NANO LETTERS
Effect of the Chemical Functionalization on Charge Transport in Carbon Nanotubes at the Mesoscopic Scale
2009 Vol. 9, No. 3 940-944
Alejandro Lo´pez-Bezanilla,† Franc¸ois Triozon,‡ Sylvain Latil,§ X. Blase,| and Stephan Roche*,† CEA, Institut of Nanosciences and Cryogenics, INAC/SPSMS/GT, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France, CEA, LETI-Minatec, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France, CEA-DSM/IRAMIS/SPCSI, CEA-Saclay, 91191 Gif-sur-YVette, France, and Institut Ne´el, CNRS and UniVersite´ Joseph Fourier, B.P. 166, 38042 Grenoble Cedex 09, France Received September 15, 2008; Revised Manuscript Received January 6, 2009
ABSTRACT We present first-principles calculations of quantum transport in chemically functionalized metallic carbon nanotubes with lengths reaching the micrometer scale and random distributions of functional groups. Two typical cases are investigated, namely, a sp2-type bonding between carbene groups (CH2) and the nanotube sidewalls and a sp3-type bonding of nanotubes with paired phenyl groups. For similar molecular coverage density, charge transport is found to range from a quasi-ballistic-like to a strongly diffusive regime, with corresponding mean free paths changing by orders of magnitude depending on the nature of the chemical bonding.
Owing to their unique physical properties,1-4 carbon nanotubes (CNTs) stand as exceptional materials for the exploration of new quantum phenomena in low dimensionality or to envision innovative applications. In particular, the (bio)chemical sensing capability of carbon nanotube or nanowire based devices has been the focus of many recent studies with possible applications in the field of nanobioelectronics.5-8 The modifications of carbon nanotubes transport properties can indeed evidence molecular adsorption events onto the tube sidewalls.9-11 In turn, nanotube functionalization has become a genuine way for tailoring CNTs electronic properties or providing to the device a novel functionality. For instance, by grafting photoactive molecules onto the nanotube sidewalls, the resulting nanotube-based devices can be optically controlled.12,13 Two types of chemical functionalization have been investigated, namely, physisorption and chemisorption. At the moment, it is not clear which method allows the best compromise between a weak degradation of intrinsic nanotube properties upon tube functionalization and the required efficient monitoring of the added nanotube functionality.11 As demonstrated by Latil and co-workers,14 the scattering efficiency that results from physisorbed molecules onto the nanotube surface is vanishingly small, owing to the weak interaction between both systems. †
CEA, Institut of Nanosciences and Cryogenics, INAC/SPSMS/GT. CEA, LETI-Minatec. § CEA-DSM/IRAMIS/SPCSI, CEA-Saclay. | Institut Ne´el, CNRS and Universite´ Joseph Fourier. ‡
10.1021/nl802798q CCC: $40.75 Published on Web 02/03/2009
2009 American Chemical Society
Noncovalent adsorption has therefore the advantage of enabling CNT functionalization while still preserving their electronic structure, since the original sp2 hybridized bondings and aromaticity remain unaltered, to the prize of a further weaker direct control of nanotubes conductance.15 In contrast, covalent functionalization of CNT involves the formation of saturated sp3 bonds breaking the pz network symmetry of the nanotube surface,16,17 which suggests a stronger alteration of clean nanotube properties. This particular difference on the electrochemical nature of the bonding turns out to be critical for the preservation of good transport properties over large distances. The impact of covalent functionalization on the tube conductance can however be significantly reduced by a suitable choice of the addends. A transport study based on a nonorthogonal tightbinding Hamiltonian by Park and co-workers18 reported strong differences between monovalent and divalent additions in short length nanotubes. Using ab initio calculations, Lee and Marzari19,20 further demonstrated that cycloaddition reactions leading to the grafting of, e.g., dichlorocarbene groups (CCl2), could preserve most of the conductance of C(5,5) metallic nanotubes, in contrast to phenyl-type functionalization that would result in strong damping of conduction ability. These calculations were however limited to a short nanotube segment with length below 50 nm. To date, the relative impact of covalent sp3 versus sp2-type functionalization on realistically long nanotubes remains to be
Figure 1. Top panel: Atomic structures of the building blocks which are the starting point for the study of longer structures for both kind of functionalization: (a) paired sp3 phenyl groups and (b) sp2 divalent addition of carbene groups.40 The bottom panel (c) represents a CNT with paired phenyl groups functionalization obtained by assembling individual sections, including translational and rotational disorders, as well as external leads (pristine semi-infinite nanotubes).
explored in-depth. In particular the clarification of the transport regime and extraction of transport length scales would be highly suitable for a quantitative understanding of experiments.21 In this Letter, we present a first principles computational study of charge transport in metallic single-walled CNTs with random distribution of paired phenyl groups and carbene functional groups bonded to the tube sidewalls. The disorder introduced by the grafted groups breaks both translational and rotational symmetries, and the study is performed for long nanotubes from a few hundred nanometers to the micrometer scale. The conductance changes and related conduction regimes (from quasi-ballistic to strongly diffusive) are investigated as a function of both incident electron energy and functional group coverage density. Our first principles mesoscopic transport study on disordered nanotubes allows us to go beyond prior work on short nanotubes19,20 and shows that carbene cycloaddition preserves ballistic conduction up to the micrometer scale, whereas the grafting of paired phenyl groups yields mean free path in the nanometer scale, leading to strong localization regime. Our computational approach is based on the following strategy.22-25 A large set of first principles calculations are first performed to obtain the ab initio Hamiltonian and overlap matrix associated with small tube sections functionalized by single groups. Such a set of Hamiltonian building blocks further allows the reconstruction of a micrometer long Hamiltonian tube formed by a random succession of functionalized sections and pristine tube portions to mimic rotational and translational disorder (see Figure 1 for illustration). The standard techniques to calculate the nondiagonal elements of Green’s function associated with the Nano Lett., Vol. 9, No. 3, 2009
sparse Hamiltonian are then used to include recursively the contribution of additional sections, leading to an order (N) scheme with respect to tube length. This scheme is now welldocumented and yields an efficient first-principle calculation of the Landauer-Bu¨ttiker conductance in the coherent regime.26,27 A detailed analysis of the CNT electronic structure is first instructive for understanding the interplay between functional groups and the nanotube backbone. To that end, following ref 28, self-consistent electronic structure calculations and functionalization-induced atomic relaxations are performed using the SIESTA code, which implements a density functional method by means of a numerical linear combination of an atomic-like orbital basis set.30,31 The calculations are performed using the local density approximation for the exchange-correlation term with Perdew and Zunger32 parametrization. Standard norm-conserving Troullier-Martins pseudopotentials33 are used to describe the interaction between ionic cores and localized pseudoatomic orbitals. Split-valence double-ζ basis34 ensures a good computational convergence with respect to the basis set. When the small block Hamiltonian associated with one scattering center is built, periodic boundary conditions are used with one functional group per unit cell. The length of the building block is chosen such that geometric and energetic perturbations induced by functional groups vanish as we reach its edges.29 Atomic positions are relaxed up to residual atomic forces smaller than 0.02 eV/Å. Once the Hamiltonian (H) and overlap (S) matrix have been extracted from the optimal geometry, transport calculations are achieved as follows (see ref 27 and citations therein). The total system is made by two semi-infinite pristine CNTs, acting as leads, to which the region of interest is coupled. The conductance 941
Figure 2. Left panel: Conductance (in units of G0 ) 2e2/h) of a 300 nm long C(10,10) nanotube functionalized with paired phenyl groups. The conductance has been averaged over 40 different random configurations. Together with the case of pristine nanotube (dashed lines), the averaged conductance is shown for an increasing number of grafted groups to the surface. Right panel: Same as in left panel, but for a 1000 nm long nanotube functionalized with divalent addition of carbene groups. All insets give the same information but for the (6,6) nanotubes.
can be computed on the basis of Landauer’s theory for transport G(ε) )
2e2 Tr(ΓLG rΓRG a) h
(1)
where G is the conductance expressed in terms of Fisher and Lee formula.25 G r(a) is the retarded (advanced) Green’s r function of the functionalized CNT, while ΓR(L) ) i(ΣR(L) a ΣR(L)) takes into account the coupling to the right (left) leads through their self-energy expression ΣR(L). Considering the nonorthogonal nature of the basis set, G is defined by G () ) (S - H)- 1 with ) E ( iη, η being an arbitrary small quantity. The crucial point in determining transport properties of the system is the calculation of its Green’s function for each value of energy E, taking into account the influence of electrodes. This is done by means of the expression G () ) [S - H - ΣL() - ΣR()]-1
(2)
which includes the Hamiltonian and overlap matrices calculated in the procedure outlined above. The description of the hybridized bonds needs to be done with a model that takes into account, beyond the simple π-π* approach, the contribution of several orbitals per atom. The basis chosen for the ab initio ionic relaxation and construction of the building blocks self-consistent Hamiltonian consists of two sets of s and p orbitals. The renormalization procedure used in this study takes advantage of the locality of this orbital basis set, allowing us to consider the system as formed by nearest-neighbors interacting sections. As depicted in Figure 1 (bottom panel), an armchair CNT is divided in segments, so that the Hamiltonian H is partitioned in on-site energy diagonal blocks and nearest-neighbors coupling blocks. When functionalized and pristine building blocks are coupled in a random way, we are able to buildup CNTs as long as desired. The present approach is limited to the low bias linear regime but allows us to study systems of realistic lengths as usually investigated in the experiments.21 942
We focus on the case of metallic armchair C(10,10) tubes for which it has been shown that the cycloaddition of carbene induces the breaking of the bridged sidewall carbon bond.20,35 The stability of this “open” geometry with respect to the “closed” one decreases when the tube diameter increases. Closed configurations introduce significant backscattering, and the advantage of cycloaddition is therefore lost for tubes large enough so that the closed geometry is favored. Further, it was shown on the basis of activation energy calculations that the desorption barrier for carbene on graphene is too small to prevent thermally activated desorption at room temperature, suggesting that carbene may not be stable at 300 K on tubes much larger than the C(10,10).36 Concerning the phenyl moeties, we study paired configurations in the 1,4-geometry (para) where two phenyls are grafted as third-nearest neighbor, keeping intact the aromatic properties of the CNT. Several arguments suggest that such a configuration is the most likely to be found on nanotubes: (a) the grafting of a first radical is known to enhance the reactivity of a carbon atom at an odd number of bonds away from it;37 (b) the 1,4-configuration is slightly more stable than the 1,2-configuration (ortho); (c) isolated phenyls have been shown to spontaneously diffuse or desorb at room temperature on standard diameter tubes.36 These considerations are consistent with the experimental observation based on a careful nuclear magnetic resonance analysis39 that, in the case of nanotube alkylation, the 1,4-addition (para configuration) appears to be more common than the 1,2addition (ortho configuration). In Figure 2 (right and left panels), the impact of an increasing number of grafted phenyl pairs is shown for 300 nm long (6,6) and (10,10) nanotubes. It is first interesting to note the effect of a single group on the otherwise clean nanotube (dashed curves). A single sp3 bond between such a single phenyl pair and the tube surface induces a decay of conductance in the whole spectrum with the occurrence of two symmetric peaks (in the first plateau) that are related Nano Lett., Vol. 9, No. 3, 2009
Figure 3. Left panel: Disorder average conductance at the charge neutrality point (mainframe) and estimated elastic mean free path (inset) as a function of grafted phenyl groups density for a nanotube 300 nm long. Right panel: Same as in left panel but for grafted carbene groups and for a nanotube length of 1 µm.
with the suppression of one conduction channel. The increase of the coverage density results in stronger damping of the conductance pattern, which however roughly follows the initial single group signature. For a fixed molecule density, the effect is found to be further enhanced with reducing nanotube diameter, which is expected because of lower transport dimensionality. As shown in Figure 3 (left panel mainframe), the decay of the conductance at the charge neutrality point appears inversely proportional to the coverage density. With a conventional phenomenological law the disorder average transmission coefficient can be related to the elastic mean free path as41 〈G 〉
( )
⁄ G0 ) N⊥ 1 +
L le
-1
(3)
which gives some approximated range and scaling behavior of the mean free path (Figure 3 left panel inset). With a more academic disorder model (such as Anderson-type onsite disorder), analytical forms and scaling behavior of elastic mean free paths were derived for carbon nanotubes.42,43 In particular, le was shown to upscale linearly with tube diameter for a fixed disorder strength. For our computed 300 nm long (6,6) nanotube, the mean free path quickly decreases with coverage density to reach le ∼ 15 nm when 40 groups are attached to the sidewalls. The same number of functional groups on a larger diameter nanotube (10,10) will however yield le ∼ 100 nm. Our calculated mean free path thus also shows some upscaling with nanotube diameter, although the scaling behavior cannot be extracted in detail. As evidenced by these results, sp3 bonds are clearly not in favor of good conduction efficiency of hybrid nanotubes, which therefore jeopardize their use for any application. In contrast, the cycloaddition of carbene groups yields a small downscaling of the conductance in the first plateau, with respect to the coverage density (Figure 3, right panel). Different from the case of phenyl groups, an asymmetry of the conductance decay is already observed for the single Nano Lett., Vol. 9, No. 3, 2009
molecule case. This is seen as a small dip in the conductance just before the transition to the first higher subband (red curve). The much weaker change of conductance indicates a quasi-ballistic regime, and by using the phenomenological formula eq 3, a rough estimate of the corresponding elastic mean free path is achieved, after computing the average conductance at the charge neutrality point (Figure 3 right panel, mainframe). The values of le are reported in Figure 3 (right panel, inset), and are in the range of ∼2-9 µm depending on coverage density and nanotube diameter. In this case a rough linear scaling of le is observed with tube diameter. One also notes that for the chosen parameters the coverage density in the case of carbene groups is larger than that for the phenyl case. This demonstrates the noninvasive effects of cycloadditions inducing open geometries, which is crucial for further use of long hybrid nanotubes. In conclusion, by using a fully ab initio transport approach, we have been able to explore for the first time the transport regimes in chemically functionalized long carbon nanotubes, comparing two different and important types of chemical bonding. The results provide evidence of good conduction ability in the case of carbene cycloadditions, whereas paired phenyl addends were found to yield a strongly diffusive regime, with estimated mean free path ranging from a few tens to the nanometer scale, depending on the coverage density and incident electron energy. One can expect thus to observe localization phenomena as reported in irradiated nanotubes, even at room temperatures.44 Acknowledgment. We thank the CEA/CCRT supercomputing facilities for providing computational resources and technical support. Financial support from the ANR/PNANO project ACCENT is acknowledged. A.L.B. acknowledges support from the Marie-Curie fellowship CHEMTRONICS program and R. Avriller and L. E. F. Foa-Torres for helpful discussions. 943
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NL802798Q
Nano Lett., Vol. 9, No. 3, 2009
