Bulk versus percolative behavior

Researchers have been studying nanostructured materials for transparent electrodes for over six years.Reference Hu, Hecht and Gruner^15, Reference Wu, Chen, Du, Logan, Sippel, Nikolou, Kamaras, Reynolds, Tanner, Hebard and Rinzler^16, Reference Saran, Parikh, Suh, Munoz, Kolla and Manohar⁶⁶ The technology is now mature enough to compare and contrast the various materials under study. It is often helpful to use figures of merit (FoM) to express the performance of a given film type and facilitate comparison with other data sets. The most useful FoMs are those that are based on the physics of the system under study. For example, FoMs for TCs should link the film transmittance, T, to its sheet resistance, R _s. This is done by combining physical expressions for transmittance and sheet resistance of thin films, both as a function of thickness, t. For example, the transmittance can be expressed asReference Dressel and Gruner⁶⁷

(1)

where Z ₀ is the impedance of free space (377 Ω), and σ_op is the optical conductivity (related to the absorption coefficient α as σ_op ≈ α/Z ₀).Reference Dan, Irvin and Pasquali¹² By combining this expression with the definition of sheet resistance (for a bulk-like film):

(2)

(σ_dc,B is the bulk dc conductivity of the film) to eliminate t, one obtains a relationship between T and R _s:

(3)

Note that this expression is appropriate for a bulk-like material (i.e., one whose dc conductivity is invariant with sample thickness). The conductivity ratio σ_dc,B/σ_op is often used as a FOM for transparent conductors as high values of σ_dc,B/σ_op lead to films with high T and low R _s. In fact, a material with a given value of σ_dc,B/σ_op can be formed into films with a range of T and R _s values depending on the thickness. Thus in practice, researchers generally prepare films with a range of thicknesses, measure R _s and T for each thickness, and then fit the (T, R _s) data to Equation 3 to extract σ_dc,B/σ_op. Examples of this procedure are shown in Figure 1. Here, transmittance is plotted as a function of sheet resistance for thin films of graphene,Reference Tien, Huang, Yang, Wang and Ma⁵⁵ SWNTs,Reference Southard, Sangwan, Cheng, Williams and Fuhrer³⁴ Cu nanowires,Reference Wu, Hu, Rowell, Kong, Cha, McDonough, Zhu, Yang, McGehee and Cui⁴¹ and Ag nanowires.Reference De, Higgins, Lyons, Doherty, Nirmalraj, Blau, Boland and Coleman³⁶ In each case, the bottom left portion of the data (corresponding to thicker films) is fit to Equation 1 (solid lines), giving values of σ_dc,B/σ_op of 0.25, 1.5, 110, and 415, respectively. Usually these numbers are assumed to encode all the relevant information about the optoelectrical properties of these films. In order to achieve the target of T ≥ 90% and R _s ≤ 100 Ω/sq, Equation 3 can be used to show that σ_dc,B/σ_op ≥ 35 is required. This is the first criterion for a high-performance transparent conductor.

Figure 1.

Typical graph of transmittance (generally measured at ∼550 nm) plotted versus sheet resistance for thin films of nanostructured materials. Note that each curve can be divided into two regimes, the bulk regime (the solid line is a fit to Equation 3) and the percolation regime (the dashed line is a fit to Equation 5).Reference Southard, Sangwan, Cheng, Williams and Fuhrer^34, Reference Wu, Hu, Rowell, Kong, Cha, McDonough, Zhu, Yang, McGehee and Cui^41, Reference Tien, Huang, Yang, Wang and Ma^55, Reference Green and Hersam⁶³

However, looking carefully at the fits in Figure 1, it is clear that the data tend to deviate severely from the fits for thinner (more transparent) films. This deviation has been observed beforeReference De, Lyons, Sorrel, Doherty, King, Blau, Nirmalraj, Boland, Scardaci, Joimel and Coleman^13, Reference Geng, Kim, So, Lee, Chang and Lee^14, Reference Doherty, De, Lyons, Shmeliov, Nirmalraj, Scardaci, Joimel, Blau, Boland and Coleman^35, Reference De, Higgins, Lyons, Doherty, Nirmalraj, Blau, Boland and Coleman^36, Reference De, King, Lotya, O’Neill, Doherty, Hernandez, Duesberg and Coleman^46, Reference De, King, Lyons, Khan and Coleman⁶⁸ and tends to occur for films with T between 50% and 92% (see later in text). This is important because applications requiring TCs usually require T ≥ 90%. Thus, σ_dc,B/σ_op fails to describe the relationship between T and R _s in the technologically relevant regime. This means that in many cases, σ_dc,B/σ_op, or indeed any FoMs based on bulk properties, are not entirely appropriate, particularly when high transparency is required. Recently, it has been shown that the deviation from bulk-like behavior as described in Equation 1, can be explained by percolation effects.Reference De, King, Lyons, Khan and Coleman⁶⁸ Such effects become important for very sparse networks of nanoconductors. When the number of nanoconductors per unit is very low, a continuous conducting path from one side of the sample to the other will generally not exist. As more nanoconductors are added, at some point (the percolation threshold) the first conducting path will be formed. As more material is added, more conductive paths are formed, and the conductivity of the network increases rapidly. Eventually it reached a “bulk-like” value above which it remains constant. Percolation theory describes how the dc conductivity of sparse networks depends on network thickness and predicts a non-linear, power law dependence:Reference Stauffer and Aharony⁶⁹

(4)

where t is the estimated thickness of the network, t_c is the thickness associated with the percolation threshold, and n is the percolation exponent. This leads to a new relationship between T and R _s, which applies to thin, transparent networks:Reference De, King, Lyons, Khan and Coleman⁶⁸

(5)

where Π is the percolative FoM:

(6)

Here, t _min is the thickness below which the dc conductivity becomes thickness dependent (i.e., Equation 5 applies below t _min but Equation 3 applies above t _min). Analysis of these equations shows that large values of Π but low values of n are desirable to achieve low R _s coupled with high T.Reference De, King, Lyons, Khan and Coleman⁶⁸ Furthermore, networks of nanostructures show values of t _min, which scale closely with the nanostructures’ smallest dimension, D: t _min ≈ 2.33D.Reference De, King, Lyons, Khan and Coleman⁶⁸ The high T portion of data in Figure 1 were fitted using Equation 5. In all cases, good fits allow the calculation of both n and Π. It is clear from this analysis that knowledge of n and Π is critical to understanding the behavior of thin networks of nanostructured materials (i.e., those in the technologically relevant regime). However, while n is a fundamental property of the network, looking at Equation 6, we note that Π is a composite parameter that is related to σ_dc,B/σ_op. This means that while σ_dc,B/σ_op does not directly control T and R _s in the high T regime, it is certainly of indirect importance. In this article, we describe both σ_dc,B/σ_op and the percolation parameters, n and Π, for a range of previously reported nanostructured TCs.

Analysis of literature data

The aim of this article is to assess the relative merits and to understand the limitations of the nanomaterials commonly used to prepare TCs. To this end, we extracted data for (R _s, T) from a range of studies describing transparent conducting properties of thin films of graphene, SWNTs, and metallic nanowires (tabulated data are included in the supporting information). The graphene data included films of solution-cast reduced graphene oxide and surfactant exfoliated graphene but no vapor grown graphene. In the majority of cases, the data contained both bulk and percolation regimes. Each regime was fitted to the relevant equation (3 or 5) and values for σ_dc,B/σ_op, n and Π were extracted. We note that we have not analyzed every paper in the literature; however, we hope that the range of papers studied and cited herein roughly captures the current state of the field.

We begin by looking at the extracted values of σ_dc,B/σ_op. For each material, the conductivity ratio was ranked from highest (i.e., number 1) to lowest (i.e., number 45 for nanotubes). The position in this ranking list is known as the ratio ranking. The conductivity ratio values are plotted as a function of ratio ranking in Figure 2. In the case of solution-processed graphene, there is significant variation with σ_dc,B/σ_op ranging from 0.01 to ∼15. The two highest ranked results were for networks of silver nanoparticle decorated graphene sheets, so these are not, strictly speaking, graphene films. Of these samples, the most promising film had T ∼ 80% and R _s ∼ 250 Ω/sq. We note that the highest graphene-only film displayed σ_dc,B/σ_op ∼ 1.3. The relatively poor performance of graphene networks is due to a combinationReference De and Coleman⁴⁵ of low dc conductivity due to resistive inter-flake junctionsReference Nirmalraj, Lutz, Kumar, Duesberg and Boland⁷⁰ and the high optical conductivity of graphene.Reference Nair, Blake, Grigorenko, Novoselov, Booth, Stauber, Peres and Geim⁷¹ In the case of nanotubes, the spread is significantly less, with σ_dc,B/σ_op values ranging from ∼0.1 to ∼60. The best sample was for nanotubes deposited from superacids whose most promising film displayed T ≈ 91% and R _s ≈ 60 Ω/sq.Reference Hecht, Heintz, Lee, Hu, Moore, Cucksey and Risser⁷² Looking at the 10 best values (Figure 2 inset), it is clear that the spread is very tight, with the top 10 varying in σ_dc,B/σ_op from 12 to 60. This reflects the relative maturity of nanotube-based transparent conductor research and may suggest that this technology is approaching physical limits, possibly set by the high inter-nanotube junction resistance.Reference Nirmalraj, Lyons, De, Coleman and Boland⁷³ Turning to the data for metallic nanowire networks, while there are relatively few papers on these materials, the σ_dc,B/σ_op values are very impressive, ranging from 106 to 453. We note that the sample with the highest σ_dc,B/σ_op was a spray cast silver nanowire network, whose best film displayed T ≈ 85% and R _s ≈ 33 Ω/sq.Reference Madaria, Kumar and Zhou³⁹ However, we note that a sample with lower σ_dc,B/σ_op (but no percolative regime, see later in text) included one film with even more impressive properties: T ≈ 89% and R _s ≈ 20 Ω/sq.Reference Madaria, Kumar, Ishikawa and Zhou³⁸ The high values observed here are due to a combination of low inter-wire junction resistanceReference De, Higgins, Lyons, Doherty, Nirmalraj, Blau, Boland and Coleman^36, Reference Lee, Connor, Cui and Peumans³⁷ and low optical conductivity of these networks.Reference De, Higgins, Lyons, Doherty, Nirmalraj, Blau, Boland and Coleman³⁶

Figure 2.

Bulk dc to optical conductivity ratio plotted in order of ratio ranking for solution-processed thin films prepared from metallic nanowires, single-walled carbon nanotubes, and graphene. The horizontal dashed line shows the value of σ_dc,B/σ_op = 35 required to reach the minimum industry standards of T = 90% and R _s = 100 Ω/sq. The inset shows the best performing samples for each material type.

This analysis clearly shows that only networks created from SWNTs produced using non-trivial processing (i.e., superacid-based processing) or metallic nanowires have surpassed the first requirement for a high-performance TC (σ_dc,B/σ_op ≥ 35). As superacid dispersions will not readily lend themselves to industrial processing, we suggest that only metallic nanowire networks are suitable candidates for solution-processed high-performance TCs. It is worth noting that σ_dc,B/σ_op is largely a material property; it is primarily controlled by the optical and electrical properties of the nanoconductors and the potential profile of the junction between them (and thus the junction resistance). This suggests that metallic nanowires qualify as candidates for high-performance TCs, while SWNTs and graphene networks currently do not, due to their intrinsic material properties. This is important, as such intrinsic properties will not be easy to modify. On the face of it, this suggests the metallic nanowire networks to be the obvious choice. However, we note that as the properties of high transparency films of metallic nanowires may be controlled by percolation, σ_dc,B/σ_op may not accurately describe their properties. Thus, it is important to know whether technologically relevant films (i.e., T > 90%) lie in the bulk or percolative regime.

To test this, we first needed to know where the crossover from bulk to percolative behavior occurred. From the published data we extracted the transmittance, T_x, and sheet resistance, R_x, at the crossover between the bulk and percolative regimes. These data are plotted in Figure 3. From this it is clear that for all materials, the crossover occurred at transmittances between 50% and 92%. In almost all cases, T_x was below 90%, meaning that any film displaying T = 90% almost always falls in the percolative regime. In addition, with the exception of the metallic nanowire data, almost all of the R_x data were >100 Ω/sq, showing that T = 90% and R _s < 100 Ω/sq cannot be achieved for the vast majority of samples.

Figure 3.

The crossover between bulk and percolative behavior. The boundary between the bulk and percolative regimes can be characterized by the transmittance, T_x, and sheet resistance, R_x, at the crossover. These are plotted in this figure for all three materials. The meaning of this crossover is illustrated for the point marked by the arrow. To the left of each data point, T is related to R_s for that material by the bulk expression (Equation 3), as shown by the solid line. However, to the right of each point, the T(R _s) data are described by the percolative expression (Equation 5), as shown by the dashed line. The red box encloses the region required by industry for electrodes for high-performance displays or solar cells.

In addition, we used the published data to estimate the resistance for a film with T = 90%, , for each data set (in some cases by extrapolation). We also noted whether the 90% film occurred in the bulk or percolative region of the (R _s, T) graph. These data are plotted in Figure 4 as a function of the conductivity ratio ranking given in Figure 2. In each case, the data points where T = 90% film occurred in the bulk regime and are marked by a solid data point, while the percolation data are open circles. A number of things are of interest. First, for the various types of nanowire, the T = 90% film occurred in the percolative regime seven times out of eight. The one nanowire type that displayed no percolation regime also displayed the best individual film, reaching T ≈ 89% and R _s = 20 Ω/sq, despite having a relatively low value of σ_dc,B/σ_op = 173.Reference Madaria, Kumar, Ishikawa and Zhou³⁸ For the SWNT data, the T = 90% film occurred in the percolative regime 66% of the time, while for graphene films, the rate was only 16%. This can be explained by noting that the percolative regime applies for films with a thickness t < t _min, where t _min ≈ 2.33D. Here, D is the thickness of the smallest dimension of the nano-material (i.e., the diameter of the nanowire or nanotube bundle or the thickness of the graphene flake). As this dimension is generally large for nanowires (D > 50 nm), even relatively thick nanowire films can be percolative. However the thickness of a graphene flake is very small (∼nm), which means only very thin films should be percolating. SWNTs fall between these two extremes. In addition, we note that when the T = 90% film is in the bulk regime, scales smoothly with the ranking, as expected from Equation 3. However, for films where the T = 90% film is in the percolating regime, shows much more scatter (because it also depends on t _min and n) and critically is always larger than would be expected for the equivalent bulk film. This illustrates the fact that percolation is very common and, when present, limits film performance.

Figure 4.

The sheet resistance expected for a film with T = 90%, , plotted as a function of the rankings for σ_dc,B/σ_op (see Figure 2). In each plot, the solid data represent films where T = 90% would fall in the bulk regime, while the open symbols represent films where T = 90% would fall in the percolation regime. Note that the presence of percolation always increases above what would otherwise be expected. To avoid clutter, we have divided the data between nanowires and nanotubes (a) and graphene (b).

Given how often percolation is the limiting factor in nanostructured TCs, it is worth looking in more detail at Π and n, which were obtained by fitting published (R _s, T) data as described previously. In Figure 5, we plot Π as a function of n, for networks of graphene, SWNTs, and metallic nanowires. The data for SWNTs are widely spread, perhaps suggesting considerable variability between nanotube types and processing methods. In contrast, the metallic nanowire and graphene data follow elongated patterns sloping downward and upward, respectively. These patterns are consistent with Equations 5 and 6 and occur for values of σ_dc,B/σ_op that are relatively high or low, respectively (typically the case for metallic nanowires or graphene networks). More importantly, by inserting T = 90% and R _s = 100 Ω/sq and rearranging Equation 5, we can plot a curve that separates films with T > 90% and R _s < 100 Ω/sq from those with T < 90% and R _s > 100 Ω/sq. This is shown as the dashed line (i.e., for a film to display T > 90% and R _s < 100 Ω/sq, its [n, Π] data must sit above this line). It is clear from these data that only the metallic nanowires surpass minimum industry standards for films where percolation controls the film properties. This is a very interesting result. It means that not only are metallic nanowires the only nanoconductors to fulfill the material requirements (i.e., σ_dc,B/σ_op ≥ 35), but they are also the only nanoconductors to form sparse networks that are capable of achieving T ≥ 90% and R _s ≤ 100 Ω/sq. This is further evidence that metallic nanowires are the most promising nanoconductors for solution-processed TCs.

Figure 5.

Percolation fit constants; the percolation figure of merit, Π, plotted as a function of the percolation exponent, n, for films of metallic nanowires, single-walled carbon nanotubes, and graphene. To achieve T > 90% and R _s < 100 Ω/sq, the data must fall above the dashed line on this graph (plotted using Equations 5 and 6).

A number of papers have speculated on the dependence of the dc conductivity of networks of nanotubes on the nanotube bundle diameter. It is generally agreed that smaller diameter bundles result in more conducting paths, and therefore higher dc conductivity.Reference Hecht, Hu and Gruner^74, Reference Lyons, De, Blighe, Nicolosi, Pereira, Ferreira and Coleman⁷⁵ Two articles in particular have predicted power law diameter dependences: σ_dc,B ∝ D ⁻ⁿ, where n has been suggested as either 2 (Reference Reference Hecht, Hu and Gruner74) or 3 (Reference Reference Lyons, De, Blighe, Nicolosi, Pereira, Ferreira and Coleman75). These are important predictions, as they suggest that films with higher σ_dc,B, and hence higher σ_dc,B/σ_op, could be achieved by using smaller nanotube bundles (i.e., better exfoliated tubes). However, such power law dependence has not been conclusively demonstrated.Reference Shin, Shim, Song, Kim and Hana⁷⁶ The published data collected during the preparation of this review allowed the examination of this question. As shown in Figure 3, data have been collected on the transmittance at the crossover from bulk to percolative behavior, T_x. This transmittance can be related to the thickness of the network at crossover, t _min via Equation 1, so long as the optical conductivity, σ_op, is known. While σ_op is not known particularly well for networks of metallic nanowires and is very scattered for graphene networks,Reference De and Coleman⁴⁵ it has been measured by a number of groups to be between 15,000 and 22,000 S/m for nanotube networks.Reference Doherty, De, Lyons, Shmeliov, Nirmalraj, Scardaci, Joimel, Blau, Boland and Coleman^35, Reference Geng, Lee, Kim, Han, Park and Lee^64, Reference Ruzicka, Degiorgi, Gaal, Thien-Nga, Bacsa, Salvetat and Forro^77, Reference Ugawa, Hwang, Gommans, Tashiro, Rinzler and Tanner⁷⁸ Taking σ_op = 18,500 S/m, this allows us to use Equation 1 to estimate t _min for a range of SWNT networks, obtaining values between 11 and 130 nm. This is of interest, as t _min should approximately scale with nanotube diameter, D, as t _min ≈ 2.33D. This allows us to use t _min as a proxy for bundle diameter. In this way, we can examine the diameter dependence of σ_dc,B via that of σ_dc,B/σ_op, which we now know. Shown in Figure 6a is σ_dc,B/σ_op plotted as a function of t _min for a range of SWNT networks (on the top axis is the estimated bundle diameter). Although the data are scattered (due to varying nanotube types, network morphologies), it is reasonably clear that smaller values of t _min, and so smaller bundle diameter, lead to larger σ_dc,B/σ_op. In addition, it can also be seen that the data are reasonably consistent with a power law of the form σ_dc,B / σ_op ∝ D ⁻ⁿ with 1 < n < 4. These data are consistent with both power laws previously proposed.

Figure 6.

(a) The conductivity ratio, σ_dc,B/σ_op, and (b) the sheet resistance expected for a film with T = 90%, , plotted as a function of t _min. Here, t _min is the film thickness where the transfer from bulk to percolation occurs. This was found by estimating the transmittance where the transition occurs, T_x, and using Equation 1 to convert to t _min (σ_op = 1.85 × 10⁴ S/m in all cases). On the top axis is the estimated bundle diameter, D, calculated using t _min ≈ 2.33D. In (a), the dashed lines illustrate D ^–4 and D ^–1 dependence. Note that only single-walled carbon nanotube films were considered.

However, it is important to remember that the technologically relevant parameter is not σ_dc,B/σ_op, but . In most cases, is not controlled solely by σ_dc,B/σ_op but also depends on t _min, σ_op, and n (Equations 5 and 6). Thus we test the dependence of on diameter in Figure 6b. From this graph, it is clear that smaller values of t _min (and so smaller D) lead to lower values of for SWNT networks, clearly illustrating the importance of good exfoliation. Previously, we have speculated that similar rules should apply to metallic nanowire networks. However, we note that the difference here is that we simply do not know enough about the optical conductivity of such networks. For example, if the optical conductivity increases with decreasing wire diameter, this effect may cancel out much of the advantage of using low diameter nanowires.

Concluding remarks

We have reviewed current state-of-the-art transparent conducting (TC) thin films of nanoscale conductors prepared by solution processing. We have analyzed the published data for transmittance as a function of sheet resistance for networks of graphene, nanotubes, and metallic nanowires. In the majority of cases, there was a transition from bulk-like behavior to percolation behavior as the film thickness was reduced. In most cases, the technologically relevant films with T ≥ 90% lie in the percolative regime. By fitting to known models, we extracted both bulk and percolative figures of merit for all samples. We found that only networks of metallic nanowires possess high enough figures of merit to qualify as high performance TCs (i.e., T ≥ 90% and R _s ≤ 100 Ω/sq). Finally, we have shown that at least for nanotube networks, the film performance scales with bundle diameter.

Networks of metallic nanowires are now very promising candidates for ITO replacement. A number of authors have reported metallic nanowire films with T ≥ 90% and R _s ≤ 100 Ω/sq.Reference Lee, Connor, Cui and Peumans^37, Reference Madaria, Kumar, Ishikawa and Zhou^38, Reference Wu, Hu, Rowell, Kong, Cha, McDonough, Zhu, Yang, McGehee and Cui⁴¹ In addition, these films can now be produced by industrially compatible methods, such as spray coating.Reference Madaria, Kumar and Zhou³⁹ However, a number of hurdles remain. Adhesion of solution processed Ag nanowire films to substrates has been poor,Reference De, Higgins, Lyons, Doherty, Nirmalraj, Blau, Boland and Coleman³⁶ an issue that must be resolved before industrialization. In addition, scattering of light due to the relatively large diameter of typical nanowires (50–100 nm) results in hazeReference Hu, Kim, Lee, Peumans and Cui⁴² which, while good for solar cells, will be detrimental to the performance of displays. In addition, commercially available Ag nanowires tend to be coated with an organic stabilizer. This is displaced locally from the junction region during network formation, giving excellent electrical contact at junctions while the remaining organic coating protects the nanowire surface from tarnishing (i.e., oxidizing). However, the temporal stability of these coatings is unknown, and if it degrades over time, the network conductivity is likely to suffer. Finally, although some work has begun,Reference Rathmell, Bergin, Hua, Li and Wiley^40, Reference Wu, Hu, Rowell, Kong, Cha, McDonough, Zhu, Yang, McGehee and Cui⁴¹ it will be necessary to demonstrate temporally stable, high-performance networks of nanowires from cheap metals such as copper. Thus, if these hurdles can be addressed, the path forward is clear; formation of networks from low diameter nanowires may result in better optoelectrical propertiesReference Lyons, De, Blighe, Nicolosi, Pereira, Ferreira and Coleman⁷⁵ and should reduce haze. If the wires can be produced from copper with an appropriate, long-lasting, surface stabilizer, the next generation of high-performance TCs may be based on metallic nanowires.