Optimization and simulation of a carbon nanotube arrangement for transparent conductive electrodes with record-high direct current to optical conductive ratios

Shau-Liang Chen; Chen-Chieh Yu; Sih-Wei Chang; Yang-Chun Lee; Hsuen-Li Chen; Hsuen-Li Chen

doi:10.1364/OME.416257

1. Introduction

Transparent and conducting films are important components of various optoelectronic devices. A simple quality factor [the ratio of the direct current (DC) conductivity to the optical conductivity (σ_DC/σ_op)] is commonly used to evaluate the photoelectric properties of transparent conductive electrodes (TCEs) quantitatively [1]. For many industrial applications, a potential TCE will require a transparency (T) of greater than 90%, a sheet resistance (R_s) of less than 100 Ω/sq, and a value of σ_DC/σ_op of at least 35 [2]. Indium tin oxide (ITO) is currently a standard TCE because of its high transparency (T > 90%), low sheet resistance (R_s < 10 Ω/sq), and high value of σ_DC/σ_op (up to 220) [1,2]. Nevertheless, the scarcity of indium and its intrinsic brittleness, coupled with the costly high-temperature processing of ITO, greatly limit its scope in flexible electronics [3,4]. For this reason, there is a search for replacements for ITO to provide superior TCEs; for example, carbon nanotubes (CNT) [5], graphene [6–8], metal nanowire networks [1,9], metal meshes [10], ultrathin metal films [11], and two-dimensional (2D) transition metal carbides and nitrides (MXenes) [2] are potential alternatives having low sheet resistances and high transparencies and flexibilities.

Carbon-based materials are particularly excellent potential alternatives to ITO because of their impressively high optical transparencies and low sheet resistances [5–8]. In addition to superb optical and electrical properties, these carbon-based materials are light in weight, flexible, highly chemically stable, mechanically strong, and relatively inexpensive to manufacture, meeting many of the needs of transparent electrodes for flexible electronics [5–8]. One such carbon-based material, graphene, has attracted much attention for its high transparency and exceptional charge transport properties. An ideal monolayer of graphene has a transmittance of 97.7% in the visible regime; the sheet resistance of undoped graphene is on the order of 1 kΩ/sq [12–14]. A chemical reduction method has been used to fabricate transparent conducting graphene displaying a transmittance of 80% and a sheet resistance of 2 kΩ/sq [15]. There appears to be much room for improving the sheet resistance. Although the conductivity of graphene can be improved by stacking several layers, thereby adding channels for charge transport, each additional layer of graphene degrades the overall optical transparency by 2.3% [16]. Without modifying the synthesis and transfer procedures, the sheet resistance of graphene grown through chemical vapor deposition (CVD) is on the order of several hundreds of ohms per square cm [16–19]. Graphene displays good electrical and optical properties in theory, suggesting that it could be an ideal material for TCE applications; for practical applications, however, its sheet resistance remains too high.

Another major class of carbon-based materials is the CNTs, which can be thought of as one or more sheet of graphene rolled up into a hollow cylindrical shell. CNT meshes are emerging as novel low-dimensional materials composed of large numbers of horizontal random mixtures of CNTs; they have significant potential as TCEs displaying transparency, conductivity, and flexibility [5]. CNT meshes can be prepared using several methods, including vacuum filtration [20] and transfer printing [21]. A CNT mesh deposited on polyethylene terephthalate (PET) exhibited a sheet resistance of 120 Ω/sq and a transmittance of 80% [22]; a CNT mesh prepared using a filtration/transfer method displayed a sheet resistance and transmission (at 550 nm) of 60 Ω/sq and 90.9%, respectively [23]. Compared with graphene, CNT meshes provide higher transmittances and lower sheet resistances; therefore, they have potential applicability in flexible TCE-based devices.

Although the circuit modeling of CNT meshes has been studied widely [24,25], relatively few research efforts have been made to address the optical behavior horizontally aligned CNT (HACNT) arrays. In particular, the optical properties of CNT meshes are affected by many factors, including the characteristics of the individual CNTs and their collective arrangement. The individual characteristics of a CNT include length, diameter, and metallic or semiconducting type; the collective arrangement of CNTs in a mesh can be characterized in terms of density, thickness, direction of arrangement, and morphology. CNTs have better electrical conductivity than silver; when they are suitably arranged, they can be highly transparent and flexible. The crossing of CNTs in a CNT mesh forms a large number of pores; incident light may pass through these pores with only a slight degree of absorption. The open areas of these pores can be controlled by adjusting the density and thickness of the CNTs. The electrical and optical properties of a CNT network depend on the density and thickness of its CNTs. At present, there remains much room for improving the optical and electrical performance of CNT meshes for use as TCEs. Although great progress has been made in horizontally aligned CNTs (HACNTs) [26], optimizing the geometrical arrangements in CNT meshes would presumably realize superior TCEs.

In this study, with the goal of designing a superior TCE, we investigated the arrangement of CNTs that would achieve high electrical conductivity without degrading the optical transmission. Fundamentally, a trade-off exists between the two key parameters of a TCE: light transmittance and sheet resistance. In this paper, by adjusting the spacing of regular arrays of CNTs, we propose a concept of optical thresholds, which provide not only effective charge transport channels but also a sufficient open area. Compared with TCEs based on various other materials, as well as the theoretical value of σ_DC/σ_op of graphene, we conclude that CNT meshes can display superior performance.

2. Simulation setup

Single-walled CNTs (SWCNTs) are akin to one shell of graphene having rolled up into a tube of small diameter. The physical properties of SWCNTs—either metallic or semiconducting—are highly dependent on the direction in which they have been rolled up (chirality) [27]. Nevertheless, the chirality-controlled growth of CNTs is difficult to perform experimentally, while the process of separating metallic and semiconducting CNTs is also difficult and expensive. The randomly distributed chirality in clusters of CNTs complicates any analysis of their electrical and optical properties. In contrast, multi-walled CNTs (MWCNTs) feature two or more concentric nested cylindrical shells with diameters ranging from several to hundreds of nanometers. Because the variations in the physical properties of each shell are averaged out, MWCNTs are always metallic [28,29]. In addition, MWCNTs can be mass-produced reproducibly with less concern about their chirality, making them ideal materials for various applications. Accordingly, MWCNTs are being investigated extensively for interconnecting applications and as TCEs. In this present study, we investigated meshes of MWCNTs for TCEs, because they tend to be mostly metallic and have more regular and uniform physical properties.

Graphene is highly optically anisotropic in the directions parallel and perpendicular to the normal axis of the graphene sheet. MWCNTs can be considered as multiple sheets of graphene rolled up into a hollow cylindrical shell; the local dielectric tensor is given by [30]:

(1)$$\varepsilon (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \theta } ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over z} ) = {\varepsilon _e}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} + {\varepsilon _o}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \theta } \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \theta } + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over z} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over z} )$$

where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r}$, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \theta }$, and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over z}$ are the base vectors of the cylindrical coordinate system and ${\varepsilon _e}$ and ${\varepsilon _o}$ are the principal components of the dielectric tensor of graphite for the extraordinary and ordinary rays, respectively [Fig. 1(a)]. Converting to the cylindrical coordinate system by using the effective medium approximation [30,31], the dielectric tensor of an individual MWCNT is given by

(2)$${\varepsilon _o} = {\varepsilon _\parallel } = {\varepsilon _{\textrm{1 }\parallel }} + i{\varepsilon _{\textrm{2 }\parallel }}$$

(3)$$\sqrt {{\varepsilon _o}{\varepsilon _e}} = {\varepsilon _ \bot } = {\varepsilon _{\textrm{1 } \bot }} + i{\varepsilon _{\textrm{2 } \bot }}$$

where ${\varepsilon _\parallel }$ and ${\varepsilon _ \bot }$ are the principal components of the dielectric tensor parallel and perpendicular to the CNT axis, respectively; ${\varepsilon _{\textrm{1 }\parallel }}$ and ${\varepsilon _{\textrm{2 }\parallel }}$ are the real and imaginary parts of ${\varepsilon _\parallel }$; and ${\varepsilon _{\textrm{1 } \bot }}$ and ${\varepsilon _{\textrm{2 } \bot }}$ are the real and imaginary parts of ${\varepsilon _ \bot }$. We can convert the effective dielectric tensor of an individual MWCNT to the refractive index ($n$) and extinction coefficient ($k$) by using Eqs. (4) and (5):

(4)$$n = \sqrt {{\raise0.7ex\hbox{${(\sqrt {{\varepsilon _{\textrm{1 }}}^2 + {\varepsilon _\textrm{2}}^2} + {\varepsilon _{\textrm{1 }}})}$} \!\mathord{\left/ {\vphantom {{(\sqrt {{\varepsilon_{\textrm{1 }}}^2 + {\varepsilon_\textrm{2}}^2} + {\varepsilon_{\textrm{1 }}})} 2}}\right.}\!\lower0.7ex\hbox{$2$}}}$$

(5)$$k = \sqrt {{\raise0.7ex\hbox{${(\sqrt {{\varepsilon _{\textrm{1 }}}^2 + {\varepsilon _\textrm{2}}^2} - {\varepsilon _{\textrm{1 }}})}$} \!\mathord{\left/ {\vphantom {{(\sqrt {{\varepsilon_{\textrm{1 }}}^2 + {\varepsilon_\textrm{2}}^2} - {\varepsilon_{\textrm{1 }}})} 2}}\right.}\!\lower0.7ex\hbox{$2$}}}$$

Highlighting the highly anisotropic optical behavior of CNT meshes, Fig. 1(b) displays the values of n and k parallel and perpendicular to a CNT axis, plotted with respect to the wavelength. The values of n and k parallel to the CNT axis are larger than those perpendicular to it.

Fig. 1. (a) Anisotropic dielectric tensor of a monolayer graphene; an MWCNT as a roll of graphene. (b) Refractive index (n) and extinction coefficient (k) parallel and perpendicular to a CNT axis, plotted with respect to wavelength. (c, d) Schematic representations of (c) 1D closely connected regular arrays of CNTs and (d) 1D regular arrays of CNTs separated by a distance P, where P is greater than D.

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We performed simulations based on these calculated refractive indices and extinction coefficients. We used the three-dimensional finite difference time domain (3D-FDTD) method to model the optical properties of 1D regular array of CNTs. Maxwell’s equations describe a situation in which the temporal change in the electric field of an electromagnetic wave is dependent on the spatial variation of the magnetic field, and vice versa. In the 3D-FDTD method, Maxwell’s equations were discretized by a central difference in time and space, allowing the equations to be solved numerically. Thus, the direct solutions of Maxwell’s equations provided insight into the near-field optical behavior.

According to the compact physical models of MWCNTs, achieving low resistance requires not only that all of the shells in a MWCNT contribute to the conductivity but also that the contact resistance for each shell should be suitably low [32–34]. Assuming that the spacing between two adjacent shells of a MWCNT is approximately 0.34 nm, and that the ratio of the diameters of the outermost and innermost shells (${D_{\max }}/{D_{\min }}$) is 2, we can estimate the number of shells per MWCNT. The number of channels per shell is given by the equation

(6)$${N_{chan/shell}}(D) \approx 0.0612D + 0.425$$

when D is larger than 3 nm. The total number of channels per MWCNT can then be obtained. The conductance per channel is given by the equation

(7)$$G = {G_0}/(1 + \ell /{L_0})$$

where ${G_0}$ is the quantum conductance (1/12.9 kΩ), $\ell $ is the length of the MWCNT, and ${L_0} = {\ell _0}D$. For MWCNT with D of 10 nm, the mean free path ${L_0}$ of the order of 10 µm [35], therefor ${\ell _0}$ is equal 1000. For extremely long MWCNTs ($\ell > > {L_0}$), the total conductance can be determined using the equation

(8)$$\sigma = aD({\Omega ^{ - 1}}n{m^{ - 2}}) + b({\Omega ^{ - 1}}n{m^{ - 1}})$$

where a is 0.02035 and b is 0.181686 for extremely long MWCNTs (physical length >> electron mean free path), the conductivities are several times larger than those of single-walled carbon nanotubes (SWCNTs) and copper wires [33]. We have also considered the conductivities of two CNTs that are in contact or are penetrating each other. The separation or tunneling distance between two crossed CNTs was set at 0.5 nm, and then a tunneling resistance was added between them. The effective electrical resistance of a MWCNT depends on the resistance of the MWCNT itself and the tunneling effect on resistance [36].

In general, the sheet resistance of a uniform conductor thin film can be written as

(9)$${R_{sh}} = \rho /t\;\;\;\;(\textrm{units}\;\textrm{of}\;\mathrm{\Omega}/\textrm{sq})$$

where t is the film thickness and $\rho$ is the resistivity. A nanostructured conducting film having a 1D periodicity P, width W, and thickness t will have a sheet resistance equal to

(10)$${R_{sh}} = \textrm{(}\rho \textrm{P)}/\textrm{(}t\textrm{W)}$$

for an electrical current along the wire [37]. As displayed in Fig. 1(c), the CNTs were closely connected one after another in parallel so that the same voltage would be applied to each CNT. In this 1D regular array, the CNTs were closely connected in the horizontal plane without any gaps between them. The sheet resistance of a 1D regular array is analogous to the sheet resistance in a uniform conductor thin film, determined by grouping the resistivity with the thickness. In our case, we assumed the unit thickness of the 1D regular array to be the diameter of the CNTs. The parallel resistance of the 1D regular array of CNTs can be regarded as the effective sheet resistance of 1D regular array of CNTs for an electrical current passing along the axis of the CNTs. Regarding a square 1D regular array of CNTs, where $Y = W$ and $W = mD$ ($m$ is the number of CNTs in the array; D is the diameter of a CNT), the effective sheet resistance of the closely connected regular array of CNTs is expressed by

(11)$${R_{sh,eff,close}} \equiv \frac{1}{m}{R_{CNT}} = {\rho _{CNT}}\frac{Y}{{\pi ({{{D^2}} / 4})}}\frac{1}{m} = \frac{{4{\rho _{CNT}}}}{{D\pi }}$$

where ${R_{CNT}}$ is the resistance of one CNT along the axial direction; $Y = mD$; ${\rho _{CNT}}$ is the resistivity of a CNT determined using Eq. (6) [33]; and $\pi ({{{D^2}} / 4})$ is the cross-sectional area of the CNT. As displayed in Fig. 1(d), the CNTs in the closely connected 1D regular arrays of CNTs were separated, forming gaps in the arrays. The 1D regular arrays of CNTs had a periodicity P, where P was greater than D; using the unit area $Y = P$, the effective sheet resistance for an electrical current along the CNTs can be expressed as

(12)$${R_{sh,eff,open}} \equiv {\rho _{CNT}}\frac{Y}{{\pi ({{{D^2}} / 4})}} = \frac{{4{\rho _{CNT}}P}}{{{D^2}\pi }}$$

3. 1D regular arrays of CNTs

We first investigated the anisotropic optical behavior of 1D regular arrays of CNTs that were closely connected one after another in the horizontal plane, without gaps between them, meaning that the period P was equal to the CNT diameter, as displayed in Fig. 2(a). In this study, we chose the diameter of the CNTs to be 10 nm, the length to be infinite, and the wavelength to be in the range 350–2000 nm, because most LEDs, solar cells, and optoelectronic devices operate within this spectral regime. As displayed in Fig. 2(a), for this 1D regular arrays of CNTs, we independently analyzed the transverse electric (TE, electric field perpendicular to the axial of CNTs) and transverse magnetic (TM, electric field parallel to the axial of CNTs) polarizations. We investigated the transmittance spectra of the TE, TM, and averaged (TE and TM) polarizations, separately. The transmittance spectra for TE polarization were much more intense than those of the TM-polarized light [Fig. 2(b)]. This anisotropic phenomenon was due to the geometry of the CNT arrays. When TE-polarized light was incident, its electric field direction was perpendicular to the longitudinal axis of CNT; it would encounter a groove filled with air between two adjacent CNTs and, therefore, experience a much lower effective refractive index and extinction coefficient, leading to lower reflectance [Fig. 2(c)] and absorbance [Fig. 2(d)] and, thus, greater transmittance. Furthermore, the calculated extinction coefficient along the transverse axis of a CNT was lower than that along its longitudinal axis, again leading to a lower effective extinction coefficient of TE-polarized light. TE-polarized light achieved a maximum transmittance of approximately 90% in the near-IR regime, with an average transmittance of approximately 85% in the spectral regime from 400 to 2000 nm. On the other hand, because the electric field of the TM-polarized light was always parallel to the longitudinal axis of a CNT, it exhibited larger reflectance and absorbance. The maximum transmittance of TM polarization was approximately 65%; its average transmittance was approximately 60%.

Fig. 2. (a) Schematic representation of a 1D regular array of CNTs; TM is the electric field parallel to the axis of the CNTs; TE is electric field perpendicular to the axis of CNTs. (b) Transmittance, (c) reflectance, and (d) absorbance of TE- and TM-polarized light, simulated through 3D-FDTD calculations of the 1D regular array of CNTs.

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Notably, a monolayer of 1D regular array of CNTs, having a thickness of 10 nm, reflected approximately 5% of TM-polarized light and 1% of TE-polarized light [Fig. 2(c)]. This large difference in reflectance is indicative of anisotropic effective refractive indices in the array of CNTs, as mentioned above. Figure 2(d) displays the absorbance spectra of this regular array of CNTs. The absorbance spectrum of the TM-polarized light was much more intense than that of the TE-polarized light. Comparing the absorbance and transmittance spectra under the two types of polarized light, this 1D array of CNTs can be considered as an absorber of TM-polarized light, while the TE-polarized light underwent a high transmission. We quantified these variations for light of various incident angles. The 1D regular arrays of CNTs having a diameter of 10 nm were closely connected without any gaps between them, as displayed in Fig. 3(a). We fixed the incident wavelength at 550 nm and investigated the angle-dependent transmittance, reflectance, and absorbance. The transmittances of the TE- and TM-polarized light both increase slightly upon increasing the incident angle [Fig. 3(b)]. The transmittance of TE-polarized light was approximately 83%; for TM-polarized light it was approximately 65%. The higher transmittance of the TE-polarized light arose from both the Brewster angle effect and the anisotropic behavior of the CNTs. The FDTD results for the 1D regular array of CNTs revealed that the reflectance of the TM-polarized light increased from 5 to 12% as the incident angle increased from 0 to 40°, whereas the reflectance of TE-polarized light decreased from 2 to 0.1% as a result of the Brewster angle effect [Fig. 3(c)]. The angle-dependent absorbance of the TM-polarized light decreased upon increasing the incident angle because of increasing reflectance. For the TE-polarized light, the absorbance initially increased slightly upon increasing the incident angle, reaching a maximum of 17% near 30°, but decreased thereafter [Fig. 3(d)]. The initial increase in absorbance arose from decreased reflectance. As the incident angle increased further, the direction of the electric field of the TE-polarized light became more perpendicular to the longitudinal axis of the CNT, leading to decreased absorbance as a result of the anisotropic absorption of the CNT.

Fig. 3. (a) Schematic representation of a 1D regular array of CNTs and its incident angle–dependent (b) transmittance, (c) reflectance, and (d) absorbance of TE- and TM-polarized light (wavelength: 550 nm).

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4. 2D regular arrays of CNTs

Next, we investigated the optical behavior of the 2D regular arrays of CNTs, featuring a set of CNTs aligned along one direction and an intersecting set of CNTs aligned perpendicularly. The distance from the center of one CNT to another ($P$) was equal to or greater than the diameter of the CNTs. A schematic representation of a 2D regular array of CNTs with variable P, simulated using the 3D-FDTD method, is displayed in the inset to Fig. 4(a). We chose a CNT diameter of 10 nm and an infinite tube length. We considered values of P ranging from 20 to 300 nm; the corresponding volume fractions (V_f) ranged from 0.0785 to 0.00523. Figures 4(a)–4(c) present the calculated transmittance, reflectance, and absorbance, respectively, of the CNT arrays subjected to light of normal incidence. The maximum transmittance at $P$=20 nm was 79% at a wavelength of 550 nm. When the value of P increased to 50 nm, the averaged transmittance reached 90%. When the value of P was increased further to 300 nm, the averaged transmittance was greater than 98%. The crossed CNTs in the CNT mesh formed a large number of pores. The open areas of these pores allow the incident light to pass through with only a very slight degree of absorption. A rapid increase in the transmittance of the CNT mesh occurred for values of P in the range from 20 to 50 nm. These open areas of the CNT mesh could be controlled by adjusting the value of P.

Fig. 4. (a) Transmittance, (b) reflectance, and (c) absorbance of 2D regular arrays of CNTs with variable distances P subject to normal incidence. (d) Plot of sheet resistance with respect to the distance P.

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The reflectance of the 2D regular arrays of CNTs was generally small. For example, when the values of P and V_f were 20 nm and 0.078, respectively, the maximum reflectance was less than 8%. When the value of P was greater than 50 nm, the reflectance was less than 1%, due to the large open areas available for light to penetrate or pass through the CNTs. Furthermore, the presence of large open areas in the arrays allowed the light to enter readily without reflectance. In addition to the open areas of the pores, the transmittance was also determined by the absorbance of the CNT arrays. The CNTs arrays would be an absorber when the incident light had its electric field parallel to the longitudinal axis of the CNTs. In our simulations, the absorbance decreased upon increasing the value of P because of the lower density of the CNTs arrays [Fig. 4(c)]. The electrical properties of a TCE can be defined by its sheet resistance. For MWCNTs having a diameter of 10 nm and an electrical resistivity (${\rho _{CNT}}$) of 2.63 Ω-nm, the sheet resistance ranged from 0.66 to 9.9 Ω/sq for values of P ranging from 20 to 300 nm. In other words, the sheet resistance increased upon increasing the value of P [Fig. 4(d)].

For 2D regular arrays of CNTs having values of P of 20, 50, 100, and 300 nm, the open area units were 100, 1600, 8100, and 84,100 nm², respectively; the transmittances were 80, 90, 94, and 98%, respectively; and the sheet resistances were 0.6, 1.65, 3.3, and 9.9 Ω/sq, respectively. Thus, when the value of P increased from 100 to 300 nm, the transmittance increased only slightly (from 94 to 98%), while the sheet resistance increased significantly (from 3.3 to 9.9 Ω/sq). This behavior is indicative of a critical distance (${P_c}$) at which an appropriate open area is formed in the regular arrays of CNTs, overcoming the absorption of the CNTs and leading to enhanced transmittance with good electrical conductance. To investigate the critical distances ${P_c}$ of the 2D regular arrays of CNTs with consideration of their application in TCEs, we adjusted the value of P from 10 to 500 nm. Figure 5 reveals a clear trend in the transmittance spectra with respect to P at a wavelength of 550 nm. When the value of P increased from 10 and 50 nm, the transmittance increased significantly, from 53 to 90%; when it exceeded 50 nm, however, the transmittance increased only moderately. The percolation threshold is highly dependent on the high aspect ratio (length-to-diameter) of objects such as conducting polymer chains, nanowires, and CNTs; therefore, CNTs have great potential for use as conductive fillers, resulting in very low percolation thresholds [38]. The percolation conductivity is dependent on the degree of alignment, while the electrical conductivity had a power-law dependence with respect to the alignment as well as the concentration [39]. In this study, we observed the transmittance of the 2D regular arrays of CNTs had a threshold value that was dependent on the distance; we name this distance the “optical threshold.” Beyond this optical threshold, the increase in transmittance was very low.

Fig. 5. Plot of transmittance with respect to distance P for 2D regular arrays of CNTs.

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For our calculations, we used the 10-nm-diameter CNTs having resistivity (ρ_cnt) of 2.6 Ω-nm and varied the value of P to determine the optimal sheet resistance and transmittance. When the value of P was 50 nm, the transmittance of the 2D regular array of CNTs was approximately 90% and the sheet resistance was 1.65 Ω/sq. The value of σ_DC/σ_op of 2077 was extremely high. Various methods have been used previously to prepare TCEs (Fig. 6), including those from Ag nanowire networks [1], ultrathin metal films [11], 2D MXenes [2], graphene grown on Ni layers [40], graphene grown on Cu foil through CVD [16], CNT films with acid treatment [40], roll-to-roll production and wet-chemical doping of graphene [19]. From those experimental studies and graphene-based theoretical calculations, the best value of σ_DC/σ_op for a graphene-based film has been 9.9 [16] and that for an acid-treated CNT film has been 32.6 [41]. The sheet resistance of graphene can be expressed as follows [42]:

(13)$${R_{sh}} = 62.4\Omega /N$$

where N is the number of monolayers of graphene. The sheet resistance of a graphene film is a function of the film thickness, and the largest value of σ_DC/σ_op is 258. When a roll-to-roll dry transfer process and a soluble polymer support wet-transfer method were applied, and a layer-by-layer stacking method was used to increase the conductance, the value of σ_DC/σ_op was 116; in this case, however, the mechanical strength of the layers was insufficient to avoid separation of the graphene films from the substrate [16]. As a result, many defects existed on the graphene films, leading to an increase in the overall sheet resistance. In this present paper, we demonstrate a method to produce thin films of a CNT mesh giving a value of σ_DC/σ_op of 2077 that is much larger even than that of ITO (σ_DC/σ_op as high as 220). Upon increasing the distance P in a 2D regular array of CNTs, the transmittance and sheet resistance both increase simultaneously, with the value of σ_DC/σ_op increasing initially (from 1543 to 2077), but decreasing thereafter (to 1928). Notably, these transmittance–sheet resistance data are superior to those of all of the data previously reported in the literature for various other materials and nanostructures.

Fig. 6. Comparison of our simulated transparent conductive performance with those of other approaches reported previously.

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A device has been fabricated previously from hundreds of parallel CNTs through AC dielectrophoresis using high-quality, surfactant-free, stable CNT solutions, displaying current outputs of up to hundreds of microamperes [43,44]. The degree of alignment and the density of the CNTs were controlled by tuning the concentration of the CNTs in the solution. Patterns of nearly perfectly aligned and long CNTs have been formed through CVD on a quartz substrate; such dense, perfectly aligned, long SWNT arrays were effective thin film semiconductors, suitable for integration into transistors and other types of electronic devices [45,46]. Dense, aligned parallel arrays of CNTs have been grown on a quartz substrate through a one-step approach to pattern uniform catalyst lines by photolithography [47]. Randomly arranged CNT films have been transferred and assembled into highly ordered, dense regular arrays of high uniformity and reproducibility by sliding the substrate during the transfer process [48]. According to those reports, many processes are available to fabricate aligned, long, and parallel arrays of CNT networks [26]. These techniques might be useful for preparing optimal regular arrays of CNT-based TCEs with extremely high DC-to-optical conductive ratios in the future.

5. Conclusions

TCEs of high transmittance and low sheet resistance can be prepared by arranging CNTs in optimal regular arrays. We have examined the effects of various parameters, including the distance between the CNTs, the working wavelength, and the polarization, on the reflection, transmission, and absorption properties of the CNT meshes. By adjusting the spacing and arrangement, we could tune and optimize the value of σ_DC/σ_op. For a regular array of CNTs having a diameter of 10 nm and a period of approximately 50–60 nm, the transmittance in the visible regime was 90% and the sheet resistance was 1.6 Ω/sq. We propose herein the concept of an “optical threshold,” which provides the most effective charge transport channels and also a sufficient open area. This concept solves the trade-off between the two key parameters of a TCE: light transmittance and sheet resistance. By adjusting the spacing in the regular array of CNTs, providing not only the most effective charge transport channels but also very high optical transmittance, the value of σ_DC/σ_op reached as high as 2077. This value is much larger than those reported previously from simulations or experimental studies of various other materials and structures. With recent technological advances in the manufacture of aligned, long, and parallel CNT network arrays, we believe that CNT meshes hold great promise for use as flexible, highly transparent, extremely electrically conductive, and durable TCEs in various optoelectronic devices.

Funding

Ministry of Science and Technology, Taiwan (108-2622-E-002 -029 -CC2, 109-2221-E-002 -104 -MY3, 109-2221-E-002 -188 -MY3, NCSIST-302-V201-109).

Disclosures

“The authors declare no conflicts of interest.”

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Optimization and simulation of a carbon nanotube arrangement for transparent conductive electrodes with record-high direct current to optical conductive ratios

Abstract

1. Introduction

2. Simulation setup

3. 1D regular arrays of CNTs

4. 2D regular arrays of CNTs

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Equations (13)

Optical Materials Express