The widespread use of copper in power and data cabling for aircraft, ships, and ground vehicles imposes significant mass penalties and limits cable ampacity. Experimental research has suggested that iodine-doped carbon nanotubes (CNTs) can serve as energy efficient replacements for copper in mass sensitive cabling applications. The high computational costs of ab initio modeling have limited complimentary modeling research on the development of high specific conductance materials. In recent research, the authors have applied two modeling assumptions, single zeta basis sets and approximate geometric models of the CNT junction structures, to allow an order of magnitude increase in the atom count used to model iodine-doped CNT conductors. This permits the ab initio study of dopant concentration and dopant distribution effects, and the development of a fully quantum based nanowire model which may be compared directly with the results of macroscale experiments. The accuracy of the modeling assumptions is supported by comparisons of ballistic conductance calculations with known quantum solutions and by comparison of the nanowire performance predictions with published experimental data. The validated formulation offers important insights on dopant distribution effects and conduction mechanisms not amenable to direct experimental measurement.

Introduction

The widespread use of copper in power and data cabling for aircraft, ships, and ground vehicles imposes significant mass penalties and can limit system electrical performance. Carbon nanotube (CNT) [1,2] based electrical conductors have attracted considerable attention, as potential replacements for pure copper, since they may offer improved specific conductivity [3] and higher ampacity [4,5]. CNT-based conductors have been studied both experimentally and computationally, as a promising new cable technology. Their relatively low electrical conductivity [6], as compared to copper, has encouraged the consideration of doped nanotubes as mass efficient replacements in weight-sensitive applications. Tables 1 and 2 compare published data on the electrical conductivity and the mass-specific electrical conductivity of doped CNT with the corresponding properties of copper.

Over the course of the last two decades, considerable experimental research has investigated the conduction performance of single wall carbon nanotubes [7], multiwall carbon nanotubes ([8], doped CNT [3,912], CNT composites [13,14], CNT junctions [15], and CNT networks [16]. Complimentary computational research on these topics has also been performed, although the ab initio computational literature has modeled rather simple systems [11], due in large part to high computational cost. Given the substantial basic knowledge base, recent experimental and computational research has increasingly focused on the most promising material candidates to replace copper in weight-sensitive engineering applications [17,18]. An example application is the development of high specific conductivity power and data cabling for civilian and military aircraft.

The most widely used approach to ballistic conductance modeling employs density functional theory and nonequilibrium Green's function methods to study the electrical transport properties of nanoscale conductors [1921]. Since a macroscopic CNT cable is necessarily composed of many nanoscale CNT conductors and junctions, modeling work on the ballistic conductance of both CNT conductors and CNT junctions is of major interest.

The previous work on CNT conductors has included studies of (1) defects (e.g., vacancies [22]), (2) chemical doping (e.g., F [23], I2, ICl, IBr [11], MoO3 [24], and AuCl3 [25]), (3) multiwall CNT systems (e.g., a double wall CNT in which each tube has different electrical properties [26] or a double wall CNT with variations in the interwall spacing [8]), and (4) the performance of nanocomposite wires (e.g., copper-CNT conductors [27] and sulfur chains positioned inside CNT [28]). Some research has investigated such parameters acting in combination. For example, Lopez-Bezanilla [29] investigated chemically doped double wall CNT, examining the effects of both interwall spacing and outer wall modification [by monovalent phenyl (-C6H5) and divalent dichlorocarbene (>CCl2) dopants] on conductor performance. Note that doping can also have negative effects on conductivity. The last cited research suggested that monovalent dopants have a stronger negative (for metallic CNT) effects on conductance than do divalent dopants. They noted that large interwall spacing can prevent the negative effects of outer tube doping from affecting the inner tube.

In the case of CNT junctions, modeling research has focused on: (1) structural effects (e.g., variations in junction overlap [30,31] or tube intersection angles [32,33]) and (2) chemical doping effects (including transition metals [34], gold nanoparticles [35], or O2 and N2 [36]) on junction performance. The computational results indicate that the junction conductance “oscillates” with the extent of tube overlap, which may explained by “quantum interference” effects [30,31]. With respect to tube intersection angles, the highest conductance has been reported to occur when the junction structure is “commensurate” [33]. In the case of transition metal doping, it appears that the best junction conductance results from chromium doping [34].

In the case of CNT network modeling, most modeling work has employed percolation theory, which accounts for both conductor and junction performance. For example, Refs. [16], [37], and [38] combine the macroscale conductance properties of the conductor with the ballistic conductance properties of the junctions to predict the overall performance of CNT composites. In this paper, the overall CNT network performance is estimated by developing a transmission line model, parametrized by the ballistic conductance properties of both the CNT conductors and the CNT junctions. Hence, in all cases, the results presented in this paper are based on electronic structure calculations.

The succeeding sections of this paper are organized as follows: Section 2 describes the computational methods employed. Section 3 presents modeling results on polyiodide-doped CNT conductors and junctions. The effects of iodine doping on both metallic [CNT(M)] and semiconducting [CNT(S)] carbon nanotubes are included in the analysis. As compared to the previous work [11,39] on iodine dopants, this section considers atom counts as high as 616, an approximate order of magnitude increase over the last cited works. The combination of high atom counts, complex junction structures, and complex dopant distributions resulted in very high computational costs. Hence, this section also describes certain modeling approximations introduced in order to perform ballistic conductance analyses at models sizes sufficient to allow for the study of nanotube interaction, dopant concentration, and dopant distribution effects. The study of these effects is a critical part of any attempt to compare ab initio performance predictions with macroscale experimental results.

Section 4 formulates a transmission line model, used to estimate nanowire performance, applying conductor and junction analysis data presented in the preceding section. The transmission line is represented as series combination [16] of CNT conductors and CNT junctions (alternative methods might be used [40]), in order to estimate nanowire performance. Using room temperature copper conductor properties as a reference, the expected performance of various CNT-based conductors are compared on a specific conductivity basis.

Section 5 discusses conclusions suggested by the computational results presented in the paper.

Computational Methods

The computational package used in this paper is the open source code SIESTA [41], which is based on density functional theory and employs atomic orbitals as a basis set. The electrical transport properties are computed using a nonequilibrium Green's function method [42], implemented in the TranSIESTA module [43] of the SIESTA package. The electrical conductance ($G$) is calculated using the Landauer formula [44]
$G=2e2h∫−∂fE∂ETEdE,TE=Trt†EtE$
(1)
where $e$ is the electron charge, $h$ is Planck's constant, $fE$ is the Fermi–Dirac distribution function, $E$ is the wave energy, $TE$ is the transmission function, $†$ denotes the conjugate transpose, and $tE$ is a matrix of transmission coefficients for waves propagating along the conductor. The calculations presented in this paper are made for zero temperature conditions, in which case [45]
$−∂fE∂E=δE−Ef$
(2)
with $Ef$ being the Fermi energy and $δ$ a Dirac delta function, so that
$G=G0TEf,G0=2e2h=7.75×10−5S$
(3)

where $G0$ is the standard quantum conductance unit. For an ideal metallic carbon nanotube, $TEf=2$ and $G=2G0$ [7].

All calculations were performed using the generalized gradient approximation for the exchange-correlation functional parameterized by Perdew et al. [46]. A single-zeta basis set was employed for all atoms to reduce computational cost. The accuracy of the single-zeta basis calculations was evaluated by comparing the computed ballistic conductance results for single and dual parallel metallic nanotubes to the “exact” conductance solutions for those systems. The integration k-points in the Brillouin zone were chosen using a Monkhorst–Pack mesh [47]. The model parameters used in the calculations are discussed in the sections which follow.

The junction conductance calculations presented in this paper were performed on atom sets obtained by removing atoms from “relaxed” models of dual, parallel, doped, and undoped nanotubes. This approximation was adopted in part due to difficulties encountered in obtaining converged equilibrium solutions for junction structures using the default SIESTA force convergence criterion (0.04 eV/Å). Note that the published work has employed force convergence criteria that vary by two orders of magnitude (0.001 eV/Å [48] to 0.1 eV/Å [36]).

The accuracy of the approximate junction models was evaluated by comparing the predictions of a nanowire performance model (which incorporates the approximate junction models) with the published experimental data on iodine-doped CNT fibers, as detailed in Sec. 4. Such indirect validation of the approximate junction models is necessitated by the absence of nanoscale experimental data measuring directly junction ballistic conduction as a function of junction geometry, nanotube type, doping concentration, and dopant distribution. The consistency of the nanowire modeling results with the published macroscale experimental data suggests that the conductance calculations presented in this paper are in general representative of the modeled physical systems. The conclusions presented in Sec. 5, which discuss junction geometry, doping mass fraction, doping distribution, and other characteristics of the system not amenable to direct experimental measurement are intended to assist experimental research on the development of new high specific conductivity cabling.

Polyiodide-Doped Carbon Nanotube

In general, the performance of iodine-doped CNT systems may be affected by iodine atom interactions. An example is the presence of iodine in polyiodide form, as described in experimental papers on both CNT [3,49] and graphene [50], which suggest that $I3−$ and $I5−$ polyiodide chains may be formed during the doping process. An iodine chain structure located inside CNT's was also observed in experiments performed by Fan et al. [51].

Transmission electron microscope images of iodine-doped CNT [3] suggest that the iodine distribution in doped CNT cables consists of (1) interstitial dopant atoms concentrated near CNT “contacts” and (2) randomly distributed dopant atoms scattered across CNT surfaces. To better understand the effects of polyiodide doping, the analysis which follows considers CNT's doped with polyiodide (note that the present model is formulated at the electronic structure level and no molecular structure is imposed). The interaction of both metallic and semiconducting CNT's with polyiodides is investigated.

The following iodine-doped system configurations were modeled, for both metallic and semiconducting CNT (in some configurations, dopant weighting was also varied):

• CNT conductors with “aligned” doping

• CNT conductors with “random” doping

• CNT conductors with “interstitial” doping

• CNT junctions with “interstitial” doping

The modeled CNTs are metallic with chirality (5,5) [CNT(5,5)] and semiconducting with chirality (8,0) [CNT(8,0)], which have diameters 7.1 Å and 6.4 Å, respectively. Note that the smallest energetically stable CNT has diameter of 4 Å [52]. The calculations employed k-points chosen using a Monkhorst–Pack mesh [47]. It is important to note that previous work has employed k-grid dimensions which varied widely. Published conductance calculations have employed k-point dimensions that range from $1×1×4$ [29,34] to $1×1×50$ [53,54]. Since the models presented here are computationally quite expensive, a relatively coarse k-grid was selected. For the relaxation calculations, the k-grid has dimensions $1×1×4$ [34]. For the conductance calculations, the k-grid has dimensions $1×1×9$. The fineness of the real space mesh was controlled by setting the energy cutoff to 200 Ry [5557].

Polyiodide-Doped Carbon Nanotube Conductors.

The polyiodide-doped conductor models investigated single nanotube and dual nanotube configurations. In the single nanotube configurations, both aligned and random doping patterns were modeled. In the dual nanotube configuration, only interstitial doping patterns were modeled. These three doping geometries are illustrated in the figures which follow.

Figure 1 shows the doped single metallic nanotube configurations considered. The first and second models assume aligned dopant atoms, with 0.7 iodine atoms per CNT unit cell (0.7/u.c.) and 1.0 iodine atoms per CNT unit cell (1.0/u.c.), respectively. The third model depicts the random doping pattern. In the case of the randomly doped CNT, the electrodes were not doped.

Figure 2 shows the computed conductance for the modeled metallic CNT's. The calculation made here for an undoped metallic CNT (shown by the bar labeled “none”) correctly returns the exact solution of 2.0 quantum conductance units. Note that Ref. [11], which employs a double zeta basis set, returns a conductance 25% lower, perhaps due to electrode effects. The present work employs an electrode whose structure matches that of the modeled device, in order to represent a segment of a much longer (as long as the material's mean free path (MFP)) conductor. The remaining three bars in Fig. 2 show conductance results for the doped metallic nanotubes. They indicate that the conductance of the doped tube is affected by both the dopant concentration and the dopant distribution. At the lowest dopant concentration, the iodine converts the metallic tube into a semiconducting tube, as reported in previous experimental and computational work [39]. At the highest doping concentration, the distribution of the modeled dopant is random and the semiconducting conversion is maintained. At the intermediate doping concentration, the dopant distribution is aligned and the metallic conductance of the system is restored. This restoration may be due to the formation of polyiodide structures, and two consequent effects: (1) the creation of p-doped conduction “pathways” (axially asymmetric doping) in the nanotubes, and (2) conduction in the polyiodide chains, via a Grothuss mechanisim [5860]. Given the highly ordered structure of the conduction paths in metallic CNT's [61], conductance sensitivity to dopant distribution is certainly plausible.

Figure 3 shows the doped single semiconducting nanotube configurations considered. The first and second models assume aligned dopant atoms, with 1.0 iodine atoms per CNT unit cell and 1.5 iodine atoms per CNT unit cell, respectively. The third model depicts the random doping pattern. In the case of the semiconducting CNT's, the electrodes were doped.

Figure 4 shows the computed conductance for the modeled semiconducting CNT's. The calculation made here for an undoped semiconducting CNT (shown by the bar labeled none) returns (as expected) negligible conductance. Note that Ref. [11], which employs a double zeta basis set, returns a conductance fully 70% of that computed (in that work) for a metallic tube, in a copper electrode configuration. The remaining three bars in Fig. 4 show conductance results for the doped semiconducting nanotubes. As in the metallic case, they indicate that the conductance of the doped tube is affected by both the dopant concentration and the dopant distribution. At the lowest dopant concentration, application of the iodine does not improve conductance. At the highest doping concentration, the distribution of the modeled dopant is random and the conductance is again negligible. At the intermediate doping concentration, the dopant distribution is aligned and a small but nonzero conductance is computed. The modeling results might again be explained by the formation of polyiodide structures: (1) the creation of p-doped conduction pathways (axially asymmetric doping) in the nanotubes and (2) the conduction in the polyiodide chains.

Although the preceding calculations on isolated nanotubes are of great interest, experimental studies of iodine-doped CNT conductors emphasize that macroscale cables are composed of nanotube bundles, and that such bundles will give rise to more complex doping patterns. The simplest doping pattern associated with interacting tubes (a pattern depicted in Ref. [3]) is the interstitial doping configuration shown in Fig. 5. The latter figure depicts dual parallel metallic nanotubes, interstitially doped at two different iodine concentrations. Figure 6 shows the computed conductance results for the modeled dual metallic CNT's. The calculation for the undoped system (shown by the bar labeled none) correctly returns the exact solution of 4.0 quantum conductance units. The remaining two bars in Fig. 6 show computed conductance results for the doped dual tubes. As in the single metallic nanotube case, low levels of iodine doping significantly reduce metallic system conductance. Note that for multinanotube bundles, alignment of the dopant atoms along the intersticial crevice might be encouraged by some manufacturing processes (e.g., extrusion). Consistent with the arguments made for the isolated tube models, the creation of p-doped conduction pathways (axially asymmetric doping) in the nanotubes and possible conduction within the polyiodide chains may be responsible for the nonmonotonic variation in conductance.

Figure 7 depicts dual parallel semiconducting nanotube configurations, interstitially doped at two different iodine concentrations. Figure 8 shows the computed conductance results for the modeled dual semiconducting CNT's. The calculation for the undoped system (shown by the bar labeled none) correctly returns a result indicating negligible conductance. The remaining two bars in Fig. 8 show computed conductance results for the doped dual tubes. Interstitial iodine doping improves system conductance, and the system conductance increases with dopant concentration. As in the metallic case, the dual nanotube geometry appears to promote the formation of polyiodides and the formation of p-doped conduction pathways. At the higher of the two modeled dopant concentrations, the computed system conductance reaches 75% of that expected for dual undoped metallic nanotubes.

Polyiodide-Doped Carbon Nanotube Junctions.

The polyiodide-doped CNT junction models investigated dual nanotube configurations, at various overlaps, in interstitial doping configurations. The dopant per unit length was varied, and both metallic and semiconducting tubes were analyzed. In general, relaxation calculations for the doped CNT junctions failed to converge. The junction models were constructed by removing carbon atoms from the relaxed models of the interstitially doped CNT's depicted in Figs. 5 and 7.

It is important to note that the junctions of interest in this paper are intended to be representative of nanotube bundles contained in macroscale cables, typically manufactured by pressure rolling [18], extrusion [62], or other mechanically intrusive processes. Given these circumstances, the junction models analyzed in this paper are perhaps far more likely to be representative of those in macroscale cables than any junction models obtained by “re-relaxation” of isolated atomic configurations of the type depicted in Fig. 9. The authors are not aware of any previous work which has performed ab initio relaxation or conductance calculations for interstitially doped junctions like those depicted in Fig. 9.

The metallic nanotube junction shown in Fig. 9 was analyzed at five different overlaps, ranging from 2 to 10 unit cells, without doping and at two different doping concentrations. Figure 10 shows the computed conductance results. As indicated in the previous work [30,31], junction conduction does not in general vary monotonically with overlap. In the undoped configuration modeled here, the junction conductance is (at best) half of that expected for an undoped metallic nanotube, emphasizing the importance of “contact resistance” in determining the performance of nanotube-based cabling. The reduced conductance computed (for all overlaps) at the low doping concentration mimics the previously discussed response of isolated metallic nanotubes to low dopant concentrations. Only at the highest levels of dopant concentration and overlap considered in the analysis does the junction conductance approach 75% of the conductance of a pristine nanotube: that result, indicated by the highlighted square in Fig. 10, is used in the nanowire performance calculations discussed in Sec. 4.

The semiconducting nanotube junction shown in Fig. 11 was also analyzed at five different overlaps, in this case ranging from 0.7 to 4.7 unit cells. Since the undoped semiconducting tubes analyzed previously performed as insulators, junction performance was modeled only with doping applied, at two concentrations. Figure 12 shows the computed conductance results. At the low doping concentration, the junction conductance was negligible at all overlaps. At the high doping concentration, junction performance was very good, peaking at an overlap of 3.7 unit cells, where the doped junction performance approached that of a pristine metallic nanotube. That result, indicated by the highlighted square in Fig. 12, is used in the nanowire performance calculations discussed in Sec. 4.

Summary.

The precise effects of polyiodide doping vary significantly with nanotube type (metallic or semiconducting) and dopant distribution (aligned, random, or interstitial). The results presented in this section suggest several conclusions:

1. (1)

At low dopant levels, metallic nanotubes are adversely affected by iodine doping. However if the dopant is properly distributed, the performance of metallic nanotubes can be recovered (at least in part) by increasing the dopant concentration. In the case of the semiconducting nanotubes, conductance improves with iodine dopant concentration, as long as the dopant is properly distributed.

2. (2)

At low dopant levels, metallic nanotube junctions are adversely affected by iodine doping. However if the dopant is interstitially distributed, the performance of metallic junctions can be recovered by increasing the dopant concentration and junction overlap. In the case of the semiconducting nanotube junctions, conductance improves with iodine dopant concentration, if the dopant is interstitially distributed.

3. (3)

In the CNT configurations modeled here: interstitial doping is broadly beneficial, aligned doping offers some benefits, and random doping is ineffective.

4. (4)

Current explanations of the effects of iodine doping on CNT conductance focus broadly on iodine as p-type dopant for the CNT [11,63]. However, the enhanced conductance offered by interstitial doping may more specifically be due to asymmetric p-doping of the nanotubes.

5. (5)

Conduction within polyiodide structures (charge transfer without mass transport [5860]) may contribute to the doped system's performance. The formation of interstitial polyiodides might be encouraged by particular fabrication processes, such as pressure rolling or extrusion.

Section 4 of the paper applies the results of the ballistic conduction calculations just discussed, in order to estimate the measured performance of iodine-doped CNT cables studied in macroscale experiments. Comparison with experiment serves to critique the assumptions made in formulating the nanoscale model, evaluating its usefulness in assisting engineering design.

Transmission Line Model

In this section, a nanowire is modeled as a transmission line consisting of a set of conductors, each with a length no greater than the electron MFP for the conductor material, joined by discrete “junction” resistors. The mass and conductivity properties of the transmission line components are taken from the ballistic conductance analysis described in Sec. 3. The assumed model, shown in Fig. 13, is inspired by experimental measurements on CNT networks [16]. Estimates of the mean free path for the nanotube-based conductors are taken from the literature [64,65].

The mass per unit length and the resistance per unit length of the transmission line are determined by the conductor resistance $Rc$, junction resistance $Rj$, conductor mass per unit length $m̂c$, and added mass per junction $mjadd$, all determined from the models described in Sec. 3, and by the mean free path ($Lm$) of an electron in the conductor. Adopting the product of mass ($m$) per unit length and resistance ($R$) per unit length as performance measures for a nanowire, one may define a performance metric ($M$) using
$1M=ρσ=mL×RL=m̂c+mjaddLmn̂Rc+RjLmn̂$
(7)
where the number of junctions per mean free path is
$n̂=nL/Lm$
(8)

and $n$ is the number of junctions in a transmission line of length $L$ (composed of segments of length $Ls$). Note that in the case of a continuum conductor, the performance metric is the mass specific conductance, defined as the ratio of electrical conductivity (σ) to mass density (ρ).

The plots which follow employ the metric $M$ to estimate the performance of nanowires fabricated using the material systems considered in Secs. 3.1 and 3.2. Specifically, they plot the relative specific conductivity $M/Mref$ versus the number of junctions per unit mean free path $(n̂)$ for each material system, where $Mref$is a reference value for the chosen metric (the specific conductivity of pure copper). Note that for the minimum value of $n̂=1$ indicated in the plots, the number of junctions is just sufficient to permit ballistic conductance. Additional junctions add parasitic mass and resistance, reducing nanowire performance. The plots which follow assume a mean free path of either 500 nm or 1000 nm for the CNT's [64,65]. The high performance combinations of polyiodide-doped CNT conductors and junctions selected for the nanowire analysis presented in this section are shown in Table 3. The junction conductance data used in the analysis are indicated by the highlighted squares shown in Figs. 10 and 12.

The upper plot in Fig. 14 shows that for the minimum junction count ($n̂=1$) and a CNT MFP of 500 nm, the relative specific conductance ($M/Mref$) for the iodine-doped CNT nanowires ranges from one to three: the estimated specific conductance of the CNT nanowire is as much three times that of pure copper. This range matches that described by published experimental data [3] on the performance of iodine-doped CNT cables. The transmission line (nanowire) model also suggests that the number of junctions per mean free path ($n̂$) should be limited, in order to obtain high performance. As indicated in the lower plot of Fig. 14, for the range of parameters considered in this analysis, performance varies approximately linearly with mean free path.

Conclusions

This section presents general conclusions on the polyiodide-doped CNT systems analyzed in this paper and offers suggestions for future research. The ballistic conductance and transmission line analysis results for the polyiodide-doped CNT nanowires suggest a number of conclusions:

• The analysis results are consistent with the published experimental data [3], which indicate that iodine-doped CNT conductors can offer specific conductivity in the range of one to three times that of copper.

• The analyses presented here considered smaller diameter nanotubes (by a factor of four) and higher dopant to carbon mass ratios (by a factor of three) than those described in published experiments [3,17]. Since the model and the experiments indicate similar mass specific conductivity, mass specific performance does not appear to depend strongly on nanotube diameter.

• Estimated CNT nanowire performance varies approximately linearly with CNT mean free path; published experimental data indicate that mean free path is reduced as temperature is increased [66].

• In the case of iodine doping, realizing high specific conductivity appears to require very mass efficient use of the dopant.

• Doping distribution is highly important and might be influenced by cable fabrication processes.

• Charge transport in within polyiodides may contribute to conductor performance.

The computational research described in this paper, and the corresponding experimental research literature, suggest many opportunities for future research. Of immediate interest are: (1) the modeling of more complex dopants, including ICl [12] and KAuBr4 [9]; (2) consideration of longer junction overlaps; (3) the modeling of multitube interactions (as computational costs permit), based on the experimentally observed complexity [67] of CNT cable architectures; (4) application of the modeling approach to graphene [50]; and (5) the development of improved computational methods for both equilibrium calculations and ballistic conduction calculations, an essential enabler if computational research is to keep pace with experimental work on the increasingly complex cable nanostructures, doping systems, and fabrication processes of interest.

Acknowledgment

Computer time support was provided by the Texas Advanced Computing Center at the University of Texas at Austin.

Funding Data

• Office of Naval Research Global (Grant No. N00014-15-1-2693).

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