Spectral Diversity in Raman G-band Modes of Metallic Carbon

J. Phys. Chem. C 2008, 112, 13017–13023

13017

Spectral Diversity in Raman G-band Modes of Metallic Carbon Nanotubes within a Single Chirality Moonsub Shim,*,† Anshu Gaur,† Khoi T. Nguyen,† Daner Abdula,† and Taner Ozel‡ Departments of Materials Science and Engineering and Physics, UniVersity of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ReceiVed: June 6, 2008

Diversity in the Raman G-band phonon modes within individual metallic carbon nanotubes of the same chirality is examined. Comparisons between Raman spectra of as-synthesized nanotubes with those obtained under electrochemical gate potential are made. We show that most of the distribution in line width and peak position of the G-band modes within a single chirality type can be explained by variations in where the Fermi level lies with respect to the band crossing point (i.e., where the nanotube is at zero charge). Varying degree of charge transfer from adsorbed O2 is likely to be the main source of ∼2 eV or larger range of Fermi level positions. On average, the Fermi level of individual metallic nanotubes lies on the order of 1 eV below the band crossing point. Both charge transfer and physical disorder are evident upon O2 adsorption. Implications of these findings on electron-phonon coupling and charge transfer processes are discussed. Introduction Raman spectroscopy has been widely used in characterizing carbon nanotubes. In addition to structural chiral index (n, m) identification,1 numerous physical and chemical processes including electron-phonon and electromechanical coupling,2 charge-transfer doping,3 and chemical reactions4 have been elucidated. The ability to measure Raman spectra at the single nanotube level especially when combined with electron transport measurements is currently providing unprecedented details.5,6 For example, unusually large softening and broadening of the G-band phonons have been observed in single metallic nanotubes5,6 with Kohn anomalies causing strong electronphonon coupling.7 However, a large distribution of behaviors is expected at the single molecule level which can make interpretations difficult to make. Hence it is important to quantify the range of distribution of behaviors and the factors that cause such distributions. In metallic nanotubes, often very broad and asymmetric G-band Raman features are observed. It is now well-established that this broad line shape can be used to identify a nanotube as metallic.8 However, some metallic tubes exhibit narrow lineshapes that are very similar to semiconducting ones and hence the presence of the broad G-band feature cannot guarantee that all metallic tubes are identified. The Raman G-band arises from in-plane C-C stretch and, to date, it has been one of the key features in spectroscopic characterization of electron-phonon interactions and chemical reactivity of carbon nanotubes.2,4 Understanding the range of diversity and where the differences in the G-band spectral features come from in individual nanotubes is therefore critical in developing robust chemistries to selectively functionalize metallic tubes as well as in elucidating the underlying physics of electron-phonon coupling in this prototypical 1D conductor. Most comparisons between experimental observations and theoretical expectations are made with the assumption that the * To whom correspondence should be addressed. E-mail: [email protected]. † Department of Materials Science and Engineering. ‡ Department of Physics.

Fermi levels of as-synthesized nantoubes (and even those that have been processed for solution suspension) lie at or very close to the band crossing point halfway between the first pair of van Hove singularities. For metallic tubes, examining the distribution of G-band features may ascertain whether or not such assumptions hold true. In doing so, two of the key factors to consider are 1) phonon softening due to the Kohn anomaly7 and 2) charge transfer doping via molecular adsorption especially from ambient O2.9 These two factors are interrelated in that the phonon softening arising from the strong electron-phonon coupling induced by the Kohn anomaly disappears upon moving the Fermi level away from the band crossing points. In this article, we first show that there is a large range of Raman G-band phonon linewidths and frequencies even within in a single chirality. This distribution is then compared to the spectral evolution of nanotubes of the same chirality under electrochemical gate potential indicating that most nanotubes are doped by O2 adsorption with their Fermi levels lying well below the zero charge point. Implications on electronically selective chemistries initiated by charge transfer as well as on comparisons between experimental observations and theoretical predictions especially those addressing electron-phonon coupling induced phonon softening are considered. Experimental Section Carbon nanotubes were synthesized on Si/SiO2 or fused quartz substrates by established chemical vapor deposition methods using either ferritin (Sigma-Aldrich) or lithographically patterned Fe(NO3)3 · 9H2O/alumina catalyst.10 Simultaneous electrochemical gating and Raman measurements were carried out as described in ref 5 using Au electrodes with Ti wetting layer deposited on top of SWNTs grown on heavily doped Si/SiO2 substrates. Electrochemical gate potential was applied through a 20 wt % LiClO4 · 3H2O solution in polyethylenimine (PEI, Aldrich).11 Raman spectra were collected with a JY LabRam HR 800 using a 1.96 eV excitation source through a 100× air objective (laser spot diameter ∼1 µm). Laser power was kept at or below ∼1 mW.

10.1021/jp8050092 CCC: $40.75  2008 American Chemical Society Published on Web 07/30/2008

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Figure 1. Raman spectra of (12, 6) and (14, 2) chiral metallic nanotubes showing diversity especially in the G-band features between 1500 and 1600 cm-1. Radial breathing modes are shown in the left panels with the dashed line indicating the expected peak position. D-band is seen at ∼1320 cm-1. Spectra are offset for clarity.

Results and Discussion It has been observed that individual metallic tubes often exhibit broad downshifted Raman G-band spectral features (with respect to semiconducting nanotubes and graphite) with asymmetric Fano line shape.12 It has also been observed that some metallic tubes exhibit narrow lineshapes that are very similar to semiconducting ones.5 Recent studies combining electrochemical gating with Raman measurements on individual metallic nanotubes have now shown that one nanotube can exhibit both broad asymmetric downshifted and narrow upshifted G-band features depending on where its Fermi level lies.5,6 It is the presence of the Kohn anomaly that leads to the broad lower frequency G-band mode (often referred to as the G- peak).7 Because of the strong electron-phonon coupling at the band crossing (Dirac) point, that is, single particle electronic excitations near the Fermi level coupling to the phonons, the G- peak has been assigned to the axial longitudinal optical (LO) mode.7,13 This is the assignment we follow here. If the metallic nanotubes are charged such that their Fermi levels lie away from the band crossing point, line narrowing and frequency upshift are expected - but only for the LO mode.7 This effect provides a starting point in examining the spectral diversity in individual metallic nanotubes. G-Band Spectral Diversity. Figure 1A shows the radial breathing mode (RBM) and D- and G-band regions of the Raman spectra of several (12, 6) and (14, 2) metallic nanotubes. The assignment of (n, m) chiral vector strictly follows the experimentally verified RBM frequencies within 2 cm-1 as reported in ref 14. The RBM frequencies are expected to vary due to the variations in the local chemical environment15 and therefore the assignment of (14, 2) for the nanotubes shown in Figure 1B needs to be justified since (15, 0) nanotube is expected to have RBM frequency difference of only 1 cm-1. All nanotubes that we assign to (14, 2) exhibit at least three well-

Shim et al. distinguished peaks in the G-band as shown in Figure 1B. A (15, 0) nanotube is an achiral zigzag tube. Both zigzag and armchair nanotubes are expected to have only one mode with TO substantial intensities (ALO 1 mode for zigzag and A1 mode for armchair) when the incident and detected light polarization is parallel to the nanotube orientation.6a,16 The chiral (14, 2) nanotube has a reduced symmetry and multiple peaks observed suggest that these tubes are not likely to be zigzag tubes. However, we note that the substrate as well as the O2 adsorption effects can cause complications and the assignment of (14, 2) is tentative at this stage. Nevertheless, the observed trends in the G-band linewidths and frequencies do not differ much within these tubes. The most striking deviations within each chirality are the linewidths and the frequencies of the peaks in the G-band. There is also a small but systematic shift in the D-band peak frequency, which is discussed later. To quantify the deviations in the G-band features, we fit all spectra to three Lorentzians and a Fano line given by I(ω) ) Io[1 + (ω - ωo)/qΓ]2/{1 + [(ω ωo)/Γ]2}, where ωo is the spectral position with intensity Io, q is the measure of phonon coupling to a continuum of states, and Γ is the width. Note that in the limit of |1/q| f 0, the Fano line reduces to a Lorentzian with Γ being the half-width at halfmaximum. Therefore, when we refer to linewidths, we use fullwidth at half-maximum for Lorentzians and 2Γ for the Fano line. We also note that the actual value of |1/q| obtained from fitting is very sensitive to small variations in the baseline but the obtained values are always less than 0.5 with most being between 0.1 and 0.2. Therefore, we limit our discussion to peak position and line width. Figure 2A shows examples of how the G-band features are fitted. Three cases shown provide the full range of G-band diversity observed with the broadest and the most asymmetric at the bottom to the narrowest and the most symmetric at the top. As seen in the lowermost curve, the intensities of the G-band of those nanotubes that exhibit broad lineshapes extend to frequencies well below the D-band. Therefore, all spectra are fitted including the D-band, which is described fairly well by a single Lorentzian. The broad line shape of the G-band is associated with asymmetry and a Fano line is necessary to fit the spectra (highlighted in red). When the G-band exhibits only narrow line features as shown in the uppermost and the lowermost spectra of Figure 1A and B as well as in the uppermost spectrum of Figure 2A, all Lorentzian fit is just as good as using a Fano line shape. However, for direct comparison, all spectra are fitted including one Fano line. The fitting results indicate that the distributions are quite large. Figures 2B shows the average peak positions (filled symbols) and the standard deviation (error bars) for the 2 main features of the G-band for three different chirality tubes. The corresponding open symbols are the maximum and the minimum frequencies observed. Figure 2C shows the average linewidths. Also shown are the standard deviation and the maximum and the minimum linewidths analogous to Figure 2B. The spread in the peak frequencies are over 20 cm-1 for both the LO and the TO modes. The spread in the linewidths of LO and TO modes are also comparable. Comparison to Electrochemical Gating. In order to explain these large distributions in the Raman G-band, we now consider the effects expected due to the Kohn anomaly7 and the possibility of ambient O2 adsorption induced charge transfer.9 To do so, we compare the spread in the Raman spectra of assynthesized nanotubes with the effects of electrochemical gating. Figure 3A shows the G- and the D-band regions of a (14, 2)

Spectral Diversity in Raman G-band Modes

Figure 2. (A) Demonstration of curve fitting for the D- and G-band spectral regions of three metallic tubes exhibiting large variations in the G-band features. Gray lines are the data. Components of the fit (as described in the text) are also shown. Fano line highlighted in red is considered as the main LO mode and the blue highlighted Lorentzian is considered as the TO mode here. (B) The distribution of LO and TO mode frequency for the indicated chiral index. Filled symbols are the average values with the error bars indicating the standard deviation. Open symbols are the minimum and the maximum values observed for the as-synthesized metallic tubes. (C) Same as in (B) but for linewidths. For the Lorentzian TO mode, fwhm is shown. For the Fano line, 2 Γ which is analogous to fwhm (see text) is shown.

nanotube under electrochemical gate potential as indicated. As expected and consistent with previous reports, softening and broadening of the G-band modes are seen around the zero charge point near 0 V. Since we need to correlate these results with phonon modes of as-synthesized nanotubes of the same chirality without any electrical contacts, we examine how D-band peak frequency shifts with the applied gate voltage as a reference point. While the softening and broadening of LO mode may be considered, frequency upshift and narrowing of the line width of this mode appears nearly symmetric away from the band crossing point. This means that we cannot distinguish between electron addition and removal using G-band peak position or its line width. While the D-band is a double resonance process involving a defect scattering,17 it is also associated with the in-plane C-C stretch much like the G-band and is therefore also sensitive to the degree of charging. Figure 3B shows the D-band region enlarged at the same voltages as in 3A. There is small but systematic downshift in the D-band peak frequency with increasing gate voltage. Figure 3C shows that this shift is essentially monotonic and nearly linear unlike the G-band. In Figure 3C, we also show

J. Phys. Chem. C, Vol. 112, No. 33, 2008 13019 the 2D peak (sometimes called the G′, D′, or D*) position. Both the D- and the 2D-bands show the same trend with the applied gate. Similar gate dependence of 2D (as well as the G) band frequencies have been observed in graphene.18 The 2D-band is also a double resonance process but it involves two phonons rather than a phonon and a defect. Hence the 2D-band may be used even in the absence of physical disorder. In this regard and the fact that the shift in 2D-band is twice as large as the D-band, 2D-band may be a better reference point to compare charging effects. However, the 2D-band intensity can be quite low in some nanotubes and we have a more complete D-band spectral data set for the current samples. Therefore, we resort to using D-band frequency for comparing electrochemical gating results with the spectral distribution in the as-synthesized nanotubes. We note that the D-band frequency is expected to be dependent on the incident excitation energy.17 Since all measurements are carried out with the same laser line here, this dependence does not affect our results. In Figure 3D, the shifts in the positions of the three main peaks of the G-band for this (14, 2) nanotube are plotted as function of the D-band peak position. The maximum softening of all three modes is seen between 1318 and 1320 cm-1. This region corresponds to applied gate potential range of 0-0.3 V. The gate potential dependence of the peak positions are similar to those shown in ref 5. Since the gate dependence of D-band (as well as the 2D-band) peak position is nearly linear, direct correlation with D-band frequency and Fermi level shift can now be made. The results of Figure 3D (filled symbols) are compared to the G-band frequencies of as-synthesized nanotubes of the same chirality without any electrical contacts (open symbols) in Figure 4A. The blue half-filled symbols are for the same nanotube that the electrochemical gating measurements were carried out on prior to the application of polymer electrolyte. The red open symbols labeled 1, 2, and 3 correspond to one nanotube under different gas ambient whose spectra are shown in Figure 4B. The effects of the ambient gases are discussed in the next subsection. Figure 4C shows the same comparison between electrochemical gating and the G-band spectral distribution of as-synthesized nanotubes all having (12, 6) chiral index. All variations in the G-band peak frequencies overlap closely or extrapolate well with electrochemical gating measurements indicating that the major factor leading to the diversity in the G-band spectra even within a single chirality is the distribution of where the Fermi level lies. In both (12, 6) and (14, 2) nanotubes, the minimum frequency for the G-band LO mode corresponds to the band crossing point (zero charge point) where the phonon softening effect due to the Kohn anomaly is the strongest. Notice that, with the exception of four nanotubes, all as-synthesized nanotubes of both chiralities lie at higher D-band frequencies than where the zero charge point lies. Referring back to Figure 3C and D, these high D-band frequencies correspond to negative voltages where the gate potential induces positive charge on the nanotube. That is, most as-synthesized metallic nanotubes are positively charged. As-synthesized metallic nanotubes of another chirality, (13, 4), shown in Figure 4D exhibit monotonic increase (i.e., no minimum) in the G-band frequencies with D-band peak position suggesting that all of these tubes are charged positive. The Fermi level lying below the band crossing point (being positively charged) may not be too surprising given that assynthesized semiconducting tubes are known to exhibit p-type behavior.9 What is striking is that a large number of assynthesized nanotubes’ Fermi levels lie well below the most

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Shim et al.

Figure 3. Spectral evolution of one (14, 2) nanotube under polymer electrolyte gate potential (Vg). (A) D- and G-band regions under indicated gate potential. (B) Enlarged D-band region in (A). Spectra are offset for clarity. (C) Gate dependence of D and 2D band frequencies (ωD and ω2D) both showing nearly linear dependence with Fermi level shift. (D) Correlation of the frequencies (ωG) of the main three peaks of the G-band with ωD. LO and TO modes are filled squares and filled triangles, respectively.

Figure 4. (A) G-band frequency distribution of (14, 2) nanotubes. Electrolyte gating on one nanotube (filled symbols) with the distribution in as-synthesized nanotubes (open symbols) are compared. Blue half-filled symbols are for the same nanotube that the electrolyte gating was carried out on but prior to polymer electrolyte applications. Open red symbols labeled 1, 2, and 3 correspond to the spectra shown in (B). (B) Raman spectra of one (14, 2) nanotube in O2 ambient (1), in Ar after annealing at 450 °C in Ar for 15 min. (2), and after re-exposure to O2 (3). All three spectra are collected at room temperature and offset for clarity. (C) Same as (A) but for (12, 6) nanotubes without the O2/Ar/O2 cycle. (D) Same as (C) but (13, 4) nanotubes without comparison to electrochemical gating.

negative PEI electrolyte gate potential of -1 V. Note that for electrolyte gating, PEI adsorption on carbon nanotubes is known to induce electron transfer.19 That is, positively charged metallic nanotubes are brought back near the zero charge point upon PEI adsorption and hence the maximum phonon softening occurs near zero gate bias. Given the nearly ideal gate efficiencies of polymer electrolyte gate,11 this implies that the average metallic tube has Fermi level more than 1 eV below the zero charge point. Furthermore, the large distribution in the G-band features indicates that the distribution in the Fermi level position may be larger than 2 eV even within a single chirality.

Effects of O2 Adsorption. In individual semiconducting nanotubes, the role of ambient O2 adsorption on the observed electrical and optical properties has been and continues to be debated.9,19 Both direct doping effects and nanotube/metal contact Schottky barrier modulations have been argued in the case of single nanotube transistors.9,20 We have previously shown that both effects are important.9a In this study, most nanotubes examined do not have any metal contacts. In fact, the very last sample preparation step for most tubes shown here is the CVD growth. Therefore, contact effects and other fabrication/processing induced effects are not applicable here

Spectral Diversity in Raman G-band Modes and we focus on possible doping effects to explain most metallic nanotubes having Fermi level significantly below the zero charge point. If O2 adsorption leads to significant charge transfer in semiconducting tubes, metallic tubes are more prone to this effect. Any charge transfer process involving carriers must overcome at least ∼0.1 to ∼1 eV barrier depending on the band gap of the semiconducting nanotubes whereas the finite density of states at the Fermi level readily provides electrons and empty states in metallic tubes. This difference has been the basis for developing several electronically selective reactions.4b,d The available electrons in metallic tubes should therefore facilitate charge transfer interaction with O2. Indeed, we have previously shown that such effect can be observed through the Raman spectra of metallic tubes.5 Here, we explore this effect further. Figure 4B shows the D- and G-band regions of a (14, 2) tube under O2 ambient (bottom curve labeled “1”), after Ar annealing and maintained under Ar (middle curve labeled “2”), and after re-exposure to O2 (uppermost curve labeled “3”). Initial relatively narrow and upshifted G-band features become broad, downshifted, and asymmetric after Ar annealing. Upon reexposure to O2, the G-band recovers back to the line shape that is nearly identical to before annealing. These observations are consistent with our previous report.5 There is also a small but significant shift in the D-band peak position which allows us to map these changes onto the G-band/D-band correlation shown in Figure 4A. The red open squares labeled “1”, “2”, and “3” in Figure 4A correspond to these changes and follow the changes due to charging via electrochemical gate potential nearly identically. The downshift in the D-band upon Ar annealing (removal or reducing the O2 induced effects) is equivalent to a change of ∼200 to 300 mV in gate potential in the direction of removing excess positive charges. In addition to the G-band broadening and D-band downshift, there is also a decrease in the D-band to G-band integrated intensity ratio. However, this particular nanotube shows very little D-band intensity initially and the changes are not obvious. To examine the possible role of O2 actually inducing physical disorder, we have carried out the same Ar annealing/O2 exposure cycle on another metallic tube that exhibits larger initial disorder. As seen in Figure 5A, there is a large reduction in the D-band intensity upon Ar annealing and a recovery when re-exposed to O2. In Figure 5B, we plot the G-band LO mode line width, which is sensitive to charging of metallic tubes, with the integrated D/G ratio. The integrated D/G ratio (ID/IG) is calculated including the asymmetry of the Fano line following ref 20. These results indicate that O2 adsorption causes both charge transfer and physical disorder in metallic nanotubes. To further examine O2 adsorption induced disorder and charge transfer, we plot the G-band LO mode line width of assynthesized metallic tubes vs their ID/IG ratios in Figure 5C. If the ambient O2 adsorbs and bonds to metallic tubes (which would lead to an increase in the actual physical disorder), the polar bond formed could lead to a reduction in electron density in the nanotube. Such effects should result in larger degree of charging in nanotubes (which would in turn cause narrower G-band line width and upshift in the D-band frequency) with increasing ID/IG ratio. This correlation appears to be present when different degree of O2 adsorption is examined on the same nanotube as shown in Figure 5B. However, when different assynthesized individual nanotubes are compared, there is no obvious correlation between G-band line width (or D-band peak position) with ID/IG ratio as shown in Figure 5C. This apparent difference is likely to arise from variations in the local chemical

J. Phys. Chem. C, Vol. 112, No. 33, 2008 13021

Figure 5. (A) Raman spectra of one metallic nanotube in O2 ambient (bottom), in Ar after annealing (middle), and after re-exposure to O2 (top). The annealing conditions are the same as in Figure 4B. LO mode line width correlation to integrated D/G intensity ratio (ID/IG) for the spectra of the nanotube in A is shown in (B) and for the as-synthesized nanotubes is shown in (C). No correlation is seen in as-synthesized nanotubes.

environment which may provide additional disorder and/or charge transfer effect different from what we can observe by Ar annealing and O2 exposure. Further work is needed and is underway to sort out these issues. Implications on Diameter Trends, Phonon Softening, and Charge Transfer Initiated Chemistries. The diversity in the Raman D- and G-band modes of individual metallic tubes and the observation that Fermi levels of as-synthesized metallic tubes are on average well below the zero charge point have several important implications. One immediate implication is on the diameter dependence of these in-plane phonon modes. The charge dependent D-band frequency combined with the fact that individual metallic tubes on average have their Fermi levels lying well below the zero charge point can explain the apparent different diameter dependence of D-band frequencies of semiconducting and metallic tubes. Pimenta et al. have shown that the diameter dependence of D-band frequencies of metallic tubes are about 10 cm-1 above the expected values based on dependence observed in semiconducting tubes and therefore follow a different trend.22 The average D-band frequency for each chirality of as-synthesized metallic tubes in Figure 4 is about 5-10 cm-1 higher than the expected D-band frequency at zero charge point. Then, it may be the charging effects due to ambient O2 exposure shifting the D-band frequencies higher

13022 J. Phys. Chem. C, Vol. 112, No. 33, 2008 rather than metallic tubes following a different diameter dependence than semiconducting tubes. The shift in the average Fermi level below the zero charge point also needs to be considered in comparing theoretical and experimental G-band LO phonon frequencies. Lazzeri et al. have shown that by assigning the lower frequency G-band peak to LO mode softened by the Kohn anomaly, the observed diameter dependence trend of G+/G- splitting can be explained well.7b However, there is a consistent offset where the experimentally obtained values of the LO mode frequencies are ∼20 to 30 cm-1 higher than those calculated by density functional theory (see Figure 3 of ref 7b). This discrepancy has been considered to arise from neglecting dynamic effects.7c However, we note that the ∼20 to 30 cm-1 offset should be expected due to the average Fermi level offset (corresponding to ∼ 5 to 10 cm-1 upshift in the D-band position) from O2 adsorption induced charging which leads to on average ∼15 to 20 cm-1 upshift in the LO frequency compared to that at zero charge. Furthermore, nonadiabatic density functional calculations predict that charge injection shifting the Fermi level away from the zero charge point should remove the LO mode softening eventually to the point where it crosses over to higher frequencies than the transverse optical (TO) mode.7d Since the polymer electrolyte gate that we use here has nearly ideal efficiencies,11 the gate voltage range should correspond closely to the actual Fermi level shift. The crossing of LO and TO is predicted to be between 0.5 and 0.6 eV away from the zero charge point in ref 7d. Our gate voltage range spans -1 to +1 V and we observe no evidence of LO/TO crossing in Figures 3D and 4A and C. This is most likely due to the higher frequency TO mode exhibiting nearly as large softening as the LO mode especially in the (14, 2) nanotube shown in Figure 3. However, the line broadening, while also significant, is not as large in the TO mode as in the LO mode suggesting that TO mode softening near the zero charge point may be a different effect. Nevertheless, all of these observations indicate that further work is necessary in understanding electron-phonon coupling and the phonon softening effect in metallic nanotubes. Finally, we point out that the large variation of ∼2 eV or larger in the Fermi level position of metallic tubes has important implications on charge transfer processes in general. Chemical reactions initiated by charge transfer are being exploited in developing electronically selective chemistries to sort out metallic tubes from semiconducting ones.4 Reactions such as those with aryl diazonium compounds4a,d that can be applied directly to nanotubes on substrates are particularly appealing in electronics applications since already established fabrication techniques can be used. However, large variations on where the Fermi level lies, at least in metallic tubes on substrates, would lead to large variations in the initial charge transfer step and therefore reduce selectivity. Conclusions We have shown that there is a large distribution of linewidths and peak positions in the Raman G-band modes of metallic nanotubes even within a single chirality. This distribution arises mainly from the variations in the degree of charging/doping caused by O2 adsorption either directly on nanotubes or indirectly via the substrate. This large variation means that the Fermi level position in metallic tubes with respect to the band crossing point can vary by as much as 2 eV or more. On average, we have found that individual metallic tubes have their Fermi levels on the order of 1 eV below the zero charge point (i.e., positively charged). There are several important implications

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