Carbon Nanotube Assemblies for Transparent Conducting Electrodes

Abstract

The goal of this chapter is to introduce readers to the fundamental and practical aspects of nanotube assemblies made into transparent conducting networks and discuss some practical aspects of their characterization. Transparent conducting coatings (TCC) are an essential part of electro-optical devices, from photovoltaics and light emitting devices to electromagnetic shielding and electrochromic widows. The market for organic materials (including nanomaterials and polymers) based TCCs is expected to show a growth rate of 56.9% to reach nearly $20.3 billion in 2015, while the market for traditional inorganic transparent electronics will experience growth with rates of 6.7% to nearly $103 billion in 2015. Emerging flexible electronic applications have brought additional requirements of flexibility and low cost for TCC. However, the price of indium (the major component in indium tin oxide TCC) continues to increase. On the other hand, the price of nanomaterials has continued to decrease due to development of high volume, quality production processes. Additional benefits come from the low cost, nonvacuum deposition of nanomaterials based TCC, compared to traditional coatings requiring energy intensive vacuum deposition. Among the materials actively researched as alternative TCC are nanoparticles, nanowires, and nanotubes with high aspect ratio as well as their composites. The figure of merit (FOM) can be used to compare TCCs made from dissimilar materials and with different transmittance and conductivity values. In the first part of this manuscript, we will discuss the seven FOM parameters that have been proposed, including one specifically intended for flexible applications. The approach for how to measure TCE electrical properties, including frequency dependence, will also be discussed. We will relate the macroscale electrical characteristics of TCCs to the nanoscale parameters of conducting networks. The fundamental aspects of nanomaterial assemblies in conducting networks will also be addressed. We will review recent literature on TCCs composed of carbon nanotubes of different types in terms of the FOM.

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4.1 Materials for Transparent Conductive Electrodes

A broad range of materials has been considered for possible applications as transparent conductive electrodes, from nanoparticles, graphene sheets, nanotubes (single to multiwalled), nanowires and their composites as well as composites with a polymer matrix. The decision whether to use one material or another depends on multiple factors such as the application requirements or the materials cost. More specifically, these can include work function, electron concentration/mobility, toxicity, deposition cost, and many others factors. For instance, in terms of production costs, the potential material may form a series in the order Cd < Zn <Ti < Sn < Ag < In, while in the toxicity series, the order would be different as follows Zn < Sn < In < Ag < Cd. Optimization of these cases is beyond the scope of this publication, since the goal is to give the background and references and instructional materials to facilitate research and development in the area of transparent conducting coatings (TCCs). The chapter is centered on the concept of the figure of merit (FOM) for transparent conductive coatings (TCC) for high-tech electro-optical applications. We will look at the TCE FOM evolution and show how the measurements of the FOM can be done for the nanomaterial-based TCCs. Furthermore, we explore how some details of the nanomaterial assembled structure could be derived from the measurements and also look at the effects of external doping on the FOM.

4.2 Indium: Cost, Supply–Demand Analysis

In most publications, the search for alternative transparent conductive electrodes to replace ITO is explained in terms of the increasing cost of indium. The price of indium has doubled in the last 10 years due to increasing demand, Fig. 4.1. The world production and consumption of indium was practically unchanged from the late 1980s to the early 1990s. The situation changed with the introduction of personal computers and consumer electronics to the market. In the last 20 years, there has been a continuous growth of consumption of TCCs and this has required increased production of indium. Most of indium, 70%, is consumed in coatings, and only 24% in electrical components, solder, and alloys [1]. The majority of indium is used in transparent conductive coatings. It is important that all indium used in the USA is imported from China, which supplies about 45%, Japan exporting 18%, and Canada providing 16%. Indium is a by-product of zinc mining and is imported to the USA where it is purified to electronic grade by two companies [1]. While Japan is the global leader in indium consumption, the growth of Chinese consumption has already led to cutting export quotas by 30% in the second half of 2010 to supply their domestic electronic industry demands.

Fig. 4.1

Historical data on apparent consumption, world production, and price of indium. Bars show the price of indium in dollars per ton. The price is adjusted by the Consumer Price Index conversion factor, with 1998 as the base year [2]

As a result, 21 Chinese indium producers exported 93 tons of indium compared to the 140 tons in the first half of 2010 [1]. Indium analysts have expressed concern of possible instability of the Chinese supply to external markets. This instability of indium supply along with its increasing price has triggered significant activity in the alternative transparent conducting coatings [3–30, 50–53].

4.3 Spectral Window Considerations

The spectral selectivity of transparent conductive coatings depends on the coating functionality (application). For instance, one of the interesting opportunities to lower the power consumption of buildings is to limit solar heating through the windows, thus reducing associated air-cooling expenses by 40–70%. In the USA, this could bring millions of dollars in savings. The Department of Energy Efficiency and Renewable Energy (EERE) Building Technology Program set the goal to achieve 50–70% of whole building energy improvement through various approaches including new windows technology. There are two configurations of windows, which could enable the solar gain control. One type of window is the passive type with static gain control, which has continuous transmittance. The other type is the chromogenic window, which has an active or dynamic transmittance control system that allows changes in transmittance depending on the solar activity [31].

Current efforts in high-performance windows target the replacement of thick, expensive infrared reflective coatings with a thin and easy to manufacture material. Candidates include reflective transition metal hydrides or suspended particles (for more details on current R&D efforts refer to the EERE Web site: http://www1.eere.energy.gov/buildings/windows_technology.html). It should be mentioned that application of transparent conductive coatings (TCCs) in architectural windows and glass facades has a substantial market with more than 4 bl. m² of floating glass produced per year. The requirements for optical properties of the coating depend on each particular window technology. Figure 4.2 summarizes the transparency requirements for the various applications, which require TCCs, including architectural windows and electronics. With the solar control technology, the coatings should be transparent below 700 nm but should have high reflectivity above 700 nm. Low emissivity windows should have broader transmittance range, up to 3 μm, and high reflectivity above this wavelength. The spectral range for TCCs in a photovoltaic cell is defined by the solar spectrum (Fig. 4.3) and the optical properties of an active medium, where photocurrent is generated. Photovoltaic TCCs should have high transmittance up to 2 μm. The TCC requirements for electronics, including flexible electronics, are that it be transparent (T ~ 1) below 750 nm.

Fig. 4.2

Spectral requirements for some applications, which require TCCs. The requirement to be transparent in the visible part of the spectrum is the same for all windows and flexible displays. The reflectance and transmittance of an ideal coating, with no absorbance (100 % T and 100 % R, is referenced in the figure as T ~ 1 and R ~ 1) for visualization. More detailed requirements for different types of displays are given in the Figure of merit section

Fig. 4.3

Normalized spectral characteristics of 1.5 air mass (AM) solar spectrum relative to the averaged sensitivity of the human eye (labeled as the photopic spectral curve), and the transmittance spectrum of ITO [33]. One can see why ITO is well suited for photovoltaic applications

4.4 Why Is Transmittance of TCCs Referenced at 550 nm?

To answer this question, we have to look at the spectral characteristics of the solar spectrum and spectral response of the human eye. For photovoltaic and smart windows applications, the solar spectrum should ideally transmit all wavelengths without any losses. The shape of the solar terrestrial spectrum, which is a result of filtering through the earth’s atmosphere, is shown in Fig. 4.3. An old definition of the air mass 1.5 terrestrial solar emissivity standard describes the solar spectral irradiance observed on a receiving surface at an inclined plane at 37° tilt towards the equator, facing the sun with a surface normal pointing to the sun at the elevation of 41.81° above the horizon. The old ASTM Standard (ASTM G159-98) was corrected in 2005 for deep UV (down to 280 nm, rather than 305 nm), also improving spectral resolution (a new standard uses 2002 wavelength compared to only 120 in the old version). This new standard resulted in constant intervals, better defined extraterrestrial spectrum and other factors while maintaining integrated irradiances standards for hemispherical and direct normal on 37° tilted surface at 1,000.4 and 900.1 W m⁻², respectively [32].

Electro-optical applications of TCCs in displays have different spectral requirements, which also depend on the spectral response of the human eye. With two basic retinal receptors: 2-μm diameter cones (concentrated in the center of the retina) and rods, our eye uses the first to sense in bright-light conditions and the second for low-intensity light conditions. The eye spectral response is directly related and influenced by the illuminescence level (light intensity) to which it is exposed. For the condition of illuminance levels (10⁻²–10⁸ Cd m⁻²) correlated to indoor-sunny day illuminance condition, the human vision depends mostly on the cones spectral response, which is described by the photopic vision mode for bright light (>1 Cd m⁻²) and mesopic vision mode for dimmer light conditions. The best color and visual acuity corresponds to illumination levels of 10⁴ Cd m⁻² for the photopic vision mode and cones sensing. The maximum of the photopic spectral responsive curve occurs at 555 nm, as shown in Fig. 4.3, for high light intensity levels. The FOM for TCCs is therefore based on transmittance/absorbance of the material at 550 nm. One should keep in mind that the spectral sensitivity shifts from ~400–730 nm to ~370–650 nm when transitioned to a low level of luminance, with corresponding transitioning of the maximum spectral response to 507 nm [34].

4.5 Development of the Figure of Merit for Transparent Conductive Electrodes

The development of the FOM is an important part of any technology, including TCC. However, in most cases, the FOM is very specific to an application, and often times it is expressed as a range of suitable properties rather than as a single number. Application-specific requirements may include environmental, mechanical stability (bending and stretching). The subject of carbon nanotube based TCC reduces FOM to the electro-optical properties of the nanomaterials. However, given the significant interest in the field of flexible electronics, we will bring an example of integration of mechanical stability into the FOM of TCCs.

Figure 4.4 shows resistance–transmittance characteristics for a series of common transparent conductive coatings, which are currently used along with ITO.

Fig. 4.4

Resistance–transmittance characteristics of some transparent conductive coatings, data adapted from [37]. The materials are produced by spray pyrolysis (SnO₂:F), sputtering (In₂O₃:Sn), chemical vapor deposition (ZnO:Ga), or by pulsed laser deposition (Cd₂SnO₄). A group of materials approaching ITO 90 % transmittance and 10 Ω/sq. sheet resistance is circled. The requirements for particular display applications of transparent coatings are shown in blocks. The toughest requirements are for flat liquid crystal display (LCD) applications, where the transmittance should exceed 87 % and the sheet resistance should be about 30 Ω/sq

In the 1970s, Haacke developed the definition of the FOM for TCCs [35, 36].

The sheet resistance, R _s, can be expressed in terms of electrical conductivity σ (Ω⁻¹ cm⁻¹) and d is the coating thickness (cm),

$$ {R_{\rm{s}}} = \frac{1}{{\sigma d}}. $$

(4.1)

The optical transmittance, T, of a thin film is given by the ratio of the radiation passed through the thin film (I) to the radiation entering the film (I ₀) or it can also be given in terms of the absorption coefficient α (cm⁻¹) and the film thickness (d):

$$ T = \frac{I}{{{I_0}}} = \exp ( - \alpha d). $$

(4.2)

The definition of the first FOM, is a ratio of the sample transmittance to the sheet resistance, ${\text{FO}}{{\text{M}}_1} = \frac{T}{{{R_{\rm{s}}}}}$, which can be rewritten in terms of the electrical conductivity, absorbance, and TCC thickness by substituting the T and R _s values from Eqs. (4.1) and (4.2):

$$ {\text{FO}}{{\text{M}}_1} = \sigma d\exp ( - \alpha d). $$

(4.3)

Equation (4.3) expresses the FOM for a coating with conductivity σ, absorbance α, and thickness d. The maximum value of FOM according to the definition can be found taking the first derivative of (4.3), equating it to zero, and solving it for minimum thickness d _min = 1/α:

$$ \frac{{\partial {\text{FO}}{{\text{M}}_1}}}{{\partial d}} = \frac{{\sigma \exp t( \propto d) - \frac{\sigma }{\alpha }\exp t(\alpha d)}}{{\exp t(2\alpha d)}} = 0. $$

(4.4)

And the transmittance at maximum FOM is then found to be, T = 1/e = 0.37. This exercise [Eq. (4.4)] demonstrates that the original approach is not adequate to define the FOM, because the best coating would transmit less than 40% of the incident light.

To circumvent this problem, Haacke proposed another definition of FOM for transparent coatings:

$$ {\text{FO}}{{\text{M}}_2} = \frac{{{T^x}}}{{{R_{\rm{s}}}}},\quad {\text{where }}x > 1. $$

(4.5)

By solving the equation $\frac{{\partial {\text{FO}}{{\text{M}}_1}}}{{\partial d}} = 0$ for the minimum thickness required to achieve the maximum value of FOM₂, d _min = 1/xα. Through the modeling of different values of x, Haacke selected x = 10 as the value leading to T = 90% [35]. One should notice that x = 20 or 100 would lead to transparency of thin coatings of T = 95% and 99%, respectively, which are not practical. Thus, the definition of FOM₂ was written for T = 90% as:

$$ {\text{FO}}{{\text{M}}_2} = \frac{{{T^{{10}}}}}{{{R_{\rm{s}}}}} = \sigma d\exp ( - 10\alpha d), $$

(4.6)

where the corresponding minimum thickness of the coating, ${d_{{{ \min }}}} = \frac{1}{{10\alpha }}$.

The right part of Eq. (4.6) was developed for TCCs with no reflection loss. If reflectivity (R) is nonzero, Eq. (4.6) changes to:

$$ {\text{FO}}{{\text{M}}_2} = \sigma d\left\{ {\left( {1 - {R^2}} \right){{\left[ {exp{{(\alpha d)}^{{ - 1}}}} \right]}^{{10}}}} \right\}. $$

(4.7)

Equation (4.7) is a more general expression for FOM₂ because it is valid for both R > 0 and R = 0. This definition of FOM is still active. Different TCCs can be compared by their conductivity/resistance on a condition that the coatings have same transmittance (90%) at 550 nm.

Haacke noticed that between metal and semiconductor materials, the highest FOM was obtained for semiconducting coatings [35]. This was explained through the following analysis.

The ratio of electrical conductivity to absorbance in terms of the velocity of light (c), the index of refraction (n), the light frequency (ν), and the relaxation time of charge carriers (τ) is given by Eq. (4.8) left part. Further, by replacing carrier lifetime τ = l/ν is by the free mean path (l), it becomes apparent that the highest ratio will be obtained for the material with the largest carrier mean free path [right part of Eq. (4.8)]:

$$ {\text{FO}}{{\text{M}}_3} = \frac{\sigma }{\alpha } = \pi cn{v^2}{\tau^2} = \pi cn{l^2}. $$

(4.8)

The carrier mean free path for semiconductors is on the order of 10⁻⁶ cm or higher compared to that of metals. Moreover, metals absorb strongly in the visible part of the spectrum, and at a thickness of a film less than 1 μm, the diffuse scattering of the charge carriers at the surface of the TCC is high, thus, significantly reducing the mean free path. Rewriting Eq. (4.8) for semiconductors, gives:

$$ \frac{\sigma }{\alpha } = \frac{{\pi cn{v^2}}}{e}{\mu^2}m_{\rm{eff}}^2\sim A{({m_{\rm{eff}}})^{{ - x}}}{({m_{\rm{eff}}})^2} = {\text{const}}\left( {{m_{\rm{eff}}}^{{0.64(75)}}} \right), $$

(4.9)

where charge mobility μ is replaced by μ = eτ/m _eff ~ (m _eff)^−x where e and m _eff are the electron charge and its effective mass and x = 1.35 for many materials. Equation (4.9) gives a recipe for semiconducting electrodes with high FOM i.e., those materials with high mobility or low effective mass.

Gordon continued the approach of using σ/α as the FOM and incorporated the theory of electrons in metals, which enables estimation of the theoretical upper limit of the FOM [37].

$$ {\text{FO}}{{\text{M}}_4} = \frac{\sigma }{\alpha } = - {\left\{ {{R_{\rm{s}}}\ln (T + R)} \right\}^{{ - 1}}} = 4{\pi^2}{\varepsilon_0}{c^3}n{\lambda^{{ - 2}}}{{\text{e}}^{{ - 2}}}{({m_{\rm{eff}}}\mu )^2}, $$

(4.10)

where ${\varepsilon_0}$ is the permittivity of free space and λ is the visible wavelength of light. The highest value of FOM₄ is expected for materials that demonstrate a high value of the product of the effective electron mass and charge mobility.

The electron-scattering processes significantly reduce the electron mobility. These processes include scattering electrons by phonons (the dominant process in single crystals) and by grain boundaries (dominant in polycrystals). In the case of doped semiconducting TCCs, scattering by ionized dopants may dominate, limiting the charge carrier mobility.

George Grüner’s group suggested [3] that the correlation between transmission (in the visible spectrum) and the sheet resistance R _s should follow a metallic skin-effect model for electromagnetic waves in thin metal films by replacing the transmittance with an expression for thin metal films, which was proposed earlier by Tinkham [38]. Assuming that the TCC film thickness is smaller than the wavelength of light and neglecting the imaginary part of the conductivity, the transmittance in the visible part of the spectrum can be written as:

$$ T = \frac{1}{{{{\left( {1 + \frac{{2\pi }}{c}{\sigma_{\rm{opt}}}d} \right)}^2} + {{\left( {\frac{{2\pi }}{c}{\sigma_{\rm{dc}}}d} \right)}^2}}} = \frac{1}{{{{\left( {1 + \frac{{2\pi }}{{c{R_{\rm{s}}}}}\frac{{{\sigma_{\rm{opt}}}}}{{{\sigma_{\rm{dc}}}}}} \right)}^2}}}, $$

(4.11)

where ${\sigma_{\rm{dc}}},{\sigma_{\rm{opt}}}$ are the DC and the optical conductivity of a film with thickness d. The ratio of DC to optical conductivities was assumed to be constant for different film thickness and equal to one for transmittance measured at 550 nm [3]. For a broad frequency range (DC to optical frequency), the ratio of $\frac{{{\sigma_{\rm{opt}}}}}{{{\sigma_{\rm{dc}}}}}$ was found to be around 3 [4].

The ratio of DC to optical conductivity, another expression of FOM for TCCs, can be written as:

$$ {\text{FO}}{{\text{M}}_5} = \frac{{{\sigma_{\rm{dc}}}}}{{{\sigma_{\rm{opt}}}}} = \frac{{2\pi }}{{c{R_{\rm{s}}}\left( {\frac{{\sqrt {T} }}{{1 - \sqrt {T} }}} \right)}}. $$

(4.12)

Thus, the sheet resistance of a thin film is expected to depend on its transparency as:

$$ {R_{\rm{s}}} = \frac{{\frac{c}{{2\pi \left( {\frac{{1 - \sqrt {T} }}{{\sqrt {T} }}} \right){\sigma_{\rm{opt}}}}}}}{{{\sigma_{\rm{dc}}}}}, $$

(4.13)

and a plot of the sheet resistance as a function of $\left( {\frac{{1 - \sqrt {T} }}{{\sqrt {T} }}} \right)$ is expected to be linear with a slope of $\frac{c}{{2\pi }}\frac{{{\sigma_{\rm{opt}}}}}{{{\sigma_{\rm{dc}}}}}$.

All FOMs described so far are applicable only to the low frequency region, where conductivity is not complex or frequency dependent. At higher frequencies, simple relationships between conductivity and transmittance are no longer valid and have to be defined by either using the Airy formula or the Kramers–Kronig transformation. Using this approach, a new FOM has been proposed by Kamaras [5]. This FOM relies on a wavelength-dependent general FOM for thick and thin nanotube TCCs. The major argument is that this model takes into account that the optical properties in the visible spectral range depend on the concentration of bound charge carriers, while DC resistivity depends on the free carriers. Applying Kramer–Kronig transformations for free standing laser ablation nanotubes, they demonstrated that due to metallic and semiconducting nanotubes contributing to absorbance at 550 nm, the optical conductivity derived from Grüner’s approach [3] overestimates it by a factor of 2. They also showed that the optical conductivity is wavelength (frequency) dependent. Therefore, the ratio of $\frac{{{\sigma_{\rm{opt}}}}}{{{\sigma_{\rm{dc}}}}}$ and Eq. (4.11) should be frequency dependent:

$$ T(w) = \frac{1}{{{{\left( {1 + \frac{{2\pi }}{{c{R_{\rm{s}}}}}\frac{{\sigma {{(w)}_{\rm{opt}}}}}{{{\sigma_{\rm{dc}}}}}} \right)}^2}}}, $$

(4.14)

where w is the wavenumber in cm⁻¹.

The authors found that the optical properties of nanotubes could be fitted by the Drude–Lorentz model, where the Drude part models the contribution of the metallic nanotubes (free charge carriers). Based on the proposed model and deconvolution of the transmittance spectra, the transmittance in Eq. (4.2) describes the far-infrared free charge carrier for w > 2,000 cm⁻¹ (below 5,000 nm), and the optical conductivity of these charge carriers is zero.

The following assumptions are introduced: the mass of electrons is the same as the mass of charge carriers, the DC conductivity can be obtained by the zero limit of the optical conductivity, the frequency w is less than the relaxation rate of free charge carriers, and that in the frequency range of the Lorentzian contribution, the reflectance of the film is negligible, and the absorbance obeys Beer’s law. Although no assumption is made about interrelation of the number of bound and free charge carriers, one can assume that they should be proportional to the number of metallic N ₁ and semiconducting N ₂ nanotubes, and for laser-grown carbon nanotubes, the ratio of N ₁/N ₂ is expected to be around 1/3. The optical conductivity, described by the Drude–Lorentz model, can be written as:

$$ \sigma {(w)_{\rm{opt}}} = \frac{{{N_1}{e^2}}}{{mV}}\frac{{{\gamma_1}}}{{({w^2} + \gamma_1^2)}} + \frac{{{N_2}{e^2}}}{{mV}}\frac{{\gamma_2^2}}{{{{(w_0^2 - {w^2})}^2} + \gamma_2^2{w^2}}}, $$

(4.15)

where V is the volume of the system, N ₁ , m, and γ ₁ are the number of free charge carriers, the mass of the charge carriers, and the width of the free carrier conductivity (the relaxation rate), respectively. The same quantities labeled with subscript 2 correspond to bound charge carriers. In the zero limit, and for frequencies w < γ ₁, Eq. (4.15) expresses the sheet conductance of nanotubes, σ _s:

$$ \sigma_{\rm{s}} = \frac{{{N_1}{e^2}}}{{mV{\gamma_1}}}d. $$

(4.16)

Beer’s law for Lorentzian contribution can be expressed in terms of the frequency-dependent extinction coefficient α(w), as follows:

$$ - \log (T(w)) = \frac{{a(w){N_2}}}{V}d. $$

(4.17)

Combining this equation and Eq. (4.16), we see that the value of σ _s depends linearly on −log(T(w)) with a slope, which expresses the ratio between the number of free charge carriers responsible for conductance to the number of bound charge carriers responsible for high frequency absorbance. This slope is used by Kamaras as the “ideal” FOM expression.

$$ {\text{FOM}}_6^{\rm{ideal}}{\text{FO}}{{\text{M}}_6} = \frac{{\sigma_{\rm{s}}}}{{ - \log (T(w))}} = \frac{{{N_{{1}}}{e^2}}}{{{N_2} \in (w)m{\gamma_1}}} $$

(4.18)

Although the expression of FOM for the nonideal case, where DC conductivity is determined by the contacts (this does not meet the assumption of the zero frequency for optical conductance), was also derived, we omit it here because information about the charge carrier concentration cannot be derived directly from the experimental data and the description is beyond the scope of this manuscript.

The FOM₆ was derived to reflect the fact that experimental data do not go through the origin, but they intercept the absorbance axes at zero conductivity. This value of transmittance, T _p(w) is related to the percolation threshold condition and is frequency dependent and has also been observed by Hu and Gruner [3]. Applying this correction to the expression for FOM₆ gives

$$ {\text{FO}}{{\text{M}}_6} = \frac{{\sigma_{\rm{s}}}}{{ - {{{\log T(w)}} \left/ {{{T_{\rm{p}}}(w)}} \right.}}}. $$

(4.19)

Based on this definition of FOM₆, unsorted CoMoCat¹ nanotubes [two grades commercial and enriched with (6, 5) and (7, 5) tubes], ark, and HiPCO² single-walled carbon tubes TCCs were compared (Fig. 6 in [5]). The comparison was conducted for 550 and 1,600 nm wavelengths (see Table 1 in [5]). The nanotubes formed the following series HiPCO, CoMoCat (commercial), arc, CoMoCat [enriched with (6, 5) and (7, 5) tubes], with HiPCO tubes showing the highest FOM₆. The numerical values of the best and worst FOM₆ for 550 nm are 0.0068 and 0.0013, respectively.

Analysis of FOM for HiPco, laser ablation-grown, and arc discharge SWNTs using optical conductivity measured across the UV-NIR spectrum gave a very different result. The sorted and laser-produced nanotubes showed the highest FOM₅: sorted and laser ablation-grown > arc discharge > HiPco SWNTs [17]. The observed difference may be related to a variety of factors, including different lengths of nanotubes and bundle-to-bundle resistance.

Flexible electronics require not only transparency and low resistivity but also flexibility of the electrode material. All the FOMs described above do not consider flexibility, focusing only on the transparency and the conductivity aspect. Kotov’s group proposed an idea for further modification of the FOM to reflect the mechanical stability of flexible electrodes [39]. This particular FOM may be beneficial for touch screen displays, wearable sensors, and many others.

$$ {\text{FO}}{{\text{M}}_7} = \frac{{\sigma {\varepsilon_{\rm{c}}}}}{\alpha },{\text{ oh}}{{\text{m}}^{{ - 1}}}, $$

(4.20)

where ε _с is the critical strain before critical failure, which can cause a device to malfunction. The strain of a component layer in a film can be estimated as a ratio of the distance from the neutral axis layer and the radius of the bending curvature. The value of ε _с depends on the thickness of the substrate, the buffer layer adhesion of a coating, and many other factors. To measure the ε _с, one could find the point of inflection on a strain–resistance curve or measure the onset of crack formation on the film’s surface. It is interesting to note that the value of ε _с (for LbL SWNT films) was 99 and 120% before and after super acid doping, while the critical strain for ITO coating on PET was 1% for tension and 1.7% for compression. Thus, the SWNT film demonstrates 100 times improvement in bending properties as compared to ITO, if the coatings are compared using the critical strain values. However, the difference is only a factor of 2 if the FOM₆ for these coatings are compared. The FOM₇ = 0.15 Ω⁻¹ for the layer-by-layer assembled SWNT film, while the FOM₇ (ITO) = 0.07 Ω⁻¹. The value of the critical strain before critical failure in (4.20) could be replaced by other mechanical property, which is more critical for the particular TCC application (for instance critical tensile strength, toughness, Young’s modulus can be used). One should probably consider a series of experiments, where both ITO and competing nanomaterial coatings are deposited on exactly the same substrate, using the same fabrication method and tested under similar conditions. Table 4.1 summarizes the definitions of FOMs described above and what conditions they apply under.

Table 4.1 The expressions of the figure of merit for transparent conductive coatings

Figure 4.5 summarizes most of the known reports on nanotube-based TCCs, where a series of coatings were tested to demonstrate a percolation behavior for nanotube networks. Showing the coordinates of sheet resistance and transmittance at 550 nm, the plotted results allow the comparison of the technological requirements for various display applications with the electro-optical properties of the nanotube coatings. At this time, the quality and the FOM of these coatings matches the requirements for resistive, capacitive, and LCD displays. The best coatings, composed of doped carbon nanotubes, show FOM values approaching those of ITO [6, 51]. However, the stability of doping or the possibility of a dedoping during the device operation should be tested before making a definitive conclusion. This suggests that the definition of the FOM can be further developed to include other parameters related to the performance of the coatings for each particular application.

Fig. 4.5

A summary of the electro-optical properties of carbon nanotube-based (SWNT, DWNT, and MWNTs) transparent conductive coatings reported in the literature. The grayed areas show possible applications including resistive and capacitive touch screen, liquid crystal, and flat panel displays. More details on the requirements for each particular display application are shown in Fig. 4.4. Carbon nanotube coatings already meet the requirement for capacitive and resistive touch screen and LC displays. Some exceptional coatings (doped SWNT) showed electro-optical properties approaching those of ITO [6, 51]. In most cases, the error bars for these values are not reported, but they could be very large

4.6 Resistance Measurements

The accurate assessment of sheet resistance is of fundamental importance for characterization of conductive films. Sheet resistance (Ω/sq.) is the two-dimensional equivalent of 1D linear resistivity (Ω/cm) and 3D bulk resistivity (Ω cm). The resistivity of a wire can be calculated by dividing its resistance by the wire length. The sheet resistivity is a property of a specific film, and it is independent of its geometry. 2D resistivity in applications for nanotube films is related to the in-plane electrical property of the nanotubes, where they are randomly oriented (isotropically). The dimension-independent nature of sheet resistance can be understood using the following example. The resistance of a rectangle with an increasing sample dimension perpendicular to the electrode will increase resistance proportionally. By increasing the sample size along the electrode length, resistance will decrease proportionally. An equal increase of sample size in all directions will therefore have no effect on the value of measured resistance. Thus, the sheet resistance of a square is independent of the square size. A measurement of sheet resistance is also a measurement of sheet resistivity. The units for sheet resistance are ohm per square (Ω/sq.).

There are two options to consider for sample design. The first option is to create a square sample of nanotube film and deposit electrodes on opposite sides of the square. The measured value is the sheet resistance of the nanotube film. Electrodes could be deposited in a multilayer structure composed of 10 A of Ti and 500 A of Au. Ti should be used to improve the contact resistance. One should consider that this is an irreversible procedure, rendering posttreatment of the films to be impossible. The second option involves using spring-loaded point probes placed in contact with the film surface for electrical measurements. The advantage of this procedure is that it does not require any irreversible sample modification.

The electrical properties of the nanotube membranes can be measured using direct and alternating current (DC and AC) techniques, which together provide complementary information and a cross-checking of the accuracy of the measurements.

A source-meter (for instance Keithley 6430) can be used to automatically choose progressively smaller currents for increasingly higher resistances, so that the current will not heat the sample during the measurement process. We estimate that the potential error due to ohmic heating may be in the range of a few degrees, if the minimum current condition is not chosen. We usually use the four-probe technique [40] to estimate the sheet resistance of our samples. The four-probe technique works by contacting four equally spaced point electrodes with the surface sample. The current is sourced between the outer two electrodes, and the voltage is measured across the inner two electrodes. Any contact resistance between the probe tip and the sample is eliminated by nature of the four-probe system; however, an additional correction factor is needed to arrive at the sheet resistance from the measured resistance in a four-probe technique. This factor depends on sample geometry, but values for many different geometries are very well tabulated [41–43].

The elimination of contact resistance in a four-probe setup is easily seen by viewing the system as a series of floating resistances between the current source and sink. In Fig. 4.6, the mutual resistance of point 2 with electrodes 1 and 4 is given by ${R_{\rm{M}}} = {R_{{21}}} - {R_{{24}}}$. For both points 2 and 3, the mutual resistance is then ${R_{\rm{M}}} = {R_{{21}}} - {R_{{24}}} - {R_{{31}}} + {R_{{34}}}$. The actual measurement of each resistance would include not only the resistance of the material itself but also the contact resistance as well: ${R_{\rm{ab}}} = {R_{\rm{sample}}} + {R_{\rm{contact}}}$. If the resistance of each contact is the same, the contact resistance will cancel out of the above equation, leaving only the sample resistances.

Fig. 4.6

Schematic of a four-probe setup. The probes are separated by equal distances, labeled S. The current is sourced between probes 1 and 4, and the voltage is measured between probes 2 and 3

AC impedance measurements can be conducted using a four- or a two-probe configuration. In our measurements, we usually use the two-probe configuration. At a sufficiently low frequency, AC measurements are expected to agree with the DC measurements, and a relation between two- and four-probe measurements is therefore necessary. Two-probe AC and four-probe AC measurements can be related to one another in two different ways.

The floating potential V at any point away from a current point source on a conductive sheet is given by Uhlir [41]:

$$ {V_{{{2}{\rm{probe}}}}} = \frac{{\rho I}}{{2\pi }}\ln r, $$

(4.21)

where ρ is the sheet resistivity and r is the distance from the current source.

The sheet resistance of a sample using two point electrodes spaced over distance s, incorporating Ohm’s law, is given by:

$$ R{(s)_{{2{\rm{probe}}}}} = \frac{\rho }{{2\pi }}\ln s $$

(4.22)

One can note that subtracting R(2s) from R(s) gives the relation:

$$ R{\left( {2s} \right)_{{2{\rm{probe}}}}} - R{(s)_{{2{\rm{probe}}}}} = \frac{\rho }{{2\pi }}\ln 2 $$

(4.23)

This is half the four-probe resistance, as given by Smits [42]:

$$ R{(s)_{{4{\rm{probe}}}}} = \frac{\rho }{\pi }\ln 2 $$

(4.24)

Therefore, one can measure the two-probe resistance at 1s spacing and then again at 2s spacing and readily arrive at the measured four-probe resistance. The advantage of this method is that, like a four-probe measurement, the contact resistance is built into the calculations such that it is automatically eliminated upon subtraction. The disadvantage is, of course, that additional measurements are required.

The second method consists of using a conversion factor between the two- and four-probe measurements. It can be shown [40–42] that the floating potentials for a four-probe and two-probe configuration are given by

$$ {V_{{4{\rm{probe}}}}} = \frac{{\rho I}}{{2\pi }}\ln \frac{{{r_2}}}{{{r_1}}};\quad {V_{{2{\rm{probe}}}}} = \frac{{\rho I}}{{2\pi }}\ln r, $$

(4.25)

where r _i is the distance from the current source for which the potential is measured. Assuming the two configurations to differ by some multiplicative expression A, one obtains ${V_{{4{\rm{probe}}}}} = A{V_{{2{\rm{probe}}}}};\quad \frac{{\rho I}}{{2\pi }}\ln 2 = A\frac{{\rho I}}{{2\pi }}\ln r$, where we have taken r ₂ = 2r ₁, which is the condition for a standard four-probe configuration with the probes equally spaced. We see that for constant tip spacing, s, the factor of conversion, A, between the four-probe and two-probe measurements is constant and independent of source current or sample resistivity.

$$ A = \frac{{\ln 2}}{{\ln s}}. $$

(4.26)

By calculating this constant for the four-probe setup used, it is possible to compare two- and four-probe measurements for all of the samples. While this calculation does not mathematically eliminate the contact resistance, a constant contact resistance would appear into an experimentally determined conversion constant A, still allowing an experimental comparison between measurements.

For the tip spacing we used, s = 15.9, measured in units of tip radii [44, 45], since the tip-to-tip spacing was 1.59 mm and the tip radius was estimated as 0.1 mm. Therefore we expect that A = 0.25.

Experimentally, it was found that the four-probe resistance was 0.22 times the two-probe resistance, for a constant distance between probes (Fig. 4.7). This is very close to the expected value of 0.25.

Fig. 4.7

Four-probe resistance of a series of SDS-dispersed (solid circles) and Triton X-100 dispersed (open circles) SWNT films as a function of two-probe resistance, as measured by low frequency AC measurements. These two values are linearly related by a slope of 0.22

If the contact resistance was a contributor to the two-probe resistance, the experimental value of A would be expected to be smaller than the calculated value, as was observed.

This “contact resistance” is often interpreted as the probe tip being in less than complete contact with the surface and is perhaps better referred to as a “constriction resistance” [44]. Neither the film nor the probe tip is uniformly smooth, so only a certain percentage of the surface area will be in contact. The resistance arises from the current being constricted to certain areas of the probe–film contact, rather than from the tip–film interface itself. Using the experimentally determined value of A = 0.22 rather than A = 0.25, and substituting it in (4.26) gives an effective tip radius of 0.07 mm rather than 0.1 mm, which implies that only 50% of the tip is in contact with the film.

The comparison between two- and four-probe measurements can then be done by simple multiplication, since this factor of 0.22 was found to be universal among all nanotube films that we measured. Furthermore, the effective tip radius should be used instead of the original tip radius when determining film properties from spreading resistance calculations.

4.7 Impedance Spectroscopy

AC measurements on nanotube networks can shed some light on the electrical transport of individual components of the network, something that would be difficult to obtain from other measurements. By viewing each part of the network as an idealized circuitry component, the contribution of network features to the overall electrical properties can be determined from AC impedance.

Impedance spectroscopy operates by sourcing an alternating potential in the form:

$$ V(t) = {V_0}{{\text{e}}^{{\left( {{\rm{i}}\omega t} \right)}}}. $$

(4.27)

The response signal is a current, shifted in phase from the potential signal:

$$ I(t) = {I_0}{{\text{e}}^{{{\rm{i}}\left( {\omega t - \varphi } \right)}}}. $$

(4.28)

The impedance of the nanotube networks can be expressed in the complex polar form of the impedance magnitude Z ₀ and phase angle ϕ as follows:

$$ Z(t) = \frac{{{V_0}{{\text{e}}^{{{\rm{i}}\left( {\omega t} \right)}}}}}{{{I_0}{{\text{e}}^{{{\rm{i}}\left( {\omega t - \varphi } \right)}}}}} = {Z_0}{{\text{e}}^{{{\rm{i}}\varphi }}} = {Z_0}\cos \varphi + {\text{i}}{Z_0}\sin \varphi. $$

(4.29)

For purely resistive (R) behavior, ϕ = 0, and the impedance consists entirely of a real term, Z′, which is frequency independent. When an inductive and/or capacitive element is introduced, a nonzero value of the phase shift ϕ ≠ 0 gives rise to the imaginary component of impedance, Z″. At sufficiently low frequency, ω, the imaginary part of the impedance approaches zero, and the real part of the complex impedance, Z′, is equivalent to the DC resistance of the system. The direction of the phase shift (the sign of ϕ) depends on whether the circuit is dominated by inductance or capacitance. A capacitive (C) circuit has negative phase angle, whereas an inductive (L) circuit has positive phase angle [46].

Impedance spectroscopy, along with knowledge of RCL circuitry, therefore allows us to determine which combination of R, L, and C appears in a given network. We know that a capacitor has impedance $Z = 1{\text{/i}}\omega C$, while an inductor has impedance of $Z = {\text{i}}\omega L$. The carbon nanotube networks of low density behave as capacitive circuits.

From the superposition principle of Ohm’s law, a complex circuit can be reduced to a simpler set of equivalent circuits. Impedance data can be further modeled to deconvolute the circuit elements that the sample consists of.

Modeling of impedance data is done by analysis of the complex plot of −Z″ vs. Z′, commonly called a Nyquist plot or a Cole–Cole plot. A simple RC circuit in parallel will result in a semicircular Nyquist plot on the complex impedance plane (Fig. 4.8) [46].

Fig. 4.8

Representative impedance of an SWNT film. |Z| and phase may be shown as a function of frequency, or broken into its real and imaginary components

From the Cole–Cole plot, the value of the circuit resistance can be obtained from the right-most value, corresponding to ω = 0. The highest point in the arc, the Z″ maximum is the circuit’s critical frequency. This critical frequency is evident as the sharp falloff of |Z| in the impedance vs. frequency plot. The capacitance can then be calculated as $C = 1{/}{\omega_{\rm{c}}}R$, the well-known critical frequency relationship for parallel RC circuits. This measure of capacitance is based on the assumption that the circuit being measured behaves as a simple parallel RC circuit, and the impedance arc is therefore truly semicircular. However, nanotube networks consist of multiple superimposed arcs [29] and require advanced fitting techniques to determine the proper equivalent circuit to completely describe the electrical response of the sample.

It should be mentioned that multiple equivalent circuits can have the same impedance, yet consist of different elements as shown in Fig. 4.9 [47]. A more extensive analysis, which is beyond the scope of this chapter, is necessary to explain the differences between them (Fig. 4.9).

Fig. 4.9

Three different equivalent circuits, which have the same impedance at all frequencies [47]

To model experimental impedance data, one could use a variety of commercially available programs, including the EIS Spectrum Analyser [48] or others. An educated guess at an equivalent circuit should always be made based on the expected structure of the sample, so that the equivalent circuit model proposed will have physical meaning.

4.8 Two-Probe Impedance Results

The impedance response from low density SWNT films are capacitive rather than inductive in nature, as the impedance trends towards zero at high frequency, a characteristic of RC circuits. It was found that impedance spectra from SWNT films were best modeled by two Voigt elements in series [29]. A Voigt element is a capacitor and resistor in parallel (see Fig. 4.10). This model makes physical sense, as not only every nanotube but also every junction or defect serves as a resistor, and each will have its own capacitance. It is also the model favored for impedance of bulk polycrystalline systems, with one element being interpreted as the grain boundary and the other as the bulk material [49, 54].

Fig. 4.10

The double-Voigt-element structure fits low density SWNT network impedance well, with one element interpreted as interbundle junctions and the other as the SWNT bundles themselves [29]

The values of R and C for these two parallel RC circuits in series were found for films of different thicknesses. The Voigt elements with the lowest R value and highest R values were plotted in two different curves, as it was assumed that the nanotube bundles and the junctions had similar responses in each film (Fig. 4.11). The corresponding values for capacitance were found to be symmetric, with the average capacitance being a constant (about 0.48 nF) for all loadings well above the percolation threshold for films prepared using Triton X-100 and SDS. If the aforementioned $C = 1{/}{\omega_{\rm{c}}}R$ relationship were to hold true, this constant network capacitance would mean that the critical frequency and low-frequency conductivity (or four-probe DC conductivity) are linearly related, which we have found to be true.

Fig. 4.11

(a) The results of modeling the impedance of SDS-dispersed SWNT networks with a double-Voigt-element circuit [29]. The R and C of nanotube junctions are shown in black and RC of nanotube bundles are shown in red. (b) Values of critical frequencies for each Voigt element of SDS-dispersed SWNT films [30]

A similar dependence on film density (loading) has been noted for the critical frequency of each circuit element in our model [30]. The critical frequency of both Voigt elements increases with loading. For SDS films, each f _c increases at different rates (Fig. 4.11), while for Triton X-100 films, each f _c increases at the same rate (not shown).

The model results indicate that independent of thickness of nanotube networks (loading of nanotubes), the resistance of bundle–bundle junctions is about 3.3 ± 0.3 times higher than the resistance of the bundle, independent of loading, SWNT purity, or dispersant used [29]. This fairly constant value of junction resistance to bundle resistance is expected from the theoretical consideration that the number of junctions increases linearly with the number of bundles [16–18].

4.9 Macroscale Approach to Evaluation of Resistance of Junctions and Bundles in SWNT Percolation Networks by Impedance Spectroscopy

The percolation behavior of the SWNT nanotube networks was investigated for the series of SWNT membranes with increasing loading of nanotubes. Critical percolation threshold and scaling constants were determined for the four-probe DC resistance and low-frequency AC impedance spectroscopy results of SWNT films.

The results indicate that both the AC and DC measurements follow a percolation scaling law, where conductivity, σ can be written in terms of percolation threshold (p _c), the dimension-dependent critical exponent (β) and the equation prefactor (σ ₀), which depends on the conductivity of a single SWNT bundle and junction as follows:

$$ \sigma = {\sigma_0}{\left( {p - {p_{\rm{c}}}} \right)^{\beta }}. $$

(4.30)

The value of critical exponent can take values of 1.33 and 2.0 for two- and three-dimensional conduction. Taking the log of both parts of the percolation equation, we can find that a plot of log(σ) as a function of log(p − p _c) should result in a straight line. Experimental results for the series of SWNT membranes in these coordinates are shown in Fig. 4.12.

Fig. 4.12

Falloff frequency from real impedance (a), low-frequency admittance (b), and DC conductivity (c), all obey the percolation scaling law, $\sigma = {\sigma_0}{\left( {p - {p_{\rm{c}}}} \right)^{\beta }}$, where σ is conductivity and is proportional to the falloff frequency, p is nanotube loading, with p _c the loading at percolation threshold, and β is the critical exponent. The percolation equation was plotted in a log–log scale. By varying p _c, and choosing the linear fit with the lowest R ² value, the best fit of the data was found. Values of β are obtained from the slope of the best fit

The value of percolation threshold, p _c, was found to be 0.18 ± 0.01 and was dispersant independent for the DC measurements. On the other hand, p _c estimated from the AC measurements was found to be dependent on dispersant and purity. Thus, the lowest value of p _c = 0.11 ± 0.01 was estimated for purified SDS-dispersed SWNTs (see Table 4.2). For the as-grown SWNT films, p _c was found to occur at a loading of 0.25 ± 0.01. The higher value of p _c in the as-grown SWNT can be explained by the contribution of low aspect ratio, carbonaceous impurities. However, purified SWNTs have low amorphous carbon content, and the lower value of p _c for the purified material should have a different cause. One of the potential reasons could be because of the change in the aspect ratio of the SWNTs after purification, which makes them shorter [30]. This is contrary to what should be expected as higher aspect ratio materials normally have lower percolation thresholds. The lower amount of amorphous carbon (with possibly fewer insulating impurities) is a more likely reason for the pc being lower.

Table 4.2 A summary of fit parameters attained for the percolation equation by fitting AC and DC experimental results

In the simplified approach, the SWNT bundles and junctions are assumed to be identical, and the critical exponent is expected to depend on the orientation of the nanotubes only [18]. For random orientation of SWNTs in two dimensions only, the critical exponent takes a value of β = 1.3, for a three-dimensional networks β = 1.9. Previous publications have reported β > 1.3 for nanotube networks [19]. If SWNT bundles and junctions are not identical across the SWNT networks, then β would deviate from the pure instance of 2D conduction [20, 50]. Unlike percolation threshold, the values of the critical exponent, β, were found to range from 1.4 to 2.0 for frequency-dependent and independent measurements, see Table 4.2.

SWNT networks can be described in terms of DC and AC percolation models with the junctions and bundles having a distribution of conductivities, which drives the value of the critical exponent, β, higher than expected for a 2D network. The contributions of junctions and bundles to the macroscopic impedance of a nanotube network can be separated by modeling AC data with a double-Voigt-element model as described earlier. This approach will provide essential information for optimization of nanotube-based transparent conductive films.

4.10 Effect of Doping on the Frequency-Dependent Impedance

For doped films of similar thickness and method of preparation, the relation between the critical frequency and low frequency (DC) conductivity also exhibits a linear dependence. Figure 4.13a shows that the critical frequency depends on the degree of doping and can be determined based on choice of dopant and method of doping. Figure 4.13b on the right displays the effect of doping type on the Nyquist plots of SWNT films of 1.4 μg cm⁻² density. This means that, just like conductivity can be selectively determined by doping, so can f _c. In-situ control of f _c would also be possible by monitoring the DC conductivity, which is experimentally much more easily determined in real time, as a sweep of many frequencies is not necessary.

Fig. 4.13

(a) The dependence of critical frequency on low frequency admittance (DC conductivity). Vapor-doped specimens are shown in red, liquid-doped in black. (b) Nyquist plot of 1.4 μg cm⁻² SWNT films exposed to vapors of dopants, which include orthodichloro benzene, nitric acid, and thionyl chloride [30]

It is notable that the shape of the arc in the Cole–Cole plot does not change upon doping, only the size of the arc changes. This means that the model applied to undoped films is still valid for doped films.

4.11 Effect of Nanotube Length on Transparency and Conductivity of CNT Electrodes

The aspect ratio of sticks comprising a conductive network is inversely proportional to the percolation threshold, the minimum amount of a material required for conduction. The high aspect ratio of carbon nanotubes results in the percolation threshold for these materials at less than 1% by volume or area. In fact, for the series of films described in Fig. 4.11 and Table 4.2, the percolation threshold was 0.18 on average [30].

Most of the SWNT networks are composed of nanotubes with some broad distribution of the aspect ratios, which in most instances, is difficult to characterize. At the same time, most models are done for a fixed aspect ratio, thus making it hard to relate the modeling to the experimental result.

The effect of nanotube aspect ratio on the transparency and conductivity of nanotube networks is also difficult to confirm experimentally as it requires separation of nanotubes by length. Assuming that the diameter of nanotubes for a particular synthesis condition is constant, nanotubes with different aspect ratio can be produced by dispersion of nanotubes through the sonication followed by dense liquid centrifugation [7]. Other techniques have been shown to be effective in fractionalization of nanotubes by length including gel electrophoresis [8] and size exclusion chromatography (SEC). For instance, using gel electrophoresis and column chromatography on cup horn sonicated nanotubes, Strano et al. resolved fractions of nanotubes with average length between 92 and 435 nm [8]. One should be thorough in analyzing the fractions of nanotubes obtained, as instead of fractionalization, chiral separation can be obtained along with length fractionalization [9, 52]. Strano also reported a concomitant enrichment of large diameter nanotubes in fractions with shorter nanotubes.

In another effort, iodixanol (5,5′-[(2-hydroxy-1-3 propanediyl)-bis(acetylamino)] bis [N,N′-bis(2,3 dihydroxylpropyl-2,4,6-triiodo-1,3-benzenecarboxamide)]) has been used to generate various density solutions and achieve length-fractioned samples of SWCNTs (laser, CoMoCat, and HiPco) [7, 10]. To achieve density modification, a surfactant (sodium deoxycholate) solution was mixed with iodixanol and layered in the centrifuge tube. The length-fractioned nanotubes showed no evidence of chirality specific separation as evident from the solution absorbance spectra [7]. Simien et al. used length fractionation to investigate the effect of nanotube length on the optical and electrical properties of SWNT films [11]. The films composed of 130, 210, and 820 nm long and mixed lengths of SWNTs (CoMoCat) were tested. It was found that the absorbance of nanotubes scales linearly with their density and the changes in conductivity can be quantitatively described by the generalized effective medium approximation.

The resulting percolation curves are shown in Fig. 4.14 (summarized from Figs. 4 and 5 in [11]). This graph shows the clear advantage of using length-separated tubes, compared to SWNT of mixed length. A detailed analysis of the properties of SWNT around the percolation threshold indicated that the networks can be described as nearly 2D networks for long SWNTs and as 3D networks for short ones. The authors also demonstrated that experimental results correlate well with percolation theory and the conductivity percolation threshold (p _c) varies with the aspect ratio L as, p _c = 1/L. It is interesting to note that the most drastic effects were observed around the percolation threshold of films made from 0.018 μg cm⁻² (820-nm long tubes). The networks prepared from separated SWNTs show similar conductivities for loading >0.2 μg cm⁻². The distinction between mixed and separated nanotubes disappears for SWNT loading exceeding 1.5 μg cm⁻².

4.12 Effect of Semiconducting and Metallic Nanotubes on the Transparency and Sheet Resistance of TCCs

As produced CNTs contain about 75% semiconducting nanotubes, which do not contribute to the conductivity but reduce the transmittance of the networks. Therefore, one would expect that TCC from the networks of pure metallic nanotubes would demonstrate best FOM. We should remember that a linear dependence of the optical properties with the amount of nanotubes exists only for the absorbance. Thus, one might expect that the network of unsorted nanotubes with sheet resistance of 10 Ω/sq. and 66% of transmittance may be possible to turn into films with 90% transmittance with the same resistance if only metallic nanotubes were used. However, most of the nanotube networks show sheet resistance above 1,000 Ω/sq., which is a factor of 10 larger.

Several approaches have been identified for sorting nanotubes into metallic and semiconducting types for postprocessing, including gel-separation [12], density gradient ultracentrifugation (DGU) [13], dielectrophoresis, chemical selection, electrical breakdown, chromatography, and selective growth; yet, the most popular method for large-scale production of metallic and semiconducting nanotubes is DGU and has been recently realized on the commercial scale by Nanointegris.

Interesting comparisons of all-metallic and unsorted (a mixture of metallic and semiconducting) SWNT were made by the Yang, Blackburn, and Hersam groups [14–16]. They demonstrated that semiconducting nanotubes and unsorted tubes show higher sheet resistances compared to networks composed of pure metallic nanotubes. The network composed of all-metallic nanotubes shows 150 Ω/sq. and 80% transmittance, while similar networks for unsorted nanotubes show 200 kΩ/sq. and 82% transmittance (see Fig. 4.14).

Fig. 4.14

Effect of nanotube length fractionation on the sheet resistance and the transmittance of their conducting networks. Data were extracted from Figs. 4 and 5 in [11]. TCE prepared from separated nanotubes show better quality factor compared to those built from mixed length tubes. The 775-nm wavelength was selected because at 550 nm CoMoCat SWNT exhibit higher absorbance due to the E ^s₁₁ transition

The absolute improvement in sheet resistance is 1,000 times [14]. Laser tubes showed better FOM (lowest R and highest T) compared to HIPCO and arc tubes [14]. The acidic and SOCl₂ doping of semiconducting and metallic networks improves their sheet resistance by a factor of 10 and 4, respectively, with almost no effect on their transmittance [15]. Green and Hersam showed that all metallic nanotubes improve the conductivity of networks by a factor 5.6, with the best network showing ~90% transmittance and sheet resistance of 500 Ω/sq. [17]. According to Miyata et al., metallic nanotube networks show relatively small difference in resistance compared to unsorted nanotubes, and a slightly larger improvement of resistance for unsorted tubes upon doping with sulfuric acid. In both cases, the transmittance of nanotubes was in the 40–48% range and showed 2–5% improvement in doped samples [18].

Tyler et al. showed that the sheet resistance of nanotube electrodes can be tuned by varying the amount of metallic tubes and the doping level (nitric acid or PEDOT:PSS) [16]. They demonstrated that TCCs deposited from 99.9% semiconducting nanotubes could get lower sheet resistance (188 Ω/sq.) compared to those produced from 99.9% metallic nanotubes (411 Ω/sq.) with close transmittance values (88–90%). However, the performance in OPV is the opposite, with metallic CNT TCCs showing higher open-circuit voltage, short-circuit current, and fill factor compared to that for doped semiconducting nanotubes. The short-circuit current showed the most drastic difference, 300 times higher for doped metallic CNT TCE [16].

We have previously noted that nanotube networks could behave differently even if they are assembled from the same type of nanotubes [29, 30]. The differences may come from the purity, the bundle size, and/or the aspect ratio, all of which can result in significant changes in the electro-optical properties. Without identifying all of the important parameters of the networks, it would be hard to make side-by-side comparison with other published results. Thus, absolute changes in the measured sheet resistance of semiconducting nanotubes upon doping may be a result of smaller aspect ratio networks with a larger number of junctions for the metallic tubes rather than nanotube type. This is one of the reasons why it is suggested that instead of reporting an absolute improvement on the sheet resistance or transmittance, a comparison of the appropriate FOM should be used. Figure 4.14 summarizes the different trends just described.

4.13 The Effect of the Number of Walls on the Quality of Carbon Nanotube-Based Transparent Conducting Electrodes

Possible damage (introduction of sidewall defects) of single-wall carbon nanotubes during their dispersion under sonication may reduce the quality of the nanotubes. One solution is to use nanotubes with larger number of walls as material for TCEs. In this case, the outer wall of the nanotube can be sacrificed/damaged to enable dispersion, but the inner wall would stay undamaged for charge carrier transport. A comparison of transparent and conductive electrodes composed of single-wall nanotubes with a diameter of 0.6–0.9 nm, double-wall carbon nanotubes with a diameter of 1.5–2.7 nm, and multiwall nanotubes with diameter of 16–35 nm was reported by Li et al. [19]. The small variation in the diameter of the nanotubes was compensated by length, resulting in relatively similar aspect ratio for the series of samples. For the same transmittance, higher conductivity was measured for SOCl₂-doped SWNTs followed by double-wall nanotubes, MWNTs, and undoped SWNTs [19]. The authors rationalized the results by ascribing them to better conductivity of individual MWNTs as compared to SWNTs. It is interesting to note that the effect of SOCl₂ doping is more pronounced (factor of 10 improvement in conductivity) for thicker coatings, and much smaller for thin coatings. Green and Hersam used GDC to separate DWNTs from a mixture of as produced SWNTs and DWNTs and measured their optical and electrical properties in solution and as coatings [20]. They reported that for the same transmittance, enriched DWNTs doped with SOCl₂ make 42% more conductive networks compared to those of undoped SWNT. Pristine, undoped, DWNTs coatings are a factor of 2.4 better conducting compared to SWNTs [20].

Li et al. reported on TCCs prepared from pristine DWNTs (Xin Nano Materials) [21]. The purification of nanotubes was done by air oxidation, followed by the hydrochloric acid wash to remove catalyst and oxide nanoparticles. Purified DWNTs (p-DWNTs) were obtained by treating purified tubes with a mixture of sulfuric and nitric acids for functionalization (f-DWNT). The electro-optical properties of nanotube films prepared from DWNTs (pristine-, p-, and f-DWNT) are shown in Fig. 4.15. Purification and functionalization of DWNTs led to better quality of material, mostly due to removal of amorphous carbon. The best f-DWNT demonstrated sheet resistance of about 1.1 kΩ/sq. and transmittance of 90% [21]. MWNTs are expected to exhibit higher absorbance for the same density of nanotubes as compared to the SWNT, DWNT, or FWNTs. Earlier reports of similar electro-optical properties of DWNT and MWNT TCCs may stem from some morphological differences (aspect ratio) or possibly a mixed character of CVD-grown DWNT [22]. Considering the much lower cost of MWNTs compared to other nanotubes, the possibility of creating an inexpensive conductive coating prompted extensive research. The electro-optical properties of MWNT TCCs, summarized by Kaempgen [23], showed a percolation-like behavior observed for coatings with the transmittance of 85–90%.

Fig. 4.15

Summary of the effect of nanotube character (semiconducting or metallic) on transmittance and sheet resistance of a nanotube transparent conducting electrodes. Circles—data adapted from [14] show the effect of film thickness (1, 2, and 3 layers) on the R–T parameters of pure (metallic and semiconducting) and mixed films. Data shown in open circles were adapted from the [15]. The effects of doping on sheet resistance shown in triangles were obtained from [16]. Data shown as semi-solid squares were taken from [18]. Broken lines connect data for all metallic nanotube TCEs from the same reference. The TCC with the best properties shown above was prepared from semiconducting nanotube doped with nitric acid [16]

Castro et al. reported on processing 10 nm MWNT (from Nanocyl) into conductive coatings without additional purification [24, 53]. Ko prepared TCCs from 40 nm diameter and 1–2 μm long MWNTs [25]. The TCCs made from the solvent containing drops of MWNT suspension produced a macroscopic structure of overlapping rings with a sheet resistance of 300 Ω/sq. and transmittance at 550 nm of about 80% [25]. The electro-optical properties of TCCs composed from nanotubes with different number of walls are summarized on Fig. 4.15. As anticipated, the tubes with a smaller number of walls exhibit better properties, and highly enriched DWNT take the lead in this group [20].

4.14 The Effect of Metal Nanoparticles on the Quality of Carbon Nanotube-Based Transparent Conducting Electrodes

While doping with metal nanoparticles seemed to be attractive for reduction of tube-to-tube resistance and sidewall defects, one we should be aware that such p-doping may also downshift the SWNT work function up to 0.42 eV by a strong charge transfer from the nanotubes to AuCl₃ [26]. Also, as with any doping, we should consider the stability of the metal nanoparticles, as well as the potential instability of ligand doping [27]. At elevated temperatures, Au nanoparticles have high mobility on carbon due to the low melting point of nanoparticles and the noncovalent character of their coordination. This will lead to growth in nanoparticle size, leading to increase of sheet resistance and reduction in transmittance due to large contribution of light scattering. Anion doping may not be stable either, even for stable ions like Cl⁻ [27]. Park et al. reported on decoration of purified and functionalized bamboo MWNT (Iljin Nanotechnology Inc.) [50]. Functionalization of MWNTs was done by the reaction with 1-butyl-3-methylimidazolium tetrafluoroborate (BMITB) in the solid phase at room temperature. MWNTs were then washed with acetonitrile and deionized water to remove excess of BMITB and dried under vacuum. Au decoration of functionalized MWNTs was achieved through a reaction with HAuCl₄·3H₂O in the solution of Et/H₂O mixed in 1:1 ratio. The mixture was briefly sonicated using 240 W at 20 kHz to initiate Au nanoparticles. The membranes of Au-decorated functionalized MWNT (Au-f-MWNT) were prepared by a filtration method, followed by dissolution of the membrane and deposition of TCC on a substrate [50]. Park et al. report more than twofold lower sheet resistance for Au-f-MWNT (Au nanoparticles with diameters of 10.3 ± 1.5 nm) compared to original MWNTs due to the well-interconnected three-dimensional structure, which incorporates Au nanoparticles. For Au-f-MWNT films, with a transmittance of 88.3%, the sheet resistance was found to be about 42.5 kΩ/sq. (see Fig. 4.16).

Fig. 4.16

Electro-optical properties of double-wall, few-wall, and multiwall nanotubes presented in coordinates of sheet resistance and transmittance. Although not all reports followed properties of nanotube networks through the percolation, a clear tend can be seen. The smaller number of walls, the lower is the position of percolation curve on the plot. The electro-optical properties of ITO are shown with solid star symbol (10 Ω/sq., 90 % T). The best FOM is shown by enriched d-DWNTs [20]

Yang et al. compared acidification of nanotube networks with the effect of Au-decoration of purified DWNT and SWNTs (HiPco and Unidyme) on the electro-optical properties of the TCCs produced from these materials [28]. The nanotubes were first dispersed in a mixture of water and propanol, with the help of Nafion, which stimulates electrostatic and steric stabilization of the nanotubes, leading to a stable dispersion of the nanotubes in the solvent mixture. For Au-decoration, nanotube films were immersed in a 1 mM gold salt solution in HAuCl₄·3H₂O in 50 vol. % for 10 min. For acidification, the nanotube films were soaked in concentrated HNO₃ (60% pure), followed by washing in deionized water. Au decorated DWNT (with a diameter of 0.9–1.2 nm and length of 10–30 μm) showed a better quality than the rest of the nanotubes (1 kΩ/sq. and 90% transmittance). The nanotubes are arranged in the following order based on the quality of the properties of the transparent conductive films: Au-DWNTs > acid-DWNTs> Au-SWNTs > pristine DWNTs > pristine SWNTs, which are summarized in Fig. 4.17.

Li et al. reported that decoration of a few wall carbon nanotubes by Pd also reduces the resistance of nanotube networks [21]. Pd decoration was achieved by reduction of palladate salts on the surface of nanotube networks with hydrogen at 500 °C for 30 min. This process produces homogeneous nanoparticles of Pd, with average diameter of 5 nm. The nanotube ink was formed by dispersion of 30 mg Pd-FWNT in polyvinylpyrrolidone in 120 mL of ethanol, followed by centrifugation and collection of the top layer of the solution. A two order of magnitude reduction in sheet resistance of Pd-FWNT was achieved, with the best TCC demonstrating about 274 Ω/sq. and 81.65% transmittance, as shown in Fig. 4.17.

Fig. 4.17

Effect of metal decoration on the electro-optical properties of carbon nanotube TCCs, as compared to uncoated MWNT, DWNT, and f-DWNT

4.15 Concluding Remarks

This chapter described the methodology for properly assessing the necessary optical and electrical properties needed to use carbon nanotube assemblies as transparent conducting coatings, with some special requirements for some specific applications. The first part of the chapter focused on defining the FOM, a quantity that can be used to determine the best combination of the optical transmission and the sheet conductance necessary to maximize the film’s performance. Various FOMs have been reviewed and used to demonstrate their validity using results available from the vast literature in this field. In the second part, a description of how to characterize the electrical properties of the nanotube thin films was presented. The procedure for how to relate four-probe sheet resistance measurements and two-probe AC measurements was also included. In the last part of the chapter, data showing the effects of varying the nanotube characteristics (SWNT, DWNT, and MWNT) was summarized in several useful sheet resistances versus transmission graphs, where the effects of nanotube length as well as fractions of metallic and semiconducting nanotubes and the effect of doping were considered.