Structural (n, M) Determination of Isolated Single-wall Carbon Nanotubes by Resonant Raman Scattering

We show that the Raman scattering technique can give complete structural information for one-dimensional systems, such as carbon nanotubes. Resonant confocal micro-Raman spectroscopy of an ͑n, m͒ individual single-wall nanotube makes it possible to assign its chirality uniquely by measuring one radial breathing mode frequency v RBM and using the theory of resonant transitions. A unique chirality assignment can be made for both metallic and semiconducting nanotubes of diameter d t , using the parameters g 0 ෇ 2.9 eV and v RBM ෇ 248͞d t. For example, the strong RBM intensity observed at 156 cm 21 for 785 nm laser excitation is assigned to the ͑13, 10͒ metallic chiral nanotube on a Si͞SiO 2 surface. Much effort has been devoted to improving the methods of nanotube production, and significant progress has been made to narrow the diameter distribution of nanotubes produced by different catalysts and growth processes [1]. However, there is as yet no method for selecting a specific chirality in the nanotube production process, since the nanotube structural energy is only weakly dependent on chirality [2]. Consequently, actual nanotube samples appear to exhibit a homogeneous chirality distribution. Thus the development of techniques to assign a chirality ͑n, m͒ to a given single-wall carbon nanotube (SWNT) is very important for the development of both future technological applications and scientific studies. Resonant confocal micro Raman spectroscopy (RCMRS) provides a powerful technique to study the quantum properties of electrons and phonons in carbon nanotubes [3], and to identify the nano-tube diameters (d t) in a sample containing a mixture of chi-ralities. We show here how RCMRS can give the complete ͑n, m͒ atomic structural assignment for an isolated SWNT. Isolated SWNTs were prepared by a chemical vapor deposition method on a Si substrate containing nanometer size iron catalyst particles. The Si substrate was oxidized to have a thin SiO 2 surface coating and, in this case, we do not expect significant charge transfer between the SWNTs and the substrate. Since the nanotubes nucleate and grow from well isolated catalyst particles, nanotube bundles are not formed. We have transferred these nanotubes from the surface to AFM probe tips and have confirmed that they are individual single-walled tubes by transmission electron mi-croscopy [4]. Figure 1 shows an atomic force microscopy (AFM) image of one of our samples exhibiting a concentration of 6 6 3 SWNTs per mm 2. Most nanotubes are individual SWNTs, …

Much effort has been devoted to improving the methods of nanotube production, and significant progress has been made to narrow the diameter distribution of nanotubes produced by different catalysts and growth processes [1].However, there is as yet no method for selecting a specific chirality in the nanotube production process, since the nanotube structural energy is only weakly dependent on chirality [2].Consequently, actual nanotube samples appear to exhibit a homogeneous chirality distribution.Thus the development of techniques to assign a chirality ͑n, m͒ to a given single-wall carbon nanotube (SWNT) is very important for the development of both future technological applications and scientific studies.Resonant confocal micro Raman spectroscopy (RCMRS) provides a powerful technique to study the quantum properties of electrons and phonons in carbon nanotubes [3], and to identify the nanotube diameters (d t ) in a sample containing a mixture of chiralities.We show here how RCMRS can give the complete ͑n, m͒ atomic structural assignment for an isolated SWNT.
Isolated SWNTs were prepared by a chemical vapor deposition method on a Si substrate containing nanometer size iron catalyst particles.The Si substrate was oxidized to have a thin SiO 2 surface coating and, in this case, we do not expect significant charge transfer between the SWNTs and the substrate.Since the nanotubes nucleate and grow from well isolated catalyst particles, nanotube bundles are not formed.We have transferred these nanotubes from the surface to AFM probe tips and have confirmed that they are individual single-walled tubes by transmission electron microscopy [4].Figure 1 shows an atomic force microscopy (AFM) image of one of our samples exhibiting a concentration of 6 6 3 SWNTs per mm 2 .Most nanotubes are individual SWNTs, but a few of them are entangled with each other, showing a "Y" shape in some AFM images.The in-set to Fig. 1 shows the wide diameter distribution of the sample (ϳ1 , d t , ϳ3 nm).The accuracy of the AFM diameter measurement is 60.2 nm at most.Resonant Raman spectra from 100 to 1900 cm 21 were obtained from single isolated SWNTs on this substrate, using a Kaiser Optical Systems, Hololab 5000R: Modular Research Micro-Raman Spectrograph (1 mm laser spot) with 25 mW power, and E ᐉ 785 nm 1.58 eV laser line excitation.
Figure 2 shows several typical Raman spectra, each taken from one resonant nanotube on the Si substrate.Inset (a) shows the spectrum obtained from one spot on the sample (Fig. 1 frequency region, where we observe the nanotube tangential G-band modes.The main part of Fig. 2 shows a zoom for the low frequency region (100 350 cm 21 ), where we observe the 303 cm 21 peak from the Si substrate and the nanotube radial breathing modes (RBM).The RBM intensity is comparable to that of the quasi-3D Si film because of the large 1D resonant Raman enhancement effect in the nanotubes.The three spectra come from three different spots on the substrate, showing the presence of only one resonant nanotube and one RBM frequency for each spot, with v RBM 148, 164, and 237 cm 21 for the three spots, each exhibiting natural line widths of 5 10 cm 21 [6,7].As discussed below, looking at the frequency and intensity of the RBM features in the RCMRS spectra, it is possible to uniquely assign ͑n, m͒ values to the observed resonant SWNTs [see ͑n, m͒ assignment on Fig. 2] based on theory.
Figure 3(a) shows a plot of the measured frequency vs intensity for the RBM peak observed at 42 different spots where we found resonant SWNTs.Figure 3(a) has 47 points, since we occasionally observe the appearance of 2 or even 3 RBMs at a single light spot.The surprising appearance of sufficient Raman intensity for a Raman signal from an isolated SWNT is due to the resonance condition for Raman spectroscopy that is obtained when an energy separation E ii between van Hove singularities (vHs) [8,9] is close to the laser excitation energy E ᐉ 1.58 eV.Therefore, although the sample has a relatively large density of isolated SWNTs (6 6 3 SWNTs per laser spot), the probability of finding a SWNT in resonance with the E ᐉ 1.58 eV laser line is only about 1͞10.Many spots on the sample must be measured before we find a resonant Raman signal from one isolated SWNT.0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Calculated energy separation E ii as a function of 1͞d t , g 0 2.9 eV, and v RBM 248͞d t .Circles are for metallic SWNTs (E M ii ), and crosses for semiconducting SWNTs (E S ii ).
Figure 3(b) shows a plot of the energy separations E ii between the vHs for a given ͑n, m͒ SWNT as a function of 1͞d t .The vHs in the density of states (DOS) are calculated using the tight binding calculation with the parameters g 0 2.9 eV and a C2C 0.144 nm, which reproduces the resonant Raman spectra of SWNTs very well [9,10].The SWNTs predicted to be resonant must have E ii within a resonant window E ᐉ 6 0.10 eV.Figures 3(a) and 3(b) can be correlated since v RBM a͞d t .With the value a ϳ 248 cm 21 nm, we obtain good agreement between the observed resonant RBM frequencies and calculated values for v RBM from the E M 11 and E S ii (i 2, 3, and 4 for semiconducting SWNTs) interband transitions (see Fig. 3).In fact, when we vary a by 5%, we no longer obtain good agreement with our measured v RBM data.
The assignment of only one SWNT is needed for a precise determination of a.We here show that this is possible by analyzing the observed RBM intensities.In the present single nanotube spectroscopy experiment, the DOS is that of an individual nanotube, and the DOS is highly singular.Because of the trigonal warping effect [9], each interband transition E M ii for metallic SWNTs is split into two DOS peaks, and there are two resonant conditions for each chiral (0 , u , 30 ± ) or zigzag (u 0 ± ) nanotube.The separation between DOS peaks decreases with increasing u, and is zero for armchair nanotubes (u 30 ± ) [9,11].Thus an especially large Raman intensity is expected when: (i) an E ii is close to E ᐉ , and when (ii) the DOS splitting is small.Thus, armchair or chiral nanotubes with a large chiral angle (u near 30 ± ) have a higher probability for giving the strongest signal in the single nanotube Raman experiment among the different chiralities in the resonant energy window (E ᐉ 6 0.1 eV).
Figure 3(a) shows one unusually high intensity RBM peak at v RBM 156 cm 21 .This peak should obey the two conditions given above for large intensity.Considering theory, the observed highest intensity v RBM 156 cm 21 comes from a metallic SWNT.When we select metallic nanotubes [where n 2 m in ͑n, m͒ is a multiple of 3] and the resonant window is 1.48 , E M 11 , 1.68 eV, 17 different chiralities are found theoretically, corresponding to 14 different calculated v RBM frequencies that, for a 248 cm 21 nm, are in the range ϳ144 , v RBM , 174 cm 21 , as listed in Table I.This is consistent with the experiments where 12 different frequencies in the range 144 , v RBM , 176 cm 21 within 61 cm 21 experimental accuracy are identified from Raman spectra taken at 42 different light spots.Since it is not always possible in our experiment to find all possible chiralities for a finite number of light spots, the number of experimental modes should always be smaller than the theoretical number.Table I shows that there are two armchair nanotubes, ͑11, 11͒ and ͑12, 12͒, within this resonant window, with E M 11 1.63 and 1.50 eV, respectively.These values are not very close to E ᐉ and the first condition for high intensity is not satisfied.Furthermore, using a 248 cm 21 nm, their calculated v RBM are 164.0 and 150.3 cm 21 , respectively, that are not close to the most intense peak observed at 156 cm 21 .Among the 17 different chiralities in Table I, there are only a few nanotubes with large chiral angles u .20 ± .When we use g 0 2.9 eV, the two E M 11 singularities for ͑13, 10͒ are 1.58 and 1.55 eV.One of the E M 11 values agrees very well with E ᐉ , and the other is also close.Therefore, it is reasonable to assign the ͑13, 10͒ chirality to the strongest observed intensity peak at 156 cm 21 .The parameter of 248 cm 21 nm in v RBM 248͞d t is thus determined, so that the v RBM value for the ͑13, 10͒ nanotube becomes 156.3 cm 21 .Furthermore, the empirical parameter of 248 cm 21 nm is justified by the other intense v RBM features with E M 11 closer to E ᐉ as shown in bold face in Table I [see also Fig. 3(a)].Therefore, the assignment of this one SWNT gives us a value for a, which is consistent with ͑n, m͒ assignments for all the other 37 RBMs that are observed (both in frequency and intensity).This assignment is unique in the sense that there is no alternative choice of ͑n, m͒ for explaining the large number of observed Raman frequencies and their relative intensities.
We emphasize here that we do not compare the calculated and experimental intensities quantitatively, since the measured intensity depends on the nanotube length and exact position of the experimental laser spot.In fact, in the experiment, we get  b Two E M 11 values for each chiral ͑n, m͒ SWNT are also given.c Ambiguity of the assignment occurs between ͑20, 2͒ and ͑21, 0͒.
4 different v RBM frequencies.This result is reasonable, first because of the smaller number of possible chiralities, and second because of the fewer small diameter nanotubes in the sample (see inset to Fig. 1).In Table II we list v RBM values for semiconducting nanotubes.The experimental values of 210, 229, 237, and 239 cm 21 for v RBM are assigned with no adjustable parameters to the ͑14, 1͒, ͑11, 4͒, ͑10, 5͒, and ͑8, 7͒, respectively.Again the agreement (Table II) is excellent, since five of the six observed SWNTs are expected to be strongly resonant, and are thus easier to find.The value a 248 cm 21 nm and a unique ͑n, m͒ assignment from v RBM is again confirmed.
Unique ͑n, m͒ assignments with Raman spectroscopy could not be made for SWNTs with large d t .By increasing d t (decreasing v RBM ), the number of resonant SWNTs within the resonant window increases, making a unique ͑n, m͒ assignment difficult.For lower observed v RBM values (below 136 cm 21 ), we calculate and find 48 semiconducting ͑n, m͒ for v RBM .100 cm 21 within our resonant window.However, when we select the energy window 1.48 , E S ii , 1.68 eV (i 3, 4), the corresponding v RBM range goes up to 132 cm 21 , consistent with experimental observations up to 136 cm 21 [see Fig. 3(a)].
Since the assignments of v RBM for many metallic and semiconducting nanotubes are very good, we conclude that the experimental value a 248 cm 21 nm, based on the theoretical assignment of v RBM 156 cm 21 for the ͑13, 10͒ chiral nanotube, is justified.This experimental empirical value is larger than the previously published value (a 223.75 cm 21 nm [12]) due to limitations on force constant model calculations, in which the force constants are taken from those of graphite.This 10% difference in a might be due to interlayer interactions, or to interactions between the nanotubes with air or with the substrate.These possible interaction mechanisms require further investigation with regard to their effect on a.Though the value of a 248 cm 21 nm should be used for samples similarly prepared and on similar substrates, other sample preparation methods and substrates may require a small change in the a value.What is shown in the present work, is a proper evaluation of a leads to a unique determination of ͑n, m͒ for a single isolated SWNT by measuring v RBM through the RCMRS technique in conjunction with theory.
A method for evaluating the reliability of the determination of a is also presented.In future work, use of a marked Si͞SiO 2 substrate will allow us to mark the position of the light spot, so that the ͑13, 10͒ chirality could be observed directly by an STM measurement, thus providing independent confirmation of the theory.We can also check our theoretical prediction by using another laser excitation energy on the same sample.However, it should be mentioned here that a good resonant condition for carbon nanotubes is not always obtained using a given laser energy.The selection of E ᐉ 1.58 eV (785 nm) provides a convenient window that is well matched for both semiconducting and metallic nanotubes in a sample containing a large d t distribution.However, if we had used a sample with a narrow diameter distribution around 1.38 6 0.10 nm, the laser excitation of 1.58 eV would become silent, with no nanotubes predicted to be resonant within the observable window [10].
In conclusion, here we show that in the case of onedimensional physics, Raman scattering from one carbon nanotube can independently provide its complete ͑n, m͒ structural information.Making ͑n, m͒ SWNT assignments with the Raman technique can result in a major advance for future SWNTs studies.For example, using this approach, transport measurements could be performed on a number of SWNTs, each of which have been independently characterized for their ͑n, m͒ values.
This work utilized MRSEC Shared Facilities supported by the National Science Foundation (No. DMR-9400334) and NSF Laser Facility Grant No. 9708265-CHE.The au-

FIG. 2 .
FIG. 1. AFM images of the sample.The small particles are iron catalysts.The inset shows the diameter distribution of the sample taken from 40 observed SWNTs.

FIG. 3 .
FIG. 3. (a)Measured frequency vs intensity for the RBM peaks observed at 42 different spots on the sample (Fig.1).(b) Calculated energy separation E ii as a function of 1͞d t , g 0 2.9 eV, and v RBM 248͞d t .Circles are for metallic SWNTs (E M ii ), and crosses for semiconducting SWNTs (E S ii ).
7 different intensities at 156 cm 21 for 7 different spots.A difference in intensity is expected since: (1) the nanotube lengths, at different light spots, are different from one another, and (2) tubes having different chiralities, such as ͑17, 5͒ and ͑13, 10͒, with different E M 11 splittings, still can have the same v RBM .When we get very strong experimental peaks, this means that the corresponding nanotube has an E M 11 close to E ᐉ .For higher v RBM values (.200 cm 21 ), semiconducting nanotubes are resonant and the number of chiralities is limited because of the small d t values.When we select semiconducting nanotubes with 1 , d t , 3 nm subject to the resonant window condition 1.48 , E S 22 , 1.68 eV, we get 8 possible chiralities.In the experiment we observed

TABLE I .
Possible chiralities predicted for metallic nanotubes and their calculated v RBM in the resonant window 1.48 , E M 11 , 1.68 eV.We also display the observed v RBM , with the number of times each appears between parentheses.
a Bold face indicates a strong intensity [see Fig.3(a)].

TABLE II .
Possible chiralities predicted for semiconducting nanotubes and their v RBM (calculated and observed) in the resonant window 1.48 , E S 22 , 1.68 eV.
a Bold face indicates a strong resonance.