On the Charge Carrier Localization in Zigzag Carbon Nanotube

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On the Charge Carrier Localization in Zigzag Carbon Nanotube Junctions Luis Rincon,*,†,‡ Rafael Almeida,† and Carlos A. Gonzalez‡ † ‡

Departamento de Química, Facultad de Ciencias, Universidad de Los Andes, La Hechicera, Merida-5101, Venezuela NIST Center for Theoretical and Computational Nanosciences, National Institute of Standards and Technology, 100 Bureau Dr., STOP 8380, Gaithersburg, MD 20899, United States ABSTRACT: In this work, we propose a novel photovoltaic device constructed from the junction of two carbon nanotubes, where the top of the valence band is mainly localized on one of the nanotubes' sides, while the bottom of the conduction one is mainly localized on the other side. This allows the separation of electrons on the side displaying the lower conduction band, from the holes, located at the other side. On the basis of self-consistent charge-density functional tightbinding calculations, we propose that this kind of device can be formed by joining two consecutive semiconducting zigzag carbon nanotubes. We have shown that, for these systems, the existence of the required band energy differences is related to the kind of variations in energy gap with the tube diameters, observed for the zigzag nanotubes. To illustrate the formation of the proposed junction, we have chosen to study a (10,0) tube joined linearly to a (11,0) one through a single pentagon
heptagon defect.

1. INTRODUCTION Given that optical and electronic properties of carbon nanotubes (CNTs) can be tuned by controlling their size and shape,1 they potentially constitute ideal materials in the design of photovoltaic (PV) cells. One of the crucial issues in the design of organic materials-based PV cells (including those based on CNTs) as well as dye-sensitized solar cells is related to the step involving the dissociation of excitons into free charge carriers. Thus, absorption of light by CNT materials results in the formation of tightly bound electron
hole pairs (i.e., Frenkel excitons) that can be highly localized. Furthermore, it has been found that the excitons in CNT-based PV cells are bounded more strongly than their equivalents in conventional solar cells based on inorganic semiconductor materials.2,3 Thus, typically in the case of CNTs, the exciton binding energy corresponding to the lowest optical state can be as large as 1 eV,2
9 an order of magnitude larger than those reported for monocrystalline Si or GaAs. In this work, we make use of quantum chemistry calculations in order to address the problem of charge-carrier separation in photovoltaic cells constructed with carbon nanotubes. Figure 1 illustrates the charge-carrier (electrons and holes) separation mechanism at the junction or interface between two different CNTs. Side A in that figure represents a CNT in which both the valence and the conduction bands lie at higher energies than their corresponding bands in the nanotube on the right side, B. Notice that, due to the relatively strong exciton binding energy, the excitation energies given by the optical band gaps (ωop(A) and ωop(B)) are smaller than the electronic band gaps (Eg(A) and Eg(B)). Thus, for the system as a whole, consisting of the junction of the two CNTs (called “excitonic junction” r 2011 American Chemical Society

henceforth), the top of the valence band is mainly localized on side A, while the bottom of the conduction band is localized on side B. In these cases, the effective band gap at the interface is given by the difference between those two energy values. Therefore, if ωop(B) is larger than or equal to this effective band gap (as in the case illustrated in Figure 1), the exciton dissociation at the interface of the two tubes results in the separation of electrons on the side displaying the lower conduction band from the holes that are localized on the other side. This mechanism for charge separation is fundamentally different from that observed in traditional p
n heterojunctions, given that the charge-carrier generation is simultaneously accompanied with the exciton dissociation.10
11 Thus, on the absorption of a photon, electrons are generated on the side of the interface with the lower energy conduction band (B in Figure 1), and at the same time, holes localize on the other side exhibiting a larger valence band energy. Additionally, in contrast with traditional junctions, in excitonic ones, the valence band energy offset must be larger than the excitonic binding energy on the electron-generating side. This requirement for charge separation constitutes one of the main challenges in the design of CNT excitonic junction PV cells. Because ωop(B) is smaller than Eg(B) by a quantity equal to the exciton binding energy of B, knowledge of the values of the optical band gaps is critical in the design of CNT excitonic junctions.

Received: December 27, 2010 Revised: April 25, 2011 Published: May 19, 2011 11727

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Table 1. Band Gap (in eV) as a Function of n for Semiconducting Zigzag CNTsa n 7

8

10

11

13

14

this work

0.92

1.13

0.77

0.82

0.62

0.65

DFT GGA32

0.19

0.59

0.78

0.92

0.65

0.72

DFT LDA33

0.21

0.59

0.77

0.93

0.64

0.72

DFT LDA34

0.24

0.64

0.75

0.94

0.63

0.74

TB35

1.11

1.33

0.87

0.96

TB36

1.00

1.22

0.86

0.89

0.69

0.70

a

Abbreviations: GGA, generalized gradient approximation; LDA, local density approximation.

Figure 1. Scheme of the energetics involved in an excitonic junction between two carbon nanotubes. The meanings of the employed quantities are explained in the text.

On the basis of the previous discussion, it can be expected that junctions of two CNTs can provide unique advantages that could help in the design of very efficient photovoltaic cells. Some of these advantages include the following: the capability to access a wide range of band gaps, the ability to match the different frequencies in the solar spectrum,1 the enhancement of optical absorption,12 and the reduction of carrier scattering, a typical phenomenon observed in hot carrier transports.13 Given that carbon nanotubes can be metallic, semimetallic, or semiconducting (a property that depends on their chiral vectors1), the combination of these materials provides an enormous potential in the manufacture of devices whose rational design takes advantage of the variations in the CNT band structure. Hence, by joining two CNTs of appropriate diameters and chirality, it is, in principle, possible to design novel and efficient onedimensional PV nanoelectronic devices. It is with this in mind that, in the present work, we engage in the exploration of the physics governing the mechanism of charge separation at the junction of two semiconducting zigzag CNTs.

2. COMPUTATIONAL DETAILS14 The calculations discussed in this work were carried out employing the self-consistent charge-density functional tightbinding (SCC-DFTB) method,15,16 as implemented in the program DFTBþ (version Snapshot 081217).17 All calculations were performed using the set of parameters pbc-0-2,18 whereas the band structures were plotted with the waveplot.0.2.119 and Molekel20 computer packages. Because periodic conditions are employed in the calculation of the one-dimensional systems treated in this work, the size of the periodic unit cells perpendicular to the tube axis was set in such a way that there was at least 10 Å of empty space between the two tubes. In addition, a 1  1  11 k-point grid was used for the calculations of total energies and structural optimizations. In previous studies, the SCC-DFTB methodology has been applied to the study of a variety of carbon systems, including amorphous state,21 fullerenes,22
26 diamond,27 and carbon

nanotubes.28 Recently, Morokuma’s group has performed a series of molecular dynamics simulations of the formation of carbon clusters in the gas phase based on DFTB energies and gradients.29
32 In all these examples, DFTB has proven to include the necessary physical ingredients to properly describe the electronic structure of carbon systems similar to the ones of interest in this work.

3. RESULTS Figure 2a shows the results for the SCC-DFTB direct band gap calculations (computed along the Γ point) as a function of the nanotube diameter corresponding to semiconducting zigzag CNTs with chiral vectors (n,m), where n varies between 7 and 20 (excluding values that are a multiple of 3) and m is fixed to 0. In general, for a fixed value of m (0 in this study), the diameter of the CNT increases with the value of n. The results in Figure 2a show that the decrease in the band gap as a function of the tube diameter significantly deviates from a singe monotonic behavior. Instead, a “jigsaw” relation is observed, which becomes more pronounced as the tube diameter becomes smaller. Similar behavior has been previously reported from density functional theory (DFT)33
35 and tight-binding (TB) studies.36,37 The band gaps in tubes following the series n = 3ν þ 1 (i.e., n = 7, 10, 13, 16, 19), where ν is an integer, are consistently lower in energy than the corresponding gaps for the next zigzag nanotube following the series n = 3ν þ 2 (i.e., n = 8, 11, 14, 17, 20). These trends in the band gaps corresponding to the two CNT series can be attributed to the trigonal shape of the lines around the K point of the graphene Brillouin zone,38 and to the mixing between σ and π orbitals due to the curvature of the tubes. Experimental band-gap values for some of the CNTs considered in Figure 2 have been reported in the literature. For instance, scanning tunneling spectroscopy measurements conducted by Wildoer and collaborators39 lead to apparent band gaps of 0.50 ( 0.05 eV for n = 16 and 0.55 ( 0.05 eV for n = 17 zigzag single-walled CNTs. Despite the relatively large noise of the measurements, the DFTB values of 0.518 eV for n = 16 and 0.535 eV for n = 17 are in very good agreement with these experimental results. Table 1 lists the values of the band gaps of semiconducting zigzag CNTs, between n = 7 and n = 14, obtained in this work (DFTTB) and those previously reported in the literature using DFT33
35 and TB calculations.36,37 In general, the band gaps computed by the three different DFT studies are in excellent agreement. The same trend is observed in the case of the two TB 11728

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Figure 2. (a) DFTB band gap (in eV) as a function of the diameter of zigzag semiconducting CNTs with chiral vectors (n,0). The cases for which n is a multiple of 3 are not included. (b) Energy (in eV) of the top of the valence band and of the bottom of the conduction band as a function of the diameter for the CNTs of (a).

studies. In addition, the results in Table 1 indicate that the DFTB band gaps computed in this work are consistently smaller than the TB ones by approximately 9
17%. For CNTs smaller than n = 10, the DFTB (and the TB) band gaps are found to be significantly larger than the corresponding gaps computed by all three DFT studies, regardless of the level of approximation employed in the DFT calculations (Table 1). These discrepancies could be the result of the relatively large curvature of the walls of these smaller CNTs characterized by a strong σ
π mixing, which is difficult to be properly described by TB methodologies, including the DFTB method used in this work. In the case of n = 10, the band gaps computed by DFTB and the three different DFT levels of theory are in excellent agreement (with the TB values being approximately 12% larger). Finally, despite the fact that, for n > 10, the DFTB band gaps become consistently smaller than the values computed by DFT (Table 1), the qualitative trends are in very good agreement. It is well known that DFT calculations within the Kohn
Sham formalism (and, in particular, LDA) lead to a substantial underestimation of the band gaps in semiconductors and insulators, a situation that, to some degree, is also present in our SCC-DFTB calculations. Thus, it has been found that, with respect to the experimental values, these differences are typically about 50%, but deviations as large as 100% have also been reported. The difficulty of DFT in predicting band gaps and excitation energies has largely been attributed to a discontinuity in the true DFT exchange correlation functional derivative with respect to the number of electrons as well as the existence of spurious electron self-energy interactions that are not corrected in the most popular DFT functionals currently available, including LDA.40,41 Because the SCC-DFTB methodology employed in this work is a DFT-based one, we expect this problem to affect

our results as well. In the case of DFTB, the problem gets amplified due to the nonexistence of a systematic study quantifying the range of the deviations. Nevertheless, it has been found that, in the case of semiconductors, even though the values of the energy of the conduction band with respect to that of the valence band are not quantitatively correct, they generally follow the right qualitative trends. From a formal point of view, the Kohn
Sham eigenvalues do not have a physical interpretation, except for the highest occupied band (related to the Fermi energy of the system);42,43 therefore, strictly speaking, any quantitative agreement between DFTB band gaps and experimental results should be considered fortuitous. A more computationally demanding theoretical treatment of excitation energies that includes dielectric screening has been developed using the many-body Green’s function theory, employing Hedin’s GW formalism44
46 in-conjunction with the Bethe
Salpeter equation.3,47,48 On the basis of this first-principles formalism, Louie’s group has been able to compute excitation energies, ωop, and exciton binding energies for small semiconducting zigzag CNTs.49,50 For CNTs (7, 0), (8, 0), (10, 0), and (11, 0), they obtained excitation energies of 1.20, 1.55, 1.00, and 1.21 eV, respectively.49 Additionally, for the lowest exciton binding energies, they reported values of 0.87, 1.03, 0.68, and 0.76 eV, respectively.50 Although these band-gap values (found from the addition of the excitation and exciton binding energies) are clearly larger than those shown in Table 1, both sets display similar qualitative trends. Moreover, the percentage variations between consecutive values in both sets are practically identical. Thus, despite the quantitative differences between the GW
Bethe
Salpeter predicted band gaps and those obtained from the SCC-DFTB methodology, the very good qualitative agreement between both sets of results is very encouraging and 11729

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Figure 4. Crystal orbitals at the band edge of a junction CNT formed from a (10,0) and (11,0) CNT intermolecular junction.

Figure 3. Optimized carbon nanotube structure constructed from the intermolecular junction of a (10,0) and a (11,0) semiconducting tube.

supports the idea that our calculations are able to capture the changes that are important in the prediction of trends in the band alignments and offsets for the systems under study. To shed some light on the source (and possible implications) of the oscillations of the band gap observed in the case of the semiconducting zigzag CNTs considered in this study (Figure 2a), the energies of the top of the valence band, EV, as well as the bottom of the conduction band, EC, have been computed as a function of the CNT diameter, and the results are presented in Figure 2b. In the case of the valence band, the SCC-DFTB calculations predict a slow, but steady, increase in EV, roughly showing a linear dependence with the CNT diameter. Similar calculations indicate that, in the case of the conduction band, the values of EC for the n = 3ν þ 1 CNT series are predicted to be systematically smaller than those corresponding to the n = 3ν þ 2 series (Figure 2b), leading to the oscillatory behavior of the band gap depicted in Figure 2a. The computed values of EC and EV for all CNTs considered in this study qualitatively follow the relative energetics described in Figure 1 and, therefore, suggest the possibility of proposing the design of a PV device based on excitonic junctions, such as the ones described in this work, involving the connection of two small consecutive semiconducting CNTs. In these proposed devices, the charge-separation effect is expected to be more predominant in the case of small-diameter CNTs. It should be mentioned that the possibility of connecting segments of CNTs with different chiral vectors to develop efficient devices for future PV applications is not a new idea and has recently generated much interest due to their special properties.51 Much progress has been achieved in their synthesis, properties, and applications in recent years.52 Several types of junctions have been experimentally and theoretically considered, and in most of the studied cases,

semiconductor
metallic tubes have been connected through pentagon
heptagon pair defects.53
60 The pentagon
heptagon pair defects are also observed in defective CNTs as well as elongated CNTs subjected to tensile strain.56,60
62 The formation process and stability of these defects have been simulated by tight-binding molecular dynamics and ab initio calculations.63,64 Experimental confirmation of the rectifying behavior of these semiconductor
metallic junctions has also been reported in the literature.60,65 Given the encouraging results previously obtained, we decided to perform SCC-DFTB calculations to study a (10,0) tube joined linearly to a (11,0) through a single pentagon
heptagon defect lying parallel to the tube axis as a prototype of a CNT-based PVC system. This defect induces zero net curvature so that the tube does not deviate from a linear configuration along the principal axis. Calculations were performed applying fictitious periodic boundary conditions. Furthermore, in order to minimize border effects, we have connected 20 asymmetric units of (10,0) tubes with another 20 asymmetric units of the (11,0) CNTs, generating two consecutive pentagon
heptagon defects separated by nearly 15 Å. The unit cell size containing 1680 carbon atoms was chosen as a compromise between the minimum size representative of a CNT system and computational expedience. The corresponding SCC-DFTB fully optimized structure is shown in Figure 3. The optimized system forms a quasi-1D semiconductor/ semiconductor junction of the type described in Figure 1. This is supported by direct inspection of the valence and conduction crystal orbitals at the band edge plotted in Figure 4a,b, respectively, where it can be observed how these bands are largely localized at different ends of the joint nanotube. In addition, M€ulliken population analysis of this quasi-1D system predicts a significant localization of the conduction band, with approximately 64% of the population localized along the smallerdiameter tube, and the remaining 36% lying around the neighborhood of the junction. In contrast, the valence band seems to 11730

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Figure 5. Density of states (DOS) near the Fermi level for the (a) (10,0) and (b) (11,0) carbon nanotubes. (c) Localized density of states (LDOS) near the Fermi level for the (a) (10,0) side and (11,0) side.

be less localized, with only 54% of its electronic population located in the larger diameter tube, 45% localized close to the junction, and the remaining 1% localized on the smaller-diameter tube. These differences in the degree of localization in the electron population are directly related to changes in the energy of both bands with the diameter of the nanotube (see Figure 2b), which is more pronounced in the conduction band. We have examined the local density of states (LDOS) on both sides of the tube of the (10,0)/(11,0) junction. The LDOS partition is generated so that the atoms shared in the fused pentagon
heptagon pair defect are considered part of both sides containing the (10,0) and (11,0) tubes. Figure 5c depicts a plot of the LDOS of the junction structure near the Fermi level. For comparison purposes, the density of states (DOS) corresponding to the free individual (10,0) and (11,0) carbon nanotubes are also shown in Figure 5a,c, respectively. It is interesting to notice

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that the width of the DOS gaps of the heterojunction is smaller than the corresponding gaps of (10,0) and (11,0) CNTs, suggesting that defects in the junction could enable electronic transport under certain conditions. The results in Figure 5c indicate a significant degree of LDOS localization along the large diameter segment (11,0) for negative energies (
2.00 to 0.00 eV) and in the smaller-diameter segment (10,0) for positive energies (0.00
2.00 eV). This relative large LDOS localization as compared with the localization of the orbitals at the band edge (Figure 4) is the result of the LDOS being an average over the whole Brillouin zone and not just at the band edge. Thus, the LDOS results show that, on the average, the states near the gap are physically localized by a factor of 10 or more (Figure 5). At this point, it is worth mentioning two important issues that have a significant influence on the effectiveness of the PV abilities of the CNT junctions discussed in this work: (1) The existence of an axial charge separation is critical for the PV activity of these quasi-1D CNT systems. A similar charge-separation process involving a quite different mechanism (i.e., partially strained nanowires) has been recently proposed for silicon (011) nanowires.66,67 However, it is worth mentioning that, based on the overlap of the states in the band edge at the pair defect shown in Figure 4, a large amount of states recombination for this simpler device could be anticipated, a situation that could clearly decrease the CNT’s photovoltaic efficiency. This overlap is the result of localized states arising as a consequence of the defect in the hexagonal array. In principle, the formation of localized states is expected to have a minor effect when the diameter of the tubes increases, given that the perturbation exerted by the formation of these defects is considerably smaller in larger nanotubes. On the other hand, the separation between the valence and conduction bands becomes far from optimal at very large CNT diameters (Figure 2), indicating that a delicate balance between size and band separation should be attained in order to develop effective PV devices based on CNT junctions similar to the ones considered in this study. (2) Due to the fact that typical CNT structures generate tightly bound excitons (corresponding to high excitonic binding energies), the corresponding excitonic transport in this kind of material has been generally found to be a slow process, and consequently, it is highly desirable that the charge-generation process occurs close to the junction zone where charge separation takes place, highlighting the importance of the localization of the electronic population in this region. Another important aspect of these junction nanotubes is related to the symmetry of their band orbitals. In zigzag nanotubes, both the valence and the conduction bands are 2-fold degenerate and are predominantly of π and π* character in nature. Even more interesting is the fact that the valence bands for the n = 3ν þ 1 CNTs are bonding in the direction perpendicular to the tube axis (i.e., in the radial direction). This is also the case for the conduction bands of the n = 3ν þ 2 CNTs. On the other hand, the conduction bands of the n = 3ν þ 1 CNTs are bonding in the direction parallel to the CNT axis, which corresponds to the bonding character of the valence bands of the n = 3ν þ 2 CNTs. Thus, in a joint tube, as a whole, the valence and conduction bands are the result of π orbitals aligned 11731

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semiconductor junctions in PV devices. It might be argued that, because the exciton formation at the junction involves a transition between states of identical symmetry, the electron excitation would involve a forbidden transition. However, let us recall that, in this proposed PV device, due to the slow exciton transport rate associated with its high binding energy, the process of exciton formation happens in the neighborhood surrounding the junction, while the charge-separation process is highly localized just at the junction between the two CNTs. As a result, the excitation actually occurs between states with different symmetries.

4. CONCLUSION In summary, in this work, we have proposed a CNT device, plausible, from the theoretical point of view, for efficient charge carrier separation. The calculations discussed in this work and the results reported for Si66,67 suggest that photogeneration in these kinds of devices leads to a significant separation of electrons and holes across the junction defects, which, as discussed above, seems to be the dominant factor for PV activity. In addition, the results of this work suggest new possibilities for designing novel solar cells, based on CNT junctions, that are conceptually different from other devices based on p
n semiconductor junctions or on heterointerfaces. We think that, in addition to the possible use as a PV device, these kinds of junctions may have potential applications for a wide variety of electronic and optical purposes. It is, in this sense, that further theoretical and experimental works on these excitonic junctions are worthwhile and could lead to new applications in the fields of nanoelectronics and nano-optics. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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dx.doi.org/10.1021/jp1122932 |J. Phys. Chem. C 2011, 115, 11727–11733