10 min, the size distribution of the necklace clusters was obtained by deposition at N = 30 where the individual clusters were sparsely distributed (Figure 3a). Briefly, the 9770 × 7330 nm2 SEM images with 2560 × 1920 pixel were converted to “black/ white” (B/W) images to identify each particle and corresponding necklace (Figure 3a, inset: “image analysis” with each necklace in false colors). In the B/W image analysis, the pixels above a certain gray level (defined as the percent coverage of the particle on the pixel and, importantly, correlated to the gray level of neighboring pixels) were considered to be white, resulting in a clear distinction between the particles in the necklaces and the background (SI, Section S.3). As the pixel size was ∼3.8 nm square, the estimation of the area coverage for individual 10 nm particles was reasonable. Furthermore, for consistent results, an area histogram of the foreground (i.e., necklaces) and background was independently calculated and compared to the total area (SI, Figure S2). After the identification of each necklace, the number of particles for each cluster was calculated by normalizing the total area of the cluster (i.e., white pixels) by the effective pixel area of a single particle. The cluster distribution was obtained by averaging over several images (Figure 3a). The second-moment average necklace size increased significantly from ∼20 to ∼34 particles/necklace as td increased from 18 to 27 h. The size distribution for td = 27 h had a longer tail resulting in a larger number of particles forming large necklaces. The larger tail at C
DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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insets A2 and B2). The isolated clusters were subtracted to obtain (ϕ−ϕV) (Figure 5a, insets A3 and B3). From the ϕ (of the total array) and (ϕ−ϕV), the void fraction (ϕV/ϕ) was obtained. (Also, see details in SI, Section S.4 and Figure S3, where the void image is also shown to obtain ϕV directly). The calculations of (ϕ−ϕV) and ϕV were similar to the estimation of ϕ described above. The relative void fraction, defined as ϕv/ϕ, decreased with ϕ (Figure 5b). The error bar was based on an analysis of at least six images on a chip at a given N and td. Two aspects of a void fraction, ϕv/ϕ, are noteworthy: first, for a given ϕ, the ϕv/ϕ for td = 18 h was higher than for 27 h. This is reasonable because smaller clusters can pack more efficiently while minimizing electrostatic repulsion, while larger clusters may be forced to touch owing to their size. As a result, for smaller td, ϕT was slightly higher than 30% (Figure 3c) and ϕv/ϕ was larger (Figure 5b). Second, although it seems fortuitous, interestingly, the linear extrapolation of ϕv/ϕ indicated that the voids vanished around 45%, which coincides with ϕc. The coincidence is hard to explain but the value being above saturation of 42% is reasonable because the self-limiting deposition (due to electrostatic repulsion) must cease once the monolayer deposition covers all possible sites leading to ϕv ≈ 0. A central consequence of voids in the network was on the topology of the percolation path and hence the electrical properties of the necklace array. It is well known that introducing voids in a 2D nanoparticle array increases the effective length of the percolation path (to circumvent the voids), resulting in larger VT, i.e., larger non-Ohmic behavior.23 The 1D nature of the necklaces conveniently allowed us to follow the percolation path. To quantify the percolation path topology, the shortest path between two virtual electrodes 10 μm apart was identified and measured (see SI, Section S.5 for details). Briefly, the B/W image was converted to a skeleton image of one-pixel-wide lines. All contiguous pixels were connected by a line (called an edge) that joined the terminal pixels (called vertex) for each (local) cluster. A typical morphological graph of vertex and edge (blue in Figure 6a) with several percolation paths (yellow and orange lines) was overlaid on an SEM image to show the correspondence. The shortest path between two points at the top and bottom of the image was computed using Dijkstra’s algorithm (details in SI, Section S.5). For example, the yellow line in Figure 6a was the shortest path of contour length R and end-to-end distance L. The topology of the percolation path was parameterized by tortuosity parameter, R/L. By choosing well over 500 paths computed over six images between the electrodes, spanning from the longest to the shortest L, a distribution of R/L was calculated (see SI, Figure S4e). The second-moment average, ⟨R/L⟩, over several images for each sample was obtained from the distribution. Interestingly, a “universal” behavior of the percolation path topology ⟨R/L⟩ for different morphologies for the two td with respect to ϕv/ϕ was observed (Figure 6b). The topology parameter, ⟨R/L⟩, was linearly dependent on ϕv/ϕ (irrespective of the processing details). The universality (found in these limited experiments) indicated that the “relative porosity” of the network was a primary parameter that determined the topology of the percolation path (and hence the electronic properties) of these disordered 1D NP necklace arrays. Furthermore, replotting the electrical transport property, VT, as a function of ϕv/ϕ (Figure 6c) unified the behavior of
Figure 4. Heterogeneity in the percolating array. (a) SEM image of the necklace network (fabricated at td = 27 and deposited at N = 300) close to a grounded electrode on a chip. (b) SEM image (at 1 and 1.5 kV electron energies) of necklaces fabricated for td = 27 h and deposited at N = 200. (c) SEM image of the necklace fabricated at td = 27 and deposited at N = 300 on a chip. The connection turns on as SEM electron beam energy is increased from 1 to 2 kV.
(occasionally) found as SEM electron energy modulates between 1 and 2 eV (Figure 4c). The heterogeneity can be quantified using the same strategy as in Figure 3c to identify isolated clusters that are disconnected with the surrounding matrix by at least one pixel (i.e., ∼4 nm). The heterogeneity is defined as the void fraction, ϕV, which is the total coverage of all of the isolated clusters that are not part of the percolating network. Similar to the analysis for Figure 3, the isolated clusters that were disconnected by at least one pixel from the surroundings in the B/W image (see SI, Section S.4) were identified (Figure 5a,
Figure 5. Quantification for heterogeneity: void fraction. (a) Set of images demonstrating the image analysis from the original image (insets, (A1) and (B1)) to the identification of isolated necklaces (insets, (A2) and (B2)) and removal of the isolated necklace clusters leaving the percolating matrix (insets, (A3) and (B3)). (b) Nonlinear and linear fits through coverage versus void fraction behavior. Linear fitness is reasonable with R2 in the 0.85−0.95 range. D
DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 6. Quantification of the topology of the percolation path. (a) Percolation path overlaid on the original SEM image of necklace array fabricated at td = 27 h and N = 200. (b) Linear fit for (ϕV/ϕ) versus ⟨R/L⟩. The error bar in ⟨R/L⟩ based on the dispersity of the distribution is smaller than the data point shown. (c) Linear dependence of VT on ⟨R/L⟩ and (ϕV/ϕ). For clarity of the figure, the error bars for ϕv/ϕ (in Figure 5b), VT (in Figure 2b), and ⟨R/L⟩ (in b) are not shown. At Ohmic condition (i.e., ϕv/ϕ = 0), there appears to be a small residual void fraction of ∼0.5%.
Figure 7. Effect of electron energy on the topology of the percolation path. From the original SEM images (panels A1 and B1), the shortest percolation path (panels A2 and B2) is calculated. The distribution of the shortest R/L based on over four images shows a significant change as electron energy increases from 1.0 to 1.5 kV (panels A3 and B3). The chip is fabricated from the necklaces at td = 27 h and deposition at N = 250.
different morphologies that appeared significantly different in Figure 2. Thus, a (central) conclusion of the study is that the electrical transport properties, specifically the non-Ohmic behavior, primarily depend on the void fraction, ϕv/ϕ, rather than simply the coverage, ϕ. Alternatively, based on Figure 6b, as ϕv/ϕ linearly depends on the percolation topology, ⟨R/L⟩, the VT is also a linear (unified) function of the percolation path tortuosity (Figure 6c). Experimental observation in Figure 4 and simulation studies (as a function of applied bias to high levels)26 indicated that the percolation paths may morph as the kinetic energy of the conduction electron increases. Simulation studies (consistent with the expectation from Figure 4) indicated that as the bias increased, the percolation path became more direct, i.e., ⟨R/L⟩ should tend toward 1. Image analysis of the same area of the chip at two different electron energies (Figure 7, panels A1 and B1) clearly indicated that the (shortest) percolation paths increased in density (Figure 7, panels A2 and B2) and the tortuosity, R/L, morphs significantly (Figure 7, panels A3 and B3). Consistent with simulation observations, the ⟨R/L⟩ became more direct (i.e., ⟨R/L⟩ → 1) and the distribution became sharper. To capture the morphing effect of the percolation path, we proposed a simple resistance network model (Figure 8a). The network was composed of two sets of parallel resistance signifying percolation paths. The resistance, R1, corresponded to the percolation paths connected right from zero bias, while the R2/Va corresponded to percolation paths that opened as
Figure 8. Simple model for percolation path morphing. (a) Resistance network model with a bias-dependent resistor. (b, c) Typical examples of differential conductance of two devices. (d) Dependence of exponent, a, on the coverage, ϕ.
the connectedness increased leading to an overall impedance decrease. For reciprocal dependence on V, the exponent, a, must be ≥0. The net conductance was given as I/V = (1/R1) + (Va/R2). As the net conductance, I/V, was sensitive to contact E
DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry C
the two processes, a nanoparticle array can be tailored with the same coverage of particles but significantly different Coulomb blockade characteristics. For example, the threshold bias, VT, can be changed from 2 to 4 V with the same particle coverage, ϕ, of ∼36%. The percolation threshold at ϕ = ϕT of 32% was well below the 45% for random 2D nanoparticle arrays, signifying the local 1D structure. The 1D necklace network allows a convenient approach to identify and quantify the percolation paths in the network and determine the topology and distribution of the (shortest) percolation paths traversing between two points on a gap of 10 μm (equal to the dimensions of the device). By the quantitative charge contrast SEM image analysis, necklace clusters were isolated that were not part of the percolating network. These isolated clusters, i.e., voids, posed a topological constraint to increasing the tortuosity of the percolation path. The analysis revealed that the Coulomb blockade effect is dependent on the topology of the network, i.e., percolation path tortuosity, and the void fraction of clusters is not connected to the percolating matrix. Charge contrast SEM further revealed the visualization and quantification of morphing of the percolation path as the kinetic energy of the conduction electron increases. With selfassembly conditions to obtain a percolating necklace network with a coverage area of 34−42% to achieve VT from 0.5 to 5.5 V, it was concluded that not the coverage (ϕ) but the tortuosity of the (shortest) percolation path and the void fraction unify the Coulomb blockade characteristics over a broad range of necklace network fabrication conditions.
resistance, the differential conductance, a much better property, dI/dV = (1/R1) + ((a + 1)/R2)Va, was more appropriate to test the model, where 1/R1 (including the contact resistance) became a simple offset. Because the array will become Ohmic at a = 0, which is not true due to known (and observed) MJTs, the model does not capture the Coulomb blockade effect. If we assume that the percolation morphing effect is independent of the Coulomb blockade effect, then the exponent, a, should semiquantitatively parametrize the nature of morphing of the percolation path. A (typical) I/V was differentiated to obtain differential conductance, dI/dV. Curves for the same device, with a slight offset in bias, were superimposed to obtain high-density data that was then numerically differentiated to obtain dI/dV as a function of bias. Two typical examples are shown to demonstrate the data density and noise levels (Figure 8b,c). From the nonlinear curve fit, the exponent showed a reasonable systematic trend between the exponent, a, and the coverage, ϕ (Figure 8d). However, the correlation was weak. As the coverage increased, the effect of the percolation path morphing, as expected, decreased. However, owing to noisy curves, a device-to-device variation for dI/dV on the same chip (for example, repeat spots at same ϕ) indicated, at best, that only a qualitative conclusion can be drawn, meaning that the morphing of the percolation path occurred over the whole bias range to affect the electrical behavior of the nanoparticle array (Figure 9).
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b05527.
Figure 9. Correlation between exponents a and ζ. The exponents are obtained by fitting various I−V curves for both td = 18 and 27 h.
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As shown by simulation and experiments, the critical exponent, ζ, increases with void fraction due to an increase in tortuosity.12,20,25 From our study, as the void fraction increase leads to an increase in tortuosity of the percolation path (Figure 6b), it will also cause an increase in exponents, a and ζ. Although no universal behavior was observed (as in Figure 6), consistently, both ζ and an increase monotonically with ϕv/ϕ (SI, Figure S6). Owing to their respective dependence on ϕv/ϕ, the correlation between the two exponents is reasonably linear irrespective of td, further indicating the importance of the topology of the percolation path on the I−V characteristics. As per the model, at a = 0, the I−V should follow Ohm’s law, i.e., ζ→1. A larger intercept of ζ ∼ 1.6 indicates, perhaps over a larger range, that the correlation is not linear.
Experimental conditions and image analysis; image analysis of all the intermediate steps of image processing, including necklace network fabrication, black/white image analysis, and identifying percolation paths (PDF)
AUTHOR INFORMATION
ORCID
Ravi F. Saraf: 0000-0002-8537-054X Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS R.F.S. would like to thank the Office of Basic Energy Science, DOE (DE-SC0001302), for financial support in the initial phase of this research. R.F.S. thanks financial support by Nebraska Center for Energy Sciences Research.
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REFERENCES
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CONCLUSIONS A network composed of an array of 1D nanoparticle necklaces was fabricated in a two-step, self-assembly process, where the size of the necklaces in solution and their subsequent deposition were controlled independently. The electrical behavior at room temperature (295 K) was studied. By tuning F
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DOI: 10.1021/acs.jpcc.9b05527 J. Phys. Chem. C XXXX, XXX, XXX−XXX
