## Abstract

Spectrofluorimetric measurements on single-walled carbon nanotubes (SWNTs) isolated in aqueous surfactant suspensions have revealed distinct electronic absorption and emission transitions for more than 30 different semiconducting nanotube species. By combining these fluorimetric results with resonance Raman data, each optical transition has been mapped to a specific (*n*,*m*) nanotube structure. Optical spectroscopy can thereby be used to rapidly determine the detailed composition of bulk SWNT samples, providing distributions in both tube diameter and chiral angle. The measured transition frequencies differ substantially from simple theoretical predictions. These deviations may reflect combinations of trigonal warping and excitonic effects.

Many of the major challenges currently facing basic and applied research on SWNTs arise from the diversity of tube diameters and chiral angles in samples formed through any preparative method. Because the electronic and optical properties of nanotubes vary substantially with structure (1), these samples show mixed characteristics that hinder basic investigations and limit some of the most promising applications. The recent discovery of band-gap fluorescence from semiconducting SWNTs in aqueous micelle-like suspensions (2) opens the door to powerful experimental approaches that can extract the spectroscopic properties of specific tube structures through bulk measurements. We report such a spectrofluorimetric study, in which distinct first and second van Hove optical transitions are measured for 33 different nanotube species and then assigned to specific structures. Our findings provide a tool for analyzing the detailed composition of bulk nanotube samples, one that can be adapted to monitor and guide processes aimed at separating tubes by type.

Figure 1A illustrates how each SWNT structure is indexed by two integers (*n*,*m*) that define the length (π times the tube diameter,*d*_{t}) and chiral angle (α) of that tube's roll-up vector on a graphene sheet. We omit structures having *n* –*m* evenly divisible by 3, because they are metals or semimetals and will not fluoresce. Figure 1B shows the qualitative pattern of sharp van Hove peaks, which arise from quasi–one-dimensionality, predicted for electronic state densities of semiconducting SWNTs. As shown in the figure, light absorption at photon energy *E*_{22} is followed by fluorescence emission near *E*_{11}. The values of*E*_{11} and *E*_{22} will vary with tube structure (3). To study these transitions, we first processed SWNTs grown in high-pressure carbon monoxide (HiPco) to obtain samples rich in individual nanotubes suspended by SDS surfactant in deuterium oxide (2). We then performed spectroscopic measurements using a commercial spectrofluorometer (a J-Y Spex Fluorolog 3-211) equipped with an indium-gallium-arsenide near-infrared detector cooled by liquid nitrogen. Emission intensity was measured as a function of both excitation wavelength (from 300 to 930 nm) and emission wavelength (from 810 to 1550 nm), with 3-nm steps and 7-nm spectral slit widths on both axes. The resulting 52,000 measurements were corrected for instrumental variations in excitation intensity and detection sensitivity to give the results shown in Fig. 2A as a contour plot.

The white oval drawn in Fig. 2A marks a region of special interest that contains discrete peaks, arising from the transitions shown in Fig. 1B, in various semiconducting nanotube species. Careful analysis reveals 33 such features. All absorption peaks in this region are also peaks for fluorescence excitation. Additional peaks at shorter excitation wavelengths are attributed to v_{3} → c_{3} excitation followed by c_{1} → v_{1}emission in the same set of nanotubes (Fig. 1B). Between these two branches of peaks lies a region that shows very low emission intensity despite strong absorption. We attribute this nonemissive region to metallic and semimetallic nanotubes in the sample, which are expected to absorb between the semiconductors' v_{2} → c_{2} and v_{3} → c_{3} branches (4) but not to emit measurable luminescence. Several of the features outside the circled region at longer excitation wavelengths are v_{1} → c_{1} vibronic bands (electronic transitions with vibrational excitation). Within the circled region of current interest, many spectral features that overlap in simple emission or absorption spectra are clearly separated in our two-dimensional excitation-emission spectrum. The coordinates of each peak in this region give the v_{2} → c_{2} optical excitation energy (*E*
_{22}
* = hc/*λ_{22}
* = hc*ν̄_{22}) and the corresponding c_{1} → v_{1} emission energy (*E*
_{11}
* = hc/*λ_{11}
* = hc*ν̄_{11}) for one semiconducting nanotube species (*h* is Planck's constant, *c* is the speed of light, λ is photon wavelength, and ν̄ is photon frequency in cm^{–}).

A crucial goal is to identify the specific (*n*,*m*) nanotube structure that gives each observed spectral peak. The circles in Fig. 2B plot experimental spectral peak positions in the region of interest. Clear patterns are evident, as shown by the solid curves drawn through the data points. To help interpret these patterns, we plotted the ratio of optical excitation to emission energies for each peak versus the peak's excitation wavelength. The result is shown in Fig. 2C. The pattern of solid lines through these experimental points qualitatively resembles the pattern shown in Fig. 2D, which displays computed findings from an extended tight-binding model calculation on SWNTs (5). In this computation, lines have been drawn through points of species that share the same value of *n – m*. We refer to these sets as nanotube families. The analogy between the simulation of Fig. 2D and the experimental pattern of Fig. 2C lets us sort the observed data points into families marked by the solid lines in Fig. 2C (or the black lines in Fig. 2B). Another step in the assignment process is recognizing that, within a family, the values of *n* and *m* for any point must both be one greater than for the adjacent point at a shorter wavelength. Also, the simulation shows that the transition energy ratios deviate increasingly from a central value as the value of*n – m* increases. The direction of this deviation depends on whether dividing *n – m* by 3 leaves a remainder (mod) of 1 or 2 (6, 7). (We refer to these two subsets as “mod 3 = 1” and “mod 3 = 2” structures, respectively; “mod 3 = 0” tubes are absent from our fluorescence data because they are metallic or semimetallic.) This pattern of family ratios provides a third step in deducing the assignment. A further important step is the linkage pattern across families illustrated by the colored curves in Fig. 2B (a subset of which are drawn as dashed curves in Fig. 2C). Simulation shows that adjacent points on these curves differ either by +1 in *n*value and –2 in *m* value, or by +2 in *n* and –1 in *m *(8).

The relations outlined above combine to define a complete connectivity that maps the whole pattern of experimental spectral features onto the array of nanotube structures symbolized by points on a graphene lattice (Fig. 1A). However, the assignment mapping remains ambiguous until it is “anchored” by determining the specific (*n*,*m*) values associated with a particular spectral transition. We found that there are a few plausible global assignments defined by closely related anchor choices. To select from among these, we measured resonance Raman spectra on our samples using several laser wavelengths. It is well known that the radial breathing mode (RBM) frequency of SWNTs has a simple monotonic relation to tube diameter, and that Raman scattering from this mode is strongly and selectively enhanced when the photon energy matches a nanotube's optical transition (9–12). From our Raman measurements, we compiled a set of 10 RBM frequencies that were correlated with specific spectral features of the data in Fig. 2A. A candidate assignment was tentatively invoked to give (*n*,*m*) values, and thereby tube diameters, for each spectral feature. Within this assignment, the set of Raman frequencies was analyzed using the deduced diameters and the relationto find optimal values for parameters*A* and *B *(13). We repeated this process of Raman frequency analysis for each candidate assignment. Figure 3A shows the resulting root mean square (rms) error in fitting the Raman frequencies as a function of parameter *A*(with *B *optimized) for the leading candidates. Assignment “a” is strongly preferred because of its sharp minimum and low error of only 1 cm^{−1}. In addition, the deduced RBM parameters of *A* = 223.5 cm^{−1} nm and*B* = 12.5 cm^{−1} for assignment “a” agree satisfactorily with prior independent reports (14–18). Figure 3B is a plot of measured RBM Raman frequencies versus inverse nanotube diameter in the chosen assignment “a,” along with a line computed from the optimized parameters. Table 1 lists the set of observed transition wavelengths, corresponding photon energies, deduced (*n*,*m*) assignments, and RBM frequencies. Figure 4A shows, on a graphene lattice, the 33 nanotube structures identified though spectroscopic analysis of our sample.

The deduced assignment implies a match within 10% in the numbers of observed fluorescing nanotube species with mod 3 = 1 and mod 3 = 2. With spectral features assigned to specific (*n*,*m*) species, the relative abundances of those species may be determined from the intensities of the corresponding features. Figure 4B shows the measured fluorescence intensity as a function of nanotube diameter and chiral angle. Lacking specific knowledge of any variations of nanotube fluorescence quantum yields with diameter or chiral angle, we presume that the distribution shown as a smoothed surface in Fig. 4B closely reflects the distribution of nanotube abundance in the sample. The measured distribution has a peak in diameter near 0.93 nm and extends from approximately 0.6 to 1.3 nm at ∼2% of the peak value. In the chiral angle dimension, intensity falls from a maximum at the largest angles (near-armchair) to a minimum at 0° (zigzag), with no zigzag species detected in our data. It has been suggested that armchair chiralities may be favored by greater stability of armchair-like edge structures during the HiPco growth process (19). In contrast to prior electron nanodiffraction measurements on SWNT bundles, which have deduced both armchair-rich and uniform chirality abundances (20, 21), our spectroscopic approach reveals distributions for isolated nanotubes contained in bulk samples.

Variations of transition energies with diameter and chirality are also of substantial interest for understanding nanotube electronic states and excitations. The simplest theoretical treatments predict that the optical transition wavelengths of semiconducting nanotubes depend linearly on diameter according to the relationswhere *d*
_{t} is the tube diameter, *a*
_{CC} is the C–C bond distance, and γ_{0} is the interaction energy between neighboring C atoms (22). In Fig. 4C, we plot measured optical transition wavelengths λ_{11} and λ_{22}versus deduced tube diameter. Deviations from linear relations are obvious and in qualitative agreement with the pattern expected from trigonal warping effects in carbon nanotubes (6,7). Such effects shift transition frequencies from the linear relation in a direction that depends on the value of (*n – m*) mod 3 and that reverses between first and second van Hove branches. Figure 4C also shows straight lines representing our best estimates of the undeviated transition wavelengths as a function of tube diameter. These lines separate the sets of transitions observed for mod 3 = 1 and mod 3 = 2 species.

To analyze spectral deviations in more detail, we computed frequency differences between the observed transitions and those indicated for the same tube diameter by the lines in Fig. 4C. These deviations, shown as open circles, are plotted as a function of nanotube chiral angle (α) in Fig. 4D. Also shown are solid lines with values computed from a simple model in which deviations are given by*A _{i}
*cos(3α)/

*d*

_{t}

^{2}. We selected this functional dependence on chiral angle and diameter from theories of trigonal warping effects (6,7), and we allowed the

*A*factors to vary with the value of (

_{i}*n – m*) mod 3 and with the van Hove transition index,

*i*, to optimize agreement with the experiment. Although there are some systematic differences between this model and the data in Fig. 4D, the qualitative agreement is excellent. In addition, the results show clear patterns within and between families, which are labeled in the figure with their

*n*–

*m*values.

By combining empirical expressions for the linear “undeviated” wavelengths with those for the deviations, we obtain the following formulas for first and second van Hove optical transition frequencies of SWNTs as a function of their diameter (in nm) and chiral anglewith *A*
_{1} = −710 cm^{−1} for (*n – m*) mod 3 = 1 or*A*
_{1} = 369 cm^{−1} for (*n – m*) mod 3 = 2; andwith *A*
_{2} = 1375 cm^{−1} for (*n – m*) mod 3 = 1 or*A*
_{2} = −1475 cm^{−1} for (*n – m*) mod 3 = 2. These simple formulas will be valuable for predicting transitions of nanotube species not observed in our samples, including larger diameter tubes produced by other growth processes. Using four adjusted parameters for each branch, the formulas describe observed transitions with average errors of less than 65 cm^{−1} (8 meV) for ν̄_{22 }and 35 cm^{−1} (4 meV) for ν̄_{11}. These values fall well within the spectral line widths found for these transitions, which are approximately 400 to 800 cm^{−1} (50 to 100 meV) for ν̄_{22 }and 200 cm^{−1} (25 meV) forν̄_{11}.

Our results provide a simple and practical spectroscopic route for finding the detailed composition of bulk nanotube samples. This method can be adapted for rapid analysis to guide processes that produce or sort nanotubes selectively by structure. In the area of nanotube physics, we find that optical transition frequencies deviate from a simple diameter dependence more strongly than expected. For example, the ν̄_{22} value of the (9,2) tube is 26% higher (by 3700 cm^{−1} or 0.46 eV) than that of (9,1), even though its diameter is 6% larger. In addition, as tube diameter increases, theν̄_{22}/ν̄_{11} ratio apparently approaches a value smaller than 2 (23), which is predicted by most theories (6). It seems likely that phenomena beyond simple trigonal warping, and possibly including strong exciton effects (24, 25), will be revealed by quantitative theoretical analysis of these nanotube spectral data.

↵* To whom correspondence should be addressed. E-mail: weisman{at}rice.edu