Atomistic Modeling of Charge Transport across a Carbon Nanotube

Article pubs.acs.org/JPCC

Atomistic Modeling of Charge Transport across a Carbon Nanotube− Polyethylene Junction Gabriele Penazzi,*,† Johan M. Carlsson,‡ Christian Diedrich,§ Günter Olf,§ Alessandro Pecchia,∥ and Thomas Frauenheim† †

Bremen Center for Computational Materials Science, Universität Bremen, Am Fallturm 1, 28359 Bremen, Germany Accelrys Ltd, Cambridge Science Park 334, Cambridge CB4 OWN, U.K. § Bayer Technology Services GmbH, Chempark Leverkusen, D-51368 Leverkusen, Germany ∥ CNR-ISMN, Via Salaria km 29.300, 00017 Monterotondo, Roma, Italy ‡

S Supporting Information *

ABSTRACT: The conduction mechanism in carbon nanotube (CNT) polymer nanocomposites is complex, and there has been a considerable amount of work invested in understanding the role of the distribution of the CNTs in the composite and how it influences the conductivity. However, less interest has been devoted to the electron transport across a single CNT−polymer−CNT junction. We present a first atomistic study of the electron transmission through a CNT−polyethylene−CNT junction. The morphology of the junction is described using classical molecular dynamics simulations, and transport properties are calculated within density functional tight binding method. The electron transmission depends noticeably on the CNT−CNT separation and on the consequent polymer wrapping. At CNT−CNT distances shorter than 6 Å, the polyethylene molecules do not penetrate in the space between the CNTs. In this near contact regime, the electron transmission proceeds via direct tunneling between the two CNTs across a vacuum region without relevant contribution from the surrounding polymer. For distances larger than 6 Å, the PE molecules enter into the junction region. The frontier orbitals of the PE molecules in the junction provide localized states, which can couple to the CNT metallic states. This resonance tail increases the electron transmission probability between the CNTs across the junction by several orders of magnitude, thus lowering the effective barrier. The gradual interpenetration of the polymer is resembled in transmission fluctuations. An averaging of the transmission in energy and time along MD trajectories allows a quantitative estimation of the junction resistance and tunneling barrier.

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INTRODUCTION Since their first systematic characterization,1 carbon nanotubes (CNTs) have been considered as the ultimate carbon fibers because of their remarkable mechanical, electrical, and thermal properties.2 This has made CNTs the most studied nanomaterials in their own right, but CNTs have also been suggested as an excellent filler for next generation polymer composites.3 The first CNT/polymer blend or carbon nanotube reinforced polymer (CNRP) was reported in 1994.4 In the following years, mechanical, thermal, and electrical properties of CNRPs have been extensively investigated.3−14 These studies included a wide range of polymer matrixes and used a variety of preparation and CNT dispersion techniques. A particular interesting effect of the addition of CNTs to the polymer matrix is the formation of conductive nanotube−polymer composites. Since CNTs can be good conductors ((2−20) × 107 S/m),3 CNRPs can exhibit dramatic improvements in the electrical conductivity compared to that of the pristine polymers. Coleman et al. were the first to show this effect in 1998, improving the conductivity of a poly(m-phenyleneviny© XXXX American Chemical Society

lene-co-2,5-dioctoxy-p-phenylenevinylene) (PmPV) matrix up to 10 orders of magnitude by adding 8% CNTs.5 This astonishing effect has been attributed to electronic percolation via the CNTs in the composite. The term percolation here refers to the fact that conduction paths with low resistivity open up in the composite if the CNTs are dispersed and ordered properly. The conductivity of the nanotube polymer composites is determined by two factors: at the microscopic level by the charge transfer between adjacent carbon nanotubes and at the macroscopic level by the topology of the percolation network determined by the distribution of the CNTs in the composite. The distribution of the CNTs in the composite can be controlled by varying the dispersion techniques in the manufacturing process.6 This has led to a large experimental effort to investigate different dispersion techniques to optimize the distribution of the CNTs in the polymer matrix, with the Received: December 16, 2012 Revised: March 25, 2013

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dx.doi.org/10.1021/jp312381k | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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the junction certainly play an important role in real systems. A slow degree of motion plays a huge role in electronic properties of soft matter, and schemes which apply a sampling of transport properties along MD trajectories have applied, for instance, to molecular junctions17 and DNA.18 In this work, we present a parameter-less calculation of the current through a single CNT−polymer junction averaged on realistic atomistic geometries. Junction conductivities are calculated taking into account the atomistic detail of the CNT−polymer−CNT configuration, the finite temperature fluctuation dynamics of the polymer matrix, and the proper quantum mechanical treatment of the electron transport. We have used a combination of classical atomistic molecular dynamics (MD) and density functional based tight binding (DFTB) transport calculations to calculate the electron transmission between two CNT tips embedded in a polymer matrix. The main approximation in this approach is that the dynamics of the electrons is much faster than the dynamics of the atoms in the systems, such that the electron transmission and the atomic motion can be decoupled. The atomic geometry of the CNT−polymer−CNT junction can then be calculated by classical MD, and the electron transmission can be calculated from a snapshot from the MD trajectory without a feedback coupling from the electronic system to the atomistic movements. We have used polyethylene (PE) as a prototype low conductivity polymer to easily separate the contributions of the CNTs and the polymer to the overall conductivity across the junction. The CNTs are randomly oriented in a real composite, such that the CNT−CNT junction may take any configuration, but we chose a junction geometry where the two tips were aligned directly toward each other, as can be seen in Figure 1. This junction geometry provides a well-defined CNT−CNT distance and a simple electron transfer path with just a single transmission channel. Snapshots from the MD trajectories were then used to calculate the electron transmission across the CNT−PE−CNT junction. The large scale of even a single CNT junction in the composite, which required between 5000 and 6000 atoms, unfortunately made full ab initio transport calculations out of reach. Density functional tight binding (DFTB) based transport calculations provide instead a good alternative, as the DFTB method can treat much larger systems at comparable accuracy as DFT methods. The transport theory in the DFTB formalism is summarized in the Theory section below.

goal to minimize the CNT concentration needed to obtain percolation and maximize the conductivity enhancement. Most theoretical investigations of CNT−polymer composites have also been concentrated on the effect of the network topology. Several authors have modeled the macroscopic conductivity in a CNT−polymer composite using Monte Carlo methods.7−10 The focus has either been studying the effects of CNT waviness7 or the CNT alignment.8 Furthermore, knowledge of the percolation network topology allows one to relate the overall conductivity of the CNT−polymer composite to the contact resistance.9,10 However, a detailed modeling of the electron transfer process between and within individual CNTs in the composite has achieved less attention in spite of its fundamental importance for the microscopic understanding of the overall conductivity of CNT−polymer composites. Different blends can exhibit different behaviors. Fluctuation induced tunneling (FIT)11 and three-dimensional12 and one-dimensional13 variable range hopping have been observed. It has been proposed that the leading mechanism will change whether the resistance between individual CNTs of the resistance of a single CNT inside bundles dominates;12 i.e., coherent tunneling (FIT) is related to transport across CNT−CNT and CNT−polymer−CNT junctions. These ambiguities are due to the structural complexity of the composite. The electron transfer depends on the combination of both macroscopic characteristics of the composite like the orientation of the adjacent CNTs and the connection to the polymer matrix on one hand and atomistic characteristics of the CNTs like defect concentration and functionalization on the other hand. A first attempt to describe the junction tunneling resistance has been described by Li et al.;10 the authors apply the analytical Simmons model15 in the intermediate voltage regime. However, Bao et al.16 already pointed out that the hypotheses underlying the Simmons model, for instance, the assumption that the junction can be modeled as a pair of plane electrodes separated by an homogeneous insulating film, are questionable in this context. According to the Simmons model, the tunneling resistance approaches a zero value when the distance between the CNTs approaches zero, while in reality the lower possible value is related to the quantum unit of resistance, with significant variations depending on the CNT chirality and relative position.14 Bao et al.16 start from a more justifiable Landauer model assuming that the transmission decrease exponentially with the CNT distance, according to Wentzel− Kramers−Brillouin (WKB) solution to the Schrö dinger equation for a rectangular barrier. In metal−insulator−metal tunneling theory, the height of the barrier depends on the difference between metal and insulator work function and can be estimated to be between 1.0 and 5.0 eV10 for polymers commonly employed. In real junctions, an effective value of the barrier is often used to include deviations from the ideal WKB behavior. These deviations can be due to the non-rectangular barrier shape, the morphology of the insulator (crystalline or amorphous), and atomistic details at the junction such as contact roughness, surface states, or molecular alignment. While in mesoscopic junctions the effective barrier can be calculated from tunneling current measurements, a single CNT−polymer junction is not experimentally accessible, the impact of atomistic details is not easily quantified by theory nor by experiment, and the barrier height is considered an unknown variable parameter. Variations in the transmission across the junction due to the geometrical configurations of the CNTs at

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THEORY

The density functional tight binding (DFTB) method19 is a scheme to calculate parameters suitable for a non-orthogonal tight binding representation of single particle wave functions starting from reference DFT calculations. Neglecting the threecenter integrals and using a LCAO ansatz, ϕi = ∑v ϕv(r − Rv) Cvi, the Kohn−Sham secular equation can be written as ∑ν HμνCvi = ∑νSμνCviεi, where Hμν = ⟨ϕμ|Ĥ |ϕv⟩ and Sμν = ⟨ϕμ|ϕν⟩ are the hopping matrix elements (if μ ≠ ν), the on-site energies (if μ = ν), and the overlap matrix elements. The basis functions ϕ(r) of the LCAO expansion are calculated with a single atom DFT Kohn−Sham equation: ⎡ ⎛ ⎞2 ⎤ ⎢T̂ + V (n(r)) + ⎜ r ⎟ ⎥ϕ (r) = ε ϕ (r) eff ν ν ⎢⎣ ⎝ r0 ⎠ ⎦⎥ ν B

(1)



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used to optimize the basis set and a value of r0 = 2·rcov, where rcov is the covalent radius of the atom, is generally considered a reliable choice to obtain an accurate description of band structures and molecular orbitals.19 However, it has been pointed out20 that such a compression potential is suppressing wave function tails and can lead to a severe underestimation of tunneling between unbonded molecules. Available carbon− hydrogen minimal sp parametrization19,21 assumes a narrow compression potential, to ensure portability. We have instead calculated the basis functions using a larger compression potential radius of 8.0 au, as previously suggested by other authors.20 Tunneling through polymer barriers is expected to be the transport process limiting the composite conductance when CNT wrapping is present.6 Charge transfer to the polymer can be safely neglected, as the LUMO level of PE lies above the vacuum level; therefore, charge transport through the junction occurs via coherent tunneling events through a CNT− polymer−CNT junction. We evaluate the resulting contact resistance on a system composed by two identical semi-infinite metallic CNTs as leads and a central junction region composed by two CNT tips embedded in an amorphous polyethylene matrix, as shown in Figure 1. This partitioning produces a block-matrix form of the DFTB Hamiltonian, from which it is possible to compute the two-terminal conductance of the junction. In the system of interest, the junction conductance can be evaluated as

G=

2e 2 ̃ T (μ)|μ= Ef h

(2)

where EF is the Fermi level of the semi-infinite CNT acting as a lead and T̃ (μ) is the thermally averaged transmission:22 T̃(μ) =

+∞

∫−∞

∂f (E , μ) T (E ) d E ∂E

(3)

f(E, μ) is the Fermi−Dirac distribution function, and T(E) is the transmission coefficient calculated with the Landauer/ Caroli formula23 (energy dependence is implicitly considered): T = Tr[Γ LGrDΓ R GaD]

where

GDr,a

(4)

is the Green’s function of the device

GrD, a = [ESD − HD − ΣrL, a − ΣrR, a]−1

Σr,a L,R are ΣaL,R] is

(5)

the self-energies of the contacts and Γ = − the effective coupling between device and contact region. Due to torsional deformation, a small energy gap can open in metallic CNTs.24,25 Therefore, the inclusion of thermal averaging in eq 2 is important, as the transmission is not always a smooth quantity on a room temperature energy range. In a tight binding representation, an explicit expression for the local transmission on every bond can be derived.26,27 In a system with identical probes, i.e., ΓL,R = Γ, we can write the local transmission on a bond as (discussion available in the Supporting Information)

Figure 1. Side view (a) and front view (b) of the CNT junction, formed by the capped (10,10) SWCNT embedded in the amorphous polyethylene (PE) matrix. The view along the CNT axis reveals that the PE molecules are repelled from the CNT surface and form an accumulation layer around the CNT. Note that the PE molecules that are visible in the vacuum region are actually located in the region between the two tips.

where Veff includes Coulomb, Hartree, and exchangecorrelation (XC) terms. The basis functions and densities are then used to compute overlap and Hamiltonian matrix elements in a two-center approximation. The results of the calculation are stored in lookup tables; therefore, no further integral evaluation is needed, making the computational effort comparable to empirical methods. The last term in eq 1 is a compression potential, where r0 is a parameter, which determines the confinement of the basis function around the reference atoms. The r0 parameter can be

Tmn =

L,R

i[ΣrL,R

∑ ∑ ∑ i(HμνGrνkΓklGlaμ − HνμGrμlΓlkGakν) μ ∈ m ν ∈ n kl

(6)

It must be pointed out that, due to the thermal fluctuations and the inherently disordered nature of the amorphous polymer, we aim to describe a meaningful statistics of the contact resistance along the MD trajectories, while we assume that the electron transfer in the junction is an adiabatic process. C



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removed fluctuations in the lead Hamiltonian ensured that the Fermi level and the band structure of the injecting contacts were constant and all the fluctuation effects were transferred to the Hamiltonian of the device region. The transmission coefficients for each MD snapshot were calculated using the Green’s function extension26,29 to the DFTB+ code30,31 presented in the Theory section. About 25 000 orbitals are needed to describe the electronic structure of a single snapshot. An iterative Green’s function scheme is applied to minimize the computational time.

Within this assumption, the electron dynamics are completely decoupled from the ion dynamics, and the transmission coefficients can be sampled along the snapshots of the MD trajectory. This assumption holds as long as the time scale of the charge transfer process, which is in the order of femtoseconds in the case of coherent tunneling, is much shorter than the time scale of the CNT−polymer junction dynamics. The details of the MD simulations and the DFTB transport calculations are described below.

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METHOD The MD simulations and the transport calculations require a different type of boundary conditions, so it is necessary to generate CNT-junction geometries that are compatible with both the periodic boundary conditions used in the MD simulations and the open boundary conditions required for the transport calculations. This was achieved by placing a capped (10,10)-armchair nanotube, which contained 580 carbon atoms, along the c-axis in a supercell using Materials Studio.28 The periodic boundary conditions were used to make the two ends point directly toward each other, forming a CNT−CNT junction in the middle of the supercell, as can be seen in Figure 1. The size of the supercell parallel to the c-axis was varied to generate CNT−CNT tip distances between 3 and 10 Å. The Amorphous cell module in Materials Studio was used to pack the polyethylene (PE) molecules, forming the amorphous polymer matrix around the CNT. The PE molecules had a molecular weight of 2022 Da and the number of PE molecules varied between 10 and 13 molecules, depending on the length of the supercell parallel to the tube axis. The equilibrium density for the amorphous polyethylene melt, obtained by a separate bulk calculation for the PE melt at T = 298 K, was 0.8 g/cm3. The generated supercells were equilibrated via a MD cycle using the COMPASS force field implemented in the Forcite module in Materials Studio. 28 The equilibration cycle comprised a combination of geometry optimization and molecular dynamics. The MD is run in NPT ensemble, keeping the number of molecules, pressure, and temperature constant until the CNT−polymer composite in the supercell had reached its equilibrium density at T = 298 K and an external pressure of 1 atm. The Berendsen barostat was used to control the pressure and the Nosé−Hoover thermostat was used to control the simulation temperature during the NPTMD. After the equilibration, MD simulations were run in the NVT ensemble, keeping the number of molecules, volume, and temperature constant in order to sample temperature induced fluctuations in the CNT−PE−CNT junction. Figure 1 provides an example of a snapshot of the CNT−PE−CNT junction. Eight MD simulations were performed for CNT separations between 3 and 10 Å. The atomic positions at the CNT junction were sampled every picosecond, and each snapshot was used for the DFTB-transport calculation. Each frame of the MD trajectory was converted to be consistent with the open boundaries needed in the transport calculations. Fixing the central layer of carbon atoms in the CNT during the MD simulations facilitated this process, as it ensured that three important conditions were met. The fixed atoms kept the CNT aligned along the z-axis, which enabled a precise control of the tip-to-tip distance. The fixed atoms also provide a reference structure to build the atomic models of the two infinite CNT leads necessary for the calculation of the lead self-energies described in the previous section. Finally, having

RESULTS We have first analyzed the molecular dynamics simulations focusing on the interaction between the polyethylene matrix and the sidewalls of CNTs in order to understand the penetration of the PE molecules into the region between two adjacent CNTs in the composite. The view along the CNT axis in Figure 1 shows that the PE molecules are repelled by the CNT wall and form an accumulation layer separated by a vacuum region. The radial distribution function (RDF) for the PE−CNT distances in Figure 2a has a cutoff at 3 Å, which indicates that the PE molecules would not get closer than 3 Å to the CNT sidewall. We have also performed a similar type of MD simulations for the interface between a PE matrix and graphite as a reference. The density profile perpendicular to the graphite surface in Figure 2b shows that the PE molecules form

Figure 2. (a) The radial distribution function (RDF) for intermolecular distances between the polyethylene molecules (PE− PE) and the intermolecular distances between the polyethylene molecules and carbon atoms in the CNT sidewall (PE−CNT). (b) The density profile at the graphite−PE interface. The reference value for the z-axis, zG, is the position of the outermost graphene layer in the graphite slab. The PE molecules are repelled by the graphite surface forming an accumulation layer, which is separated by a vacuum region and where the most dense part is ca. 4 Å from the graphite surface. D



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a similar accumulation layer next to the graphite surface at a distance of 4 Å. The origin of the formation of the accumulation layer is the atomic repulsion between the PE molecules and the carbon atoms in the CNT wall or graphite surface. The atomic repulsion between the PE molecules and the CNT side wall has an important implication for the penetration of the PE molecules into the region between two CNTs in the composite. The PE molecules keep a distance longer than the repulsion length, roughly approximate as 2 times the carbon van der Waals radius, from either of the CNTs at a CNT−CNT junction, such that the PE molecules are repelled from the interior region between the CNTs until the CNT−CNT separation is larger than 2 times the PE−CNT repulsion distance. The repulsion of the PE molecules at the CNT junctions defines a lower limit of 6 Å for the penetration of the PE molecules between the CNTs and leaves a vacuum region between the CNTs at separations shorter than the cutoff distance. The behavior of the PE molecules divides the CNT junctions into two distinct regimes that may influence the electron transport between the CNTs in the composite. In the near contact regime, defined as CNT−CNT separations between the direct contact between two CNTs (CNT separation around 3.35 Å) and up to the cutoff distance, it is likely that the electron transmission proceeds by direct tunneling across the vacuum region between the two CNTs. In contrast, in the far contact regime or the polymer coating regime, where the CNT−CNT separation is larger than the cutoff distance, it may be assumed that the PE molecules located between the CNT tips participate in the tunneling processes, thus lowering the barrier. The PE molecules and the CNTs are performing significant thermal movements, even at T = 298 K, leading to continuously changing geometries at the CNT junction. This dynamical behavior of the atoms at the junction confirms the need for atomistic molecular dynamics to generate realistic models of the CNT-junction geometries as input for the transport calculations, as a single snapshot can hardly be representative due to the amorphous nature of the polymer matrix. Figure 3 shows the calculated electron transmission across the CNT junction sampled on MD snapshots for three different CNT− CNT separations. The contribution of the polymer is underlined in the following way: we separate the Hamiltonian of the CNT−PE− CNT junction into three different contributions, such that (7)

Figure 3. Transmission coefficients as a function of time calculated along the MD trajectories at three different tip distances: (a) 3, (b) 5, and (c) 6 Å. The transmission calculations are based on a full Hamiltonian (SWCNT and PE - continuous line) or just including the electronic states of the SWCNT (dashed line).

where HCNT and HPE are the Hamiltonian in the Hilbert subspace including only the SWCNT or the PE basis wave functions. V CNT−PE is the coupling between the two contributions in the complete system. The transmission coefficients were calculated for the two different systems, either the complete CNT+PE system or just the CNT subsystem, for which HJunction = HCNT in the latter case. Figure 3a shows that the contribution from the PE molecules is negligible in the near contact regime, as exemplified at a distance of 3 Å. Both the magnitude and the dynamics of the fluctuations of the transmission coefficients are completely described by the states of the CNT itself. This is not completely true for a CNT separation of 5 Å in Figure 3b; however, the transmission rates are in the same order of magnitude. We can therefore conclude that the tunneling rate is determined by direct coupling

between the electronic states of the carbon nanotubes in the near contact regime. Figure 3a shows that the transmission coefficients of the full CNT+PE system are about 3 orders of magnitude higher than the transmission coefficient of the pure CNT system at a CNT−CNT distance of 6 Å. The peaks of the pure CNT− CNT and CNT−PE−CNT transmission coefficients also lose correlation. This indicates that the PE molecules are partly penetrating the CNT junction at this CNT−CNT separation and contribute to the electron transmission. The transmission coefficients for the direct coupling between the CNT tips at CNT separations larger than 7 Å are negligible, due to parametrization cutoff. In the intermediate region, where the CNT separation is in the range between 6 and 7 Å, the PE molecules are not steadily penetrated in the CNT junction and

HJunction = HCNT + HPE + VCNT − PE

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regime corresponds to 3.0 eV. Fluctuations with respect to the exponential fit depend on the penetration of PE strands, which occurs by discrete steps rather than resulting in a continuous barrier length. The shortest separation between the CNTs in our simulations is around 3 Å. Shorter distances correspond to mechanical contact with applied forces, a scenario which has been investigated by previous authors.14 The maximal transmission coefficient at this separation depends on atomistic details such as SWCNT diameter and relative alignment. These effects has been studied by others14 applying a similar Green’s function scheme for ground state geometries, and the transmission can differ by 2 orders of magnitude. Whether these effects are still important in a CNT−polymer junction is still to be investigated within the proposed approach. In Figure 5, we show an atomically resolved local transmission through the junction. For nearly touching CNTs, the surrounding PE molecules in the composite matrix do not contribute significantly to the electron transmission. Figure 5b shows furthermore that in the far contact regime only the PE atoms aligned exactly in between the CNT tips have a significant local transmission coefficient. The PE molecules located in the CNT junction provide localized bridge states, which couple the orbitals of the two CNT tips. The electron transmission is strongly affected by the tail of the resonance between the electronic states of CNTs with those PE molecules, which are penetrating in between the CNTs in the junction. The coupling between the CNTs and PE molecules is raising the transmission coefficient by several orders of magnitude, lowering the effective barrier. A connection to available experimental data is not trivial, as the experimental values refer to the overall conductance of the whole composite rather than the conductance between two SWCNTs embedded in a polymer matrix. Quinn et al. used high resolution transmission electron microscopy (HRTEM) measurements to confirm that the nanotubes get easily coated by the polyethylene matrix. The authors reported a conductance of 10−4 S/m for the polyethylene carbon nanotube composite.33 Comparison of experimental and theoretical values for a model based on tunneling led to the same order of magnitude for the transmission, which can be considered as a confirmation that the transport mechanism was dominated by tunneling through the insulating polymer barrier in this class of materials.6 The relationship between the overall conductance and the single contact resistance depends on topological details, and different authors have addressed this problem using Monte Carlo calculations.8−10 According to the model of Li et al.,10 the overall conductance is proportional to the single contact conductance. They connected an overall conductance of 10−4 S/m to a contact resistance of 1012 Ω. An independent result by Foygel et al.9 associates a contact resistance of 1013 Ω to an overall conductance of 10−8 S/m, an experimental value measured in a polyamide SWCNT composite.34 These values correspond, in our calculation, to average transmission values for a CNT−CNT distance between 8.0 and 9 Å, in the polymer barrier regime. Even though the actual CNT configuration will influence the junction resistance, allowing for significant deviations, the order of magnitude of the conductance seems to be consistent with measured composites. Fine details given by the particular topology will probably be smoothed by the averaging along the MD trajectory, and hopefully, a statistical ensemble of junction can be realized to increase predictive validity.

we observe a nontrivial behavior. For distances higher than 8 Å, polymer strands are steadily filling the junction and the transmission decreases almost exponentially. The magnitude of the fluctuations of the transmission coefficient is also related to the tunneling regime. For short distance, fluctuations are due to small variations in the alignment of the tips and modifications of single tip properties due to torsional deformation. For large distance, larger fluctuations are due to the dynamics of PE strands inside the junction and, consequently, alignment of frontier orbitals. The average network resistance in a percolation network of fluctuating resistances, such as a CNT−PE composite, is mainly depending on parallel connections.32 The fluctuations in parallel connections are added incoherently; therefore, the average contact conductance for each individual CNT junction is a meaningful quantity to present. The contact conductance can be written as GC = Tc × G0, where G0 = 2e/ℏ = 77.52 μS. Taking the average gives ⟨GC⟩ = ⟨TC⟩ × G0. The contact conductance is given by eq 2, but it is necessary to be careful when calculating the averages, as different results can be obtained by averaging the contact resistances and the contact conductances, as ⟨RC⟩ ≠ ⟨GC⟩−1. In either case, the large fluctuations require averaging long MD trajectories to smooth out the fluctuations to obtain well converged average values of the transmission probabilities. The average transmission coefficients in Figure 4 are obtained by averaging over 100 ps of the MD trajectory at

Figure 4. Average, minimum, and maximum transmission coefficients across the CNT−PE−CNT junction as a function of CNT−CNT separation and fitting of the average transmission with a single and double barrier WKB model.

each CNT−CNT separation. The overall transmission has an exponential-like decay; therefore, we can define an effective barrier by fitting with the WKB approximation for a rectangular barrier: T = exp[−(d/ℏ)(8meΔE)1/2], where d is the barrier width, me the free electron mass, and ΔE the barrier height. The fit is shown in Figure 4 (single barrier fit) and leads to ΔE = 2.2 eV. However, a better fit is obtained if we distinguish two different tunneling regimes corresponding to CNT−CNT distances d < 6 Å and d ≥ 6 Å. As previously stated, the two regions represent a close contact regime, where no PE molecules are penetrating the space between the CNTs, and a far contact regime with polymer coating. In fact, a better fit of the average transmission is obtained considering two distinct barriers (see double barrier fit in Figure 4) reflecting the close and far contact regimes. The barrier height for the far contact F



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resonance between the tails of the CNT wave functions and the frontier orbitals in the PE molecule located in between the CNTs is enhancing the tunneling, leading to a CNT−PE− CNT barrier of 3.0 eV and thus lowering the total effective barrier to an estimated value of about 2.2 eV. The direct measurement of the contact resistance between two adjacent CNTs in a CNT−polymer composite is not possible, but the equivalent contact resistances that are obtained by our calculations can compare with the order of magnitude expected from experimental data, suggesting a semiquantitative description of the junction resistance with atomistic detail is computationally possible. The simulation scheme can be generalized to a wide range of insulating polymer composites. A sampling and averaging along MD trajectories and the correct thermal averaging of transmission rates is needed in order to represent a meaningful statistic, as large fluctuations occur depending on the actual geometry of CNTs and amorphous polymer. The model goes beyond current analytical models used up to now to estimate the junction resistance, allowing one to include atomistic details such as different CNT chirality or functionalization.

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ASSOCIATED CONTENT

S Supporting Information *

The derivation and conservation of local transmission within the DFTB approach are discussed. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 421 218 62337. Fax: +49 421 218 62770. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Iijima, S. Helical Microtubules of Graphitic Carbon. Nature 1991, 354, 56−58. (2) Dresselhaus, M. S.; Dresselhaus, G.; Avours, P. Carbon Nanotubes: Synthesis, Structure, Properties and Applications; SpringerVerlag: Berlin, 2001. (3) Mittal, V., Ed. Polymer Nanotube Composites: Synthesis, Properties and Applications; Scrivener: Salem, MA, 2010. (4) Ajayan, P. M.; Stephan, O.; Colliex, C.; Trauth, D. Aligned Carbon Nanotube Arrays Formed by Cutting Polymer ResinNanotube Composite. Science 1994, 265, 1212−1214. (5) Coleman, J. N.; Curran, S.; Dalton, A. B.; Davey, A. P.; McCarthy, B.; Blau, W.; Barklie, R. C. Percolation-Dominated Conductivity in a Conjugated-Polymer-Carbon-Nanotube Composite. Phys. Rev. B 1998, 58, R7492−R7495. (6) Bauhofer, W.; Koyacs, J. K. A Review and Analysis of Electrical Percolation in Carbon Nanotube Polymer Composites. Compos. Sci. Technol. 2009, 69, 1496−1498. (7) Lia, C.; Chou, T.-W. Continuum Percolation of Nanocomposites with Fillers of Arbitrary Shapes. Appl. Phys. Lett. 2007, 90, 174108. (8) Du, F.; Fischer, J. E.; Winey, K. I. Effect of Nanotube Alignment on Percolation Conductivity in Carbon Nanotube/Polymer Composites. Phys. Rev. B 2005, 72, 121404(R). (9) Foygel, M.; Morris, R. D.; Anez, D.; French, S.; Sobolev, V. L. Theoretical and Computational Studies of Carbon Nanotube Composites and Suspensions: Electrical and Thermal Conductivity. Phys. Rev. B 2005, 71, 104201. (10) Li, C.; Thostenson, E. T.; Chou, T.-W. Dominant Role of Tunneling Resistance in the Electrical Conductivity of Carbon Nanotube-Based Composites. Appl. Phys. Lett. 2007, 91, 223114.

Figure 5. Volume plot of the local transmission coefficient across the CNT−PE−CNT junction for a CNT−CNT distance of 3 Å (a) and 8 Å (b). The local transmission coefficient was calculated as Tm = ∑n|Tmn|, where Tmn is given by eq 6. The atom radius is proportional to the value of the local transmission on the particular atom.

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CONCLUSIONS We have used a combination of classical atomistic MD simulations and DFTB transport calculations to get a detailed understanding of the relationship between the polymer configurations at a CNT−polymer−CNT junction in a CNT−polymer composite and the electronic transmission through the junction. Our calculations for a CNT−polyethylene (PE)−CNT junction reveal that we can separate two tunneling regimes. The near contact regime is characterized by the inability of the PE molecules to penetrate in the space between the CNTs at separations shorter than 6 Å. The electron transmission proceeds via direct tunneling between the CNTs for separation between the direct contact distance of 3.0 Å and the critical CNT distance of around 6.0 Å. As the PE molecule penetrates in between the CNTs at CNT−CNT separations longer than the critical distance of 6.0 Å, the G



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Atomistic Modeling of Charge Transport across a Carbon Nanotube

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