Introduction

Due to their unique electronic properties and one-dimensional structure, single-walled carbon nanotubes (SWNTs) have attracted considerable attention as a promising nanomaterial for future nanoelectronics and nanophotonics1,2,3. The SWNTs are a group of nanocarbon materials that have a cylindrical shape, but various chiralities. The chiral index (n,m) defines the nanotube diameter (d) and the chiral angle of the hexagons of the carbon atoms wrapped around its circumference. This index is the factor that determines the electronic properties of the SWNTs (i.e., electronic structure, Fermi levels, band gaps and redox potentials). The determination of the electronic/redox potentials of the (n,m)SWNTs has been a central issue in the science and nanotechnology of carbon nanotubes. Smalley and coworkers4 and Odom and coworkers5 reported the diameter-dependent band gaps of SWNTs using scanning tunneling microscopy and scanning tunneling spectroscopy. Redox titrimetry6, chemical doping7,8,9 and spectroelectrochemistry10,11,12,13,14,15,16 were employed to determine the redox potential of the individual (n,m)SWNTs. However, the achieved success in determining the redox potentials has been very limited17.

We have previously reported a photoluminescence (PL) spectroelectrochemical method that determines the precise redox potentials of individual (n,m)SWNTs (tube diameters (d) = 0.75–1.10 nm) embedded in a film of carboxymethylcellulose sodium salt (CMC-Na, Fig. S1a)17 (careful selection of the dispersant is important for fundamental and applied studies with soluble naotubes18,19). However, such an experimental method can only be applied to the (n,m)SWNTs whose diameters are small (d < ~1.5 nm) because the individual dissolution of the (n,m)SWNTs with d > 1.5 nm in a suitable polymer matrix, such as CMC-Na, is almost impossible. Even when we obtain such a film, it is difficult to detect an efficient PL intensity that allows a PL-spectroelectrochemical analysis for the determination of the precise redox potentials due to the low sensitivity of an IR-enhanced InGaAs detector. For a deeper understanding of the electronic properties, the determination of the precise reduction/oxidation potentials of the SWNTs with a wide range of diameters is quite important.

Thus, it is essential to establish empirical correlations that predict the chirality/diameter dependent redox potentials of many (n,m)SWNTs. The chirality/diameter dependence of the redox potential becomes greater in SWNTs with small tube diameters. The concept of the study is presented in Figure 1, in which we recognize that experimental measurements of the redox potentials are possible only for a limited range of SWNT tube diameters, indicating the importance of the empirical correlation to evaluate their electronic states.

Figure 1
figure 1

The concept of this study.

Experimental determination of the redox potentials of the SWNTs based on in situ PL electrochemical analysis is limited to the narrow diameter range (regions b and c) of the SWNTs; an empirical equation is needed in the ranges a and d due to i) the difficulty in synthesis of the SWNTs (region a) and ii) the very low sensitivity of an IR-enhanced InGaAs detector together with the difficulty in individual dissolution of the SWNTs (region d) that make in situ PL electrochemical analysis impossible.

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To achieve this objective, we first experimentally determined the redox potentials of (5,4)SWNTs (d = 0.620 nm), which are the SWNTs with the smallest tube diameter that we could obtain in a sufficient amount for the in situ PL spectroelectrochemical measurement. At the same time, in this study, we determined the redox states of the (6,4)- and (9,1)SWNTs whose diameters are 0.692 and 0.757 nm, respectively. Based on our previously reported data on redox potentials of 15 different (n,m)SWNTs17 and those of the 3 mentioned SWNTs measured in this study, we here present the chirality family dependent empirical equations that correlate the reduction (Ered) and oxidation (Eox) potentials, Fermi levels (EF) and work functions (WF) of the (n,m)SWNTs with a wide range of tube diameters.

We have discovered that based on these equations, the reduction potentials of the SWNTs show a clear chirality family dependence, while the oxidation potentials show no such noticeable chirality family dependence. We also performed quantum chemical calculations based on the Density Functional Tight Binding (DFTB) method to compare the experimentally determined redox values to the calculations.

Results

Determination of reduction and oxidation potentials and Fermi level of the SWNTs

CoMoCAT SWNTs (Southwest NanoTechnologies) were solubilized by carboxymethylcellulose sodium salt (CMC-Na)17,20 (Supplementary Fig. S1) and a CMC-Na-solubilized SWNT film was fabricated on an ITO electrode based on our previous reports17,21.

We recognized the isolated PL band from the (5,4)SWNTs (d = 0.620 nm) in the spectrum of the film measured using a Raman instrument (Supplementary Fig. S2). The chirality assignment is based on a report about the optical properties of the SWNTs with small diameters23. Based on the in situ PL spectral change as well as analysis using the Nernst equation, we determined the Ered, Eox and EF of the (5,4)SWNTs to be −0.59, 0.75 and 0.08 V vs. Ag/AgCl, respectively. We also detected the PL signals mainly from five chiral indices of the SWNTs including (9,1), (8,3), (7,5), (6,5) and (6,4) (d = 0.692–0.829 nm) in the two-dimensional (2D)-map and 3D-surface PL plot of the SWNTs film (Supplementary Fig. S3). We carried out in situ PL spectroelectrochemical measurements of the (6,4)- and (9,1)SWNTs (their redox potentials have not yet determined) according to a previously described method17 and the results are shown in Supplementary Figs. S4–S7 together with those of three other SWNTs. The determined Ered, Eox and EF of the (5,4)-, (6,4)- and (9,1)SWNTs are summarized in Table 1, in which all the data were converted into a potential scale versus the vacuum level22, where EF is defined as (Ered + Eox)/2.

Table 1 Experimentally determined electronic potentials of three (n,m)SWNTs in a CoMoCAT-SWNT sample
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Analysis of the diameter dependence of the electronic properties of the SWNTs

Together with the results previously reported for 15 different (n,m)SWNTs, we now have the redox potentials for a total of 18 different (n,m)SWNTs with the diameter range of 0.62–1.1 nm (Supplementary Fig. S8 and Table S1). In Fig. 2, Ered, Eox and EF are plotted as a function of the inverse nanotube diameter (1/d) using the data obtained from the 18 different (n,m)SWNTs. Both Ered and Eox show an approximate linear dependence on 1/d. The 1/d-dependent fitting lines for Ered and Eox in the figure are consistent with the electronic structure of graphene, from which the electronic structures of the SWNTs can be deduced. However, as can be seen in the figure, Ered was found to exhibit a chirality family pattern just like the optical transition energies of the SWNTs24,25. This tendency is different from the nanotube diameter dependence of Eox; namely Eox shows no such distinct chirality family pattern, only simply proportional to 1/d.

Figure 2
figure 2

Experimentally determined reduction potential, oxidation potential and Fermi level of (n,m)SWNTs.

The experimentally determined reduction potential (Ered), oxidation potential (Eox) and Fermi level (EF) of 18 chiralities of the SWNTs are plotted as a function of the inversed nanotube diameter (1/d). The gray solid curve lines show (2n + m) = constant family patterns. The red and blue solid lines are the fitting lines for Ered and Eox of the armchair SWNTs, respectively. Red open circles: Ered of mod(2n + m, 3) = 1-SWNTs. Red solid circles: Ered of mod(2n + m, 3) = 2-SWNTs. Blue open squares: Eox of mod(2n + m, 3) = 1-SWNTs. Blue solid squares: Eox of mod(2n + m, 3) = 2-SWNTs. Green open triangles: EF of mod(2n + m, 3) = 1-SWNTs. Green solid triangles: EF of mod(2n + m, 3) = 2-SWNTs.

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The Ered values of the mod(2n + m, 3) = 1 and 2-SWNTs are separated by the red fitting line, which corresponds to the line for the imaginary semiconducting SWNTs with the chiral angle θ = 30° (hereafter referred to as the nominal armchair line) (Fig. 2). The (2n + m) = constant family pattern of Ered is also clearly observed in Fig. 2 as shown by the gray lines. In these patterns, the smaller the chiral angles of the SWNTs, the greater the deviation of Ered from the nominal armchair line. This tendency is very similar to the well-known (2n + m) = constant family patterns of the diameter dependence of the optical transition energy of the SWNTs24,25. This result indicates the importance of the chiral angle for the empirical description of Ered. In order to consider the effect of the chiral angle, the Ered and Eox were fitted using eq. 1 with the parameters provided in Table 2 as:

where E is Ered, Eox, or EF and A, B, B′, C and C′ are constants determined by this fitting procedures. These results are shown in Fig. 3 and Table 2. The calculated Ered and Eox show good agreement with the experimental data for the SWNTs for the wide range of diameters. Using these results, we predict Ered, Eox and EF of the SWNTs for a wide range of diameters ranging from ~0.6–3 nm (Fig. 4 and Supplementary Table S2; in this table, we selected 122 chiralities of the SWNTs with the range of diameters of 0.620–1.990 nm. For the plot of the redox states vs. SWNT diameters in the wide range of ~0.6–3.4 nm, see Supplementary Fig. S9), in which we see a distinct family pattern dependence for the reduction potential (Ered), while the oxidation potential (Eox) does not show such a clear family pattern.

Table 2 Fitting parameters of eq. 1 for Ered and Eox of SWNTs
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Figure 3
figure 3

Comparison between experimentally determined and predicted reduction potential, oxidation potential and Fermi level of (n,m)SWNTs.

The experimentally determined and predicted reduction potential (Ered), oxidation potential (Eox) and Fermi level (EF) for the 18 chiralities of the SWNTs are plotted as a function of the inversed nanotube diameter (1/d). Red solid circles: Ered of mod(2n + m, 3) = 1-SWNTs. Red solid squares: Ered of mod(2n + m, 3) = 2-SWNTs. Blue open circles: predicted Ered of mod(2n + m, 3) = 1-SWNTs. Blue open squares: predicted Ered of mod(2n + m, 3) = 2-SWNTs. Blue solid circles: Eox of mod(2n + m, 3) = 1-SWNTs. Blue solid squares: Eox of mod(2n + m, 3) = 2-SWNTs. Red open circles: predicted Eox of mod(2n + m, 3) = 1-SWNTs. Red open squares: predicted Ered of mod(2n + m, 3) = 2-SWNTs. Green solid circles: EF of mod(2n + m, 3) = 1-SWNTs. Green solid squares: EF of mod(2n + m, 3) = 2-SWNTs. Orange open circles: predicted EF of mod(2n + m, 3) = 1-SWNTs. Orange open squares: predicted EF of mod(2n + m, 3) = 2-SWNTs.

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Figure 4
figure 4

Prediction of reduction and oxidation potentials and Fermi levels of (n,m)SWNTs.

Predicted reduction potential (Ered), oxidation potential (Eox) and Fermi level (EF) of the 18 chiralities of the SWNTs are plotted as a function of the inversed nanotube diameter (1/d). Red circles: predicted Ered of mod(2n + m, 3) = 1-SWNTs. Blue circles: predicted Ered of mod(2n + m, 3) = 2-SWNTs. Red squares: predicted Eox of mod(2n + m, 3) = 1-SWNTs. Blue squares: predicted Eox of mod(2n + m, 3) = 2-SWNTs. Red triangles: predicted EF of mod(2n + m, 3) = 1-SWNTs. Blue triangles: predicted EF of mod(2n + m, 3) = 2-SWNTs.

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In addition to the experiments, we also performed calculations on the same set of nanotubes using the Density Functional based Tight Binding (DFTB) method. We used the band-structure around the Fermi level to predict the redox properties of the (n,m)SWNTs and assumed a linear correlation of the energy of the conduction band minimum (CBM) and valence band maximum (VBM) with the oxidation and reduction potentials, respectively. This approach omits several important contributions to the redox potential, e.g., reorganization energies and solvent effect. However, the results show that the chirality family patterns and diameter dependence of the nanotube band structure still can reproduce most of the experimentally observed trends.

When the conduction and valence band energies are plotted as a function of the inverse diameters (1/d), the chiral family pattern is much more recognizable in the conducting band energies, while the valence band energies follows a smooth, quasi-linear 1/d dependence. Both this asymmetry and the calculated family pattern are in good agreement with the experimental data, in which only the reduction potentials show a characteristic chiral family pattern (see Supplementary Figure S11). The obtained similarities are even more pronounced when we compare the calculated band gaps with the electrochemical band gaps (see Supplementary Figures S12 and S13). The two values are very well correlated and the calculated energy gap values systematically underestimate the experimental ones by about 0.1 eV.

Discussion

We initially discuss the accuracy of the empirical formula for Ered, Eox, Eelc and Eex. The average relative errors for Ered, Eox, Eelc and Eex were 0.3, 0.3, 1.4 and 0.8%, respectively. The results show that the fitting for Ered, Eox, Eelc and Eex well reproduced the experimental data. Next, we discuss the range of applicability of the empirical formula. We expect that the empirical formula for the larger diameter range can be extrapolated with a good accuracy because the dispersion relation of the SWNTs with large diameters should be very similar to that of graphene and become a nearly linear function of 1/d. On the other hand, for the smaller diameter range, we have observed the electronic potentials of the SWNTs with very small diameters as shown in Table 1. These observations enabled us to construct a good empirical formula applicable for SWNTs with diameters as small as ~0.6 nm. For SWNTs with even smaller diameters (d < 0.6 nm), however, the fitting should not be applicable, because the π- and σ- orbitals of the carbon atoms should be significantly hybridized. In such a case, the energy dispersion relation of the SWNTs no longer simply inherits the dispersion relation of the graphene and should exhibit a strong curvature effect that cannot be expressed by our empirical formula.

We now discuss how we can understand this apparently different family pattern of Ered and Eox. It is known that the valence and conduction bands of graphene are nearly symmetric26 and the energy dispersion relation of the SWNTs inherits the electron-hole symmetry of graphene. The reduction and oxidation potentials correspond to the conduction band bottom energy and valence band top energy, respectively. Hence, assuming that the Fermi energy is identical for all the (n,m)SWNTs, we expect a nearly symmetric family pattern for the Ered and Eox; however, as already described, the experimental data show a distinct family pattern only for Ered. The key to understanding this apparent paradox lies in the chirality family dependence of WF of the SWNTs. A previous theoretical study has predicted that the WF of SWNTs with diameters less than about 1 nm strongly depends on the specific chiral structure of the SWNTs27. The WF and Fermi energy EF are directly related, therefore, if the WF depends on the diameter and chiral angle, EF may also depend on them.

The green circles and squares shown in Fig. 3 are the EF values obtained from the experimentally determined Ered and Eox. It was revealed that the EF for the “mod = 1” family is always greater than those for the “mod = 2” family, especially for the small diameter SWNTs and their deviation from the asymptotic value ( = 4.45 eV) disappears for the larger diameter SWNTs. For the small diameter SWNTs, the deviation becomes significant. A similar tendency for the WFs of SWNTs with small diameters has been reported by Barone et al.27 based on a theoretical study. They reported that the WF of the SWNTs with a diameter greater than 0.9 nm asymptotically and smoothly converges to the graphene limit of 4.6 eV. They also found that the WF of SWNTs with a smaller diameter exhibits a strong dependence on the diameter and chiral angle27, in which the SWNTs of the mod(2n + m, 3) = 1 family, such as the (6,4) SWNTs, have lower WFs in the diameter range of ~0.67–1.33 nm, which is consistent with our experimental observations. Therefore, we attributed the observed asymmetric behavior of Ered and Eox to the chirality dependence of the WFs.

Finally, the electrochemical band gap, which is the difference between the Ered and Eox, Eelc = Ered − Eox, was evaluated. The previously reported empirical Kataura plot25 describes the optical transition energy, which corresponds to the exciton energy Eex. However, the electrochemical band gap obtained in this study corresponds to the one-electron band gap without an offset (binding energy) due to the excitonic effect. We fit Eelc using eq.1, which gives the empirical predictions of the band gaps of the SWNTs. The results are shown in Supplementary Fig. S10 and Table 3.

Table 3 Fitting parameters of eq. 1 for electrochemical band gap (Eelc) and exciton energy (Eex) of SWNTs
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The results of the fitting contain important information on the electronic many-body effects in the SWNTs. We found that the constant A in eq.1 for the Eelc is zero, while we need a finite value of A = ~0.10 eV for the fitting of Eex (Supplementary Fig. S10 and Table 3). The finite A value for the exciton energy is attributed to the many-body correction energy that only weakly depends on the tube diameter in the diameter range observed in this study26. The coefficient B may correspond to the Fermi velocity of graphene and the electron hopping integral γ0 between the nearest neighbor carbon atoms. We obtained γ0 = ~2.85 eV from the coefficient B determined for Eelc. This value of γ0 is close to the conventionally predicted value of the π-electron hopping integral for graphene and the SWNTs28.

The observed values of Eelc are somewhat counter intuitive, because the quasi particle band gap in the intrinsic SWNTs is predicted to be strongly enhanced because of the many-body electronic correlation effects28,29, which may lead to much higher values of Eelc than Eex. The results thus imply that the electronic correlation energy at the electrochemically-doped sites on the SWNTs is strongly suppressed due to the screening by electrochemically injected electrons or holes and the dielectric response of the surrounding media with a very high dielectric constant (water) and the Eelc simply reflects the single particle hopping integral of the π-electrons.

The absence of the A coefficient for Eelc also indicates that there is no energy offset due to the many-body effect. In addition, in our previous study, we have found that the electrochemical band gaps of the SWNTs strongly depend on the dielectric constant of the materials surrounding the SWNTs21, which further supports the above understanding of the electrochemical band gaps. Recently, Spataru and Léonard30 have theoretically predicted that the quasi particle band gap decreases with the increasing hole density in the SWNTs due to the additional screening effect induced by the doped holes. Therefore, our results are qualitatively consistent with the theoretical predictions of the band gap change under the doped conditions.

We carried out quantum chemical calculations based on the DFTB methods to compare the experimental data to the calculations and revealed the following two points:

  1. i

    The calculations are able to reproduce all the important trends that we observed in the electrochemical experiments and confirm the new observations, such as the asymmetry of the chirality family dependence of the reduction and oxidation potentials.

  2. ii

    The accuracy of such calculations does not allow a high quality prediction of the redox properties at this point, therefore, they are not suitable to confirm the correctness of the suggested empirical relationship.

In conclusion, we established empirical equations that describe the dependence of the redox potentials of the SWNTs with a wide range of diameters and chiral angles. Without using the presented empirical equations, the determination of these potentials for SWNTs with larger diameters (>1.5 nm) is difficult or almost impossible. In this study, we discovered that the reduction potentials of the SWNTs showed a distinct chirality family dependence, while the oxidation potentials did not show such family pattern dependence. This difference is explained by the chirality family dependence of the WF of the SWNTs and also confirmed by the theoretical calculations. These calculations also showed that an accurate computational prediction of the redox properties is still a challenging task, therefore, empirical relationships between the structure and redox properties can be very useful and still have a strong predicting power. Finally, we would like to emphasize that the present study provides information on the fundamental properties of the SWNTs and graphene. This will lead to a deep understanding of their electronic states, which are essential for developing applications of these nanomaterials.

Methods

Materials

CoMoCAT-SWNTs, carboxymethylcellulose sodium salt (CMC-Na, Fig. S1a) and poly(diallyldimethylammonium chloride) (PDDA, Fig. S1b) were purchased from Southwest NanoTechnologies, Inc., Kishida Chemical and Aldrich, respectively and were used as received. All other chemicals were of reagent grade and used as received.

Fabrication of modified electrode

The CoMoCAT-SWNTs (0.3 mg) were added to an aqueous solution (8 mL) of CMC-Na (0.1 wt%) and the mixture was sonicated (ultrasonic cleaner, Branson 5510) for 1 h, followed by ultracentrifugation at 150,000 g for 5 h. The SWNTs were individually solubilized in an aqueous solution of CMC-Na. A 100 μL aliquot of the resultant solution was placed on a cleaned indium tin oxide-coated transparent quartz glass electrode (ITO) and then heated at ca. 100°C to obtain a SWNTs/CMC-Na film-modified electrode. A 100 μL portion of an aqueous solution of PDDA (20 wt%) was then placed on the film followed by rinsing with water to remove any excess PDDA to obtain a SWNTs/CMC-Na/PDDA-modified electrode. PL spectroelectrochemistry was carried out using the obtained modified electrode.

PL and raman spectroelectrochemistry

Water was purified using a Milli-Q Plus Ultrapure water system coupled with an Elix-5 kit (Millipore Co.). Its resistivity was over 18 MΩ·cm. The base electrolyte solution was 0.3 M aqueous NaCl containing 30 mM Na2HPO4 (pH 8). All the electrochemical measurements were conducted in an aqueous system using an Ag/AgCl (sat'd KCl) reference electrode and a coiled Pt wire counter electrode under an Ar gas atmosphere. The PL and Raman spectroelectrochemistry were carried out using a spectrofluorometer (HORIBA JOBIN YVON, FL3-21) and a Raman spectrophotometer (Kaiser Optical System, RXN) both equipped with a potentiostat (TOHO Technical Research Co., PS-06). The Raman scattering measurement was carried out at the excitation intensity of 10 mW.

Calculations

The calculations were performed using the DFTB+ program (ver 1.2) with the mio Slater-Koster set31. The nanotube models were calculated as the 3D periodic system with at least a 4.0 nm separation between the tubes in the x and y directions. The calculations were performed with the 2 × 2 × 8 Monkhorst–Pack grid32. The unit cell size as well as the positions of the carbon atoms were optimized with a convergence criteria of 5 meV. The electronic converge criteria was 0.27 eV. Broyden mixing was applied with the mixing parameter of 0.233.