Raman spectroscopy for carbon nanotube applications

The application of carbon nanotubes (CNTs) to advanced materials and devices has been broadly explored, maturing already in the first decade of the 21st century.^1–8 Examples are found in many fields, including reinforcement fibers,^3,9 electronics,^10,11 microelectronics,¹² electron sources,¹³ field emission displays,¹⁴ single-photon emitter,^15,16 composites for electrical applications,¹⁷ flexible electronics,¹⁸ super-capacitors,^19–21 organic light-emitting diodes,²² and solar cells,²³ among others. It also deserves to mention applications of bio-medicine and health,^24,25 including removal of contaminants in drinking water,²⁶ tissue engineering,²⁷ and biomedical²⁸ and bio-sensing^29,30 applications, for instance, CNT-based sensors for detection of dengue virus NS1 protein³¹ and tumor targeting.³² 

Many applications rely, of course, on the well-established outstanding mechanical, thermal, electronic, and optical properties of CNTs,^2,5,33–35 which have both aspects shared with all $sp 2$ related carbons, like graphite, graphene, and amorphous carbon. The application also relies on unique properties of a CNT, which come from the special one-dimensional character of CNTs that is enriched by their helical structure, as defined by diameter (⁠ $d t$⁠) and chiral angle (⁠ $θ$⁠), with an uncountable number of isomers specified by two integer $(n,m)$⁠.³³ Besides, carbon nanotube application also depends strongly on aspects that can vary among carbon nanotube-based materials, among CNT-based devices, and here is where Raman spectroscopy has been established as a commonly used, standard tool.

Raman spectroscopy (RS), in which we observe the inelastic scattering of light, is a technique that carries two important and unique aspects that make it highly relevant for the characterization of advanced materials and devices. First, since RS generally utilizes visible light as a probe, RS is a non-contact and non-invasive characterization tool that can be operated at both ambient (room temperature and in air) and controlled environment conditions, applied directly on any material or device with optical access. This operation is safe for the user and it does not disturb much the intrinsic properties of the measured material. Second, RS has access to the fine electric and vibrational properties of the materials with a high energy resolution, being able to distinguish energy resolution down to $μ$eV, a resolution generally much superior to any other nanotechnology technique based on electron microscopy such as electron energy-loss spectroscopy (EELS), scanning probe spectroscopy (SPS), or even other optical techniques relying on less well-defined energy levels, such as photoluminescence or infrared reflection spectroscopy. This sensitivity makes RS applicable not only to characterize the devices but also to serve as the relevant metric of a CNT-based sensor.

For the specific case of single-wall carbon nanotubes (SWNTs) and also other graphene-related materials, RS has an extra aspect that makes it even more powerful, which is the highly efficient and selective resonance effects.³⁶ The resonance effect makes it possible to measure one isolated carbon nanotube by RS³⁷ and makes it possible to obtain information from the vibrational properties that are usually Raman inactive.³⁸ Accessing phonons that are Raman inactive (e.g., phonons in the interior of the Brillouin zone) is important to characterize several static and dynamic properties of a solid or a molecule, but they are generally accessible only by inelastic neutron or x-ray scattering or EELS, which require expensive infrastructure and a single crystal of material, which is not suitable for the application for nanoscale devices.

The motivation of the present article comes from the fact that RS can say much more than phonon frequency of the device, such as characterizing strain, doping, defects, and interactions with the surrounding materials. Further, we could get much information by adopting advanced techniques for CNT-based materials and devices with keeping applicability to the industrial perspective. Following this motivation, this work is structured as follows: in Sec. II, we introduce the fundamentals of Raman spectroscopy in graphene and carbon nanotubes, which is necessary to understand its applications. In Sec. III, we discuss the applications of Raman spectroscopy with a focus on the following: the use of the radial breathing mode (⁠ $∼50−350 cm − 1$⁠) to characterize synthesis and CNT sorting in Sec. III A; the usage of the G (⁠ $∼1584 cm − 1$⁠) and G $′$ (⁠ $∼2600−2800 cm − 1$⁠) bands for strain, doping, and isotope characterization in Sec. III B; and the usage of defect-induced bands (focus on the D band, $∼1300−1400 cm − 1$⁠) for defect identification and quantification in Sec. III C. Notice the G $′$ is more commonly referred to as the 2D band in the most recent graphene literature, after Ferrari et al.,³⁹ although it is not an overtone of the D peak.⁴⁰ Section III D closes this section with considerations about the general applicability of these concepts with a focus on carbon nanotube composites. In Sec. IV, our discussion addresses how Raman spectroscopy can be used in Raman intensity mapping, which provides not only sample imaging but also carries local functional information. The discussion starts with the usual micro-Raman imaging in Sec. IV A, followed by examples of the use of tip-enhanced Raman spectroscopy (TERS) for nano-Raman imaging in Sec. IV B, finishing with comparison to other techniques in Sec. IV C. Finally, in Sec. V, we introduce, in a tutorial style, some complementary advanced techniques, with potential to help generating novel applications: electro-chemical doping (Sec. V A), circular dichroism (Sec. V B) and helicity-changing Raman spectroscopy using circularly polarized light (Sec. V C), and coherent phonon spectroscopy (Sec. V D). Some symmetry aspects related to group theory (GT) are utilized, and for a review on GT applied to solid state physics, see Ref. 41, or for GT specifically on carbon nanotubes, see Ref. 42. This Tutorial ends in Sec. VI where we present our conclusions and future perspectives for RS-based CNT applications.

Finally, it is important to stress that this article is limited in two ways. First, it focuses more on practical applications than on the fundamental concepts of Raman spectroscopy applied to carbon nanotubes; second, the authors try to cover the field, but generally we focus on the subjects that we have mainly contributed. To overcome these two limitations, we point the readers to other excellent articles and reviews that can be found in the literature, written by other experts in the field who contributed enormously to the advance of the concepts and applications in this field. Several references will be given along our review, but here, for instance, we can indicate Refs. 43–55.

When we characterize the Raman spectra of a SWNT, it is important to investigate the similarities and differences of the Raman spectra among the graphene-related structures. Figure 1 shows the Raman spectra of several graphene-related materials. What all these materials have in common is an atomic structure majorly defined by carbon atoms connected to each other by $sp 2$ carbon bonds, and this is why all the Raman spectra of other atoms or chemical bonds would appear at different Raman shifts. The special/general aspects of the Raman spectra from $sp 2$ carbons are listed below:

• RBM: Carbon nanotubes have a unique Raman-active mode around $ω R B M∼100−200 cm − 1$⁠, not present in any other graphene-related structure, which is the so-called radial breathing mode (RBM, see SWNT spectrum in Fig. 1). This vibrational mode exists because of the cylindrical shape of a nanotube, in which the radius of the nanotube is oscillating (like the tube breathing). The RBM is an out-of-plane bond-stretching mode whose frequency is inversely proportional to the diameter of a SWNT, $d t$⁠, as discussed in Sec. III A.

• G band: The G band is in-plane bond stretching mode of the C‒C bonds in the hexagonal lattice whose Raman peak appears at $∼1585 cm − 1$⁠. The G band commonly appears in all the $sp 2$ carbon materials (see Fig. 1). In pristine graphene, the G band is composed by a single Lorentzian peak.⁵⁸ In a SWNT, the G band splits into G $+$ and G $−$ peaks due to the strain effect, which is related to the curvature and/or due to electron–phonon coupling in a SWNT, as it will be discussed in Sec. III B. For defective carbon materials, the G band broadens due to bond in-homogeneity and shortening of phonon lifetime, as discussed in Sec. III C.

• $G ′$ band (or 2D band⁴⁰): Another peak present in the spectra of most $sp 2$ carbon materials is the so-called $G ′$ band, appearing at $ω G ′∼2700 cm − 1$ (see Fig. 1) for a 514 nm laser (the exact frequency of the $G ′$ peak depends on the excitation laser energy^38,59–62). The $G ′$ peak is related to the Raman scattering due to a vibrational mode characterized by the breathing of six carbons pertaining to a hexagon in the hexagonal lattice of graphene. The frequency for the breathing vibration is actually half of the observed value (⁠ $∼1350 cm − 1$⁠), but the hexagon-breathing mode is not Raman active in first-order Raman scattering,⁶³ being observable only as an overtone, which is Raman allowed. In pristine graphene, this $G ′$ band is composed by a single Lorentzian peak, while in all other $sp 2$ carbon structures, the line shape is deformed due to several effects, as is discussed in Sec. III B.

• D band: if the hexagonal $sp 2$ network exhibits a defect, the first-order component of the hexagon-breathing mode is activated combined with an elastic scattering of a photo-excited electron by the defect, as a double-resonance Raman peak (see below),^64,65 which appears at $ω D∼1350 cm − 1$ and it is called the D band (see the three bottom spectra of Fig. 1). Notice the presence of a sharp D peak for the spectra of damaged graphene and single-wall nanohorns (SWNHs). When the structure is too damaged, like in amorphous carbon (the bottom most spectrum in Fig. 1), the D and G bands are overlapped each other by broadening. Other defect-induced peaks are also activated (e.g., D $′$ and D+D $′$ in Fig. 1), and they are usually weak in intensity when compared to the D band. Notice the $G ′$ band is an overtone of the D mode, and for this reason it is also called 2D in the literature, although it has no direct relation to defects. The D band is discussed in Sec. III C.

• Double-resonance Raman peaks: Finally, all graphene-related materials exhibit several other peaks (IFM, $D ′$⁠, M, ItoLA, etc.), which are related to second-order, two-phonon (or one-phonon and one-elastic scattering) double resonance Raman processes involving the scattering of the photo-excited electron in the Brillouin zone by a non-zone-center (⁠ $q≠0$⁠) phonon^38,66 [e.g., IFM peaks in Fig. 2(b)], which will not be properly discussed in this article because they are usually too weak and not commonly utilized on applications. It is noted that $G ′$ and D are also double resonance Raman peaks. The double-resonance phenomenon was introduced by Thomsen and Reich⁶⁷ to explain the D band and further extended to other Raman features by Saito et al.³⁸ Reviews focusing on the phenomena from these and other authors, including pictures for the scattering diagrams can be found in the literature, see, e.g., Refs. 48, 66, and 68–72.

Having established the general aspects of the Raman spectra from graphene-related structures, let us focus our attention on the Raman spectra of (a) a metallic (top) and a semiconducting (bottom) isolated SWNT dispersed on a silicon substrate and (b) SWNT bundles as shown in Fig. 2. The main features (RBM, D, G, and $G ′$⁠) are highlighted in (a), while the peaks marked by * come from the substrate. Also highlighted in Fig. 2 are some small features (named M, iTOLA, or IFM) that are double-resonance Raman peaks.^66,73,74

In Fig. 2(a), we can see that the spectral width of the G band of a SWNT is broader for metallic SWNT than that of a semiconducting SWNT. The broadening is understood by the Kohn anomaly effect for a metallic SWNT, as is discussed in detail in Secs. III B and V A.^75–77 In Fig. 3(a), we schematically show the perturbation for a phonon in the Kohn anomaly effect in which a phonon with the energy $ℏ ω 0$ virtually excite an electron–hole pair in a metallic SWNT by electron–phonon interaction and the pair recombines back to a phonon from the left to the right of the figure. The second-order perturbation for a phonon by electron–phonon interaction makes the phonon energy lower and the lifetime of the phonon short, which gives the softening and broadening of the Raman spectra in the metallic SWNT in Fig. 2, respectively. A semiconducting carbon nanotube does not show such Kohn anomaly effect because the energy gap between the valence and conduction bands is larger than the phonon energy, and the process displayed in Fig. 3(a) does not occur. Back to the metallic SWNT, part of the process is suppressed by doping, as shown by the “ $×$”-labeled arrow in Fig. 3(a), which happens for phonon excitation energies up to twice of the Fermi energy, 2 $| E F|$⁠, which is shown by the blue shaded region in Figs. 3(a) and 3(b). In Fig. 3(b), we show the definition of energy denominator $h(E)$ of the perturbation. The change of the phonon frequency [Re(⁠ $h(E))$] and the spectral width [Im(⁠ $h(E))$] of the G band is used for characterizing the doping concentration.

The most important influence of the one-dimensional (1D) electronic structure of a SWNT on their Raman spectra is the resonance effect. Resonance Raman scattering happens when either the incident or the scattered light matches an optical transition energy, causing an enhancement of the Raman signal by many orders of magnitude (typically $10 3$⁠). Therefore, the Raman spectra from a CNT sample are dominated by the SWNTs in resonance. This fact is exemplified in Fig. 4, which shows the RBM Raman spectrum of a carbon nanotube bundle composed by SWNTs [Fig. 4(A)] within the transition energy window (⁠ $1.925±0.050 eV$ by two solid lines), as shown in Fig. 4(B). The RBM frequency depends on tube diameter $d t$ (see Sec. III A), and each Lorentzian peak utilized to fit the RBM spectrum represents the RBM peak from one resonant SWNT $d t$⁠. However, although the distribution in diameter is homogeneous, the RBM spectrum is obtained only by resonant SWNTs. In Fig. 4(B), all the optical transition energies for all SWNTs are plotted as a function of RBM frequency (⁠ $ω RBM$⁠), which is called the Kataura plot. The Kataura plot is usually displayed as a function of $d t$⁠, but here we plot the transition energy as a function of $ω RBM$⁠, since $ω RBM∝1/ d t$⁠, see Sec. III A. The horizontal dashed line marks the laser excitation energy (⁠ $E laser$⁠). Notice that for each bullet (optical transition) inside the RBM resonance window (the energy region between the two solid lines), there is one Lorentzian in the RBM spectrum in Fig. 4(A).

One can now use the RBM spectra to experimentally determine the Kataura plot, as shown in Fig. 5. On the left, we have the Kataura plot, now as a function of diameter, and on the right a Raman intensity map of the resonance RBMs as a function of $1/ ω RBM$⁠, which is obtained from the Raman spectra of a carbon nanotube bundle by changing the $E laser$ from 1.3eV to 2.7 eV. This procedure is utilized to fully understand the photophysics of SWNTs, by accurately describing the optical transition energies, including effects of tube curvature and many-body effects, such as electron–electron repulsion and electron–hole attraction (formation of excitons).^78–81 

In the application of Raman spectroscopy to characterize carbon nanotubes, the first general aspect is the classification of nanotubes when addressing (1) small diameter few-walls carbon nanotubes—single-wall (SWNT) and double-wall (DWNT) with smaller diameter tubes below 2 nm—and (2) larger diameter tubes (⁠ $d t>2nm$⁠) and many walls, more specifically multi-wall carbon nanotubes (MWNTs). In case (1), it is possible to distinguish the specific SWNT structure, i.e., the diameter (⁠ $d t$⁠) and the chiral angle (⁠ $θ$⁠), thus defining the chiral vector by the two integers $(n,m)$⁠.³³ In case (2), the properties of many $(n,m)$ SWNTs with a large $d t$ are too similar to characterize. Therefore, for small diameter tubes, each $(n,m)$ SWNT (or part of a DWNT) has a unique and characteristic Raman spectral signature, while for large diameter tubes, the results are just the Raman spectral characteristic in the limit of larger diameter tubes (⁠ $d t→∞$⁠) or graphene. For this reason, here we focus on the properties of small diameter tubes, and the characterization of, for example, MWNTs, can be considered a rational extension. For an in-depth description of the physics behind the Raman spectra of carbon nanotubes, see Ref. 36.

One of the important aspects for carbon nanotube application is the controlled synthesis and the further structural sorting, to enable engineering applications out of well-defined carbon nanotube properties, as already discussed in several review articles in the literature.^82–93 Since the radial breathing mode (RBM) is a unique vibration among graphene-related materials, and it is unique among specific SWNTs, the RBM relative intensity can be used to describe the carbon nanotube content in a sample.

Raman spectroscopy has been broadly utilized to determine the $(n,m)$ content in a sample. Examples are DNA-wrapped CoMoCAT sample dispersed in aqueous solution,⁹⁴ HiPco sample dispersed in aqueous solution with SDS surfactant,⁹⁵ carpet-like alcohol-assisted CVD,⁷⁹ phosphorus-doped SWNTs,⁹⁶ 0.4 nm diameter SWNTs grown inside the pores of zeolite crystals,⁹⁷ among others. In the case of CoMoCAT samples, it was shown that the RBM resonance Raman mapping can be used to differentiate the $(n,m)$ contents found in the as-grown purified sample, the SWNT+SDS solution sample, and the SWNT+SDS precipitate sample.⁹⁸ 

DWNTs are also interesting samples with a rich RBM resonance Raman map.^99,100 A very special technique of separation has been shown recently,¹⁰¹ where the authors sort DWNTs by semiconducting (S) or metallic (M) constituent tubes for the inner and outer walls of a DWNT. The electronic coupling between the inner and outer walls was used to alter the surfactant coating around each of the DWNT types, and an aqueous-gel permeation was then used to separate them. The enriched DWNT fractions were then transferred into either chlorobenzene or toluene using the copolymer PFO–BPy to yield the four inner@outer combinations of M@M, M@S, S@M, and S@S. The sorting is characterized by the RBM resonance Raman mapping, as shown in Fig. 6.

To rationalize this type of analysis, one has to define the relationship between the RBM properties and the $(n,m)$ structural indices. Since we have two indices, we need to define two spectral information, which are the RBM frequency (⁠ $ω RBM$⁠) and the RBM intensity. The RBM intensity becomes the maximum when the excitation laser matches the $(n,m)$-dependent, optical transition energy $E i i S , M$⁠, which is shown in the Kataura plot (Fig. 4).

By combining Eq. (2) with Eq. (3), we can build the experimental Kataura plot, like in Fig. 5, which enables all the RBM analysis described here; furthermore, we need the analysis of the environmental effects on the SWNT optical transitions.^104,109 The dielectric constants of the surrounding materials around a SWNT, such as surfactant, modify both $ω RBM$ and $E i i S , M$⁠, which is known as the environmental effect for the exciton. The environmental conditions, including strain^110–113 and temperature¹¹⁴ variations, can change the values of $E i i$ by a few tens of meV to a few hundreds of meV.^{81,104,105,115,116} Therefore, when using Eq. (3) for one specific interest, the researcher may give some room for adjusting the fitting parameters. For an as-grown vertically aligned SWNT synthesized by the chemical vapor deposition method from alcohol, $a=1.049 eV nm$⁠, $b=0.456$⁠, and $c=0.812 nm − 1$ in Eq. (3) were found to properly fit the data.⁷⁹ A good review on the environmental effect can be found in Ref. 104.

For a quantitative analysis of the relative amount of the $(n,m)$’s in a sample from the RBM spectra, it is important to consider that the resonance Raman intensity depends on diameter and chiral angle. This problem has been addressed theoretically considering curvature and many-body effects, which are crucial for properly describing the optical phenomena.^117–120 Several experimental works were also performed to define these dependencies,^80,98,121 and for a simple-to-use empirical formula, we refer the reader to Ref. 80. It is important to note that one laser-excitation energy is not sufficient for evaluating the amount of $(n,m)$’s since only the resonant $(n,m)$ SWNTs are observed in the Raman spectra, though we can only say that the resonant $(n,m)$ SWNTs exist.

Different from the RBM, which is observed only in SWNT, the G and G $′$ (also named 2D) bands are present in all $sp 2$ carbon structures. What makes these features special for applications is twofold: first, their high frequencies (the G band at $∼1584 cm − 1$ and the $G ′$ band at $∼2700 cm − 1$⁠) due to the small mass of the carbon atoms and a stiff C‒C sigma bond. This is important because 0.1% variation in frequency represents already $1.5 cm − 1$ for the G and $2.7 cm − 1$ for the $G ′$ bands, which are measurable shifts with most regular Raman spectrometers. First, a small change in the Raman shift or linewidths by $13 C$ isotope,¹²² strain, or doping can be measured from the Raman spectra; second, because metallic $sp 2$ structures exhibit a strong electron–phonon coupling for a phonon, enabling detailed sensing of the Fermi level position via Raman spectroscopy. These two effects will be described here.

In Fig. 7(a), we show the G-band spectra for graphite (HOPG, top), semiconducting (middle), and metallic (bottom) SWNTs. Although the C‒C bond stretching in graphene or graphite exhibits a single Lorentzian peak, SWNTs have two split peaks.¹²³ This happens because in the flat hexagonal lattice of graphite, the three C‒C bond stretching vibration for a carbon atom is isotropic. However, when you impose a curvature in the $sp 2$ network to form a nanotube, symmetry breaking occurs by a curvature-induced strain in the direction perpendicular to the tube axis, which lowers the phonon frequency of the C‒C vibrations which are preferentially in the circumferential direction.¹²³ On the other hand, the C‒C vibrations in the direction parallel to the nanotube axis do not show the frequency shift. Thus, the curvature-induced frequency splitting between the higher and lower frequency components (⁠ $ω G +− ω G −$⁠) is inversely proportional to the square of the tube diameter and depends on the chiral angle [see Figs. 7(b) and 7(c)]. For example, the intensity ratio of G+ and G- depends on the chiral angle.¹²⁴ Some chiral angle dependence can also be found in small diameter SWNTs,¹²⁵ and a more strict analysis of the G-band line shape actually shows that this feature is composed by more than two peaks of different symmetries related to the confinement along SWNT circumference,^126–129 but two of these peaks (the totally symmetric) are predominant and for most application purposes the two-peaks analysis is simple and effective.⁵⁶ 

The effect of strain^110,111 does not appear only due to curvature, but any tensile strain on a carbon nanotube or carbon nanotube fiber would shift the G and the $G ′$ frequencies, in this case of the G band acting mainly on $ω G +$⁠. Studies of uni-axial strain in graphene show both redshift and splitting of the G band are observed.^130–139 Studies on the effect in the $G ′$ band can also be found in Ref. 140.

In the case of semiconductor SWNTs, $ω G +$ and $ω G −$ correspond, respectively, to C‒C bond stretching in the direction parallel (LO) and perpendicular (iTO) to the nanotube axis, when we consider the phonon propagating direction in the nanotube axis. However, in the case of metallic SWNTs, the assignment of $ω G +$ and $ω G −$ becomes opposite [ $ω G +$(iTO) and $ω G −$(LO)] to that for semiconductor nanotubes [see Fig. 7(c)].⁵⁶ This inversion happens because the LO mode becomes soft by the Kohn anomaly effect (electron–phonon coupling, see below and Secs. II and V A).

As highlighted in Fig. 3(b), this electron–phonon coupling can be tuned by doping a gate electrode on a device, and the phonon-softening phenomenon of the G band is an evidence for metallic SWNT, as shown in Fig. 8. Therefore, the analysis of the G-band line shape of metallic SWNTs, as evidence in Fig. 8, can be used to monitor the Fermi level.^141,142 This effect will be further explored in Sec. V A, where the electro-chemical doping, which enables high doping levels will be addressed. It is noted that the Kohn anomaly effect is also predicted to occur for the RBM for metallic SWNTs, but the phonon softening effect is relatively small compared with that of the G band but chirality dependent.⁷⁶ 

Another aspect related to the electron–phonon coupling is its effect on electronic devices, for example, limiting the current flowing in carbon nanotube transistors. Phonon population and electrical power dissipation can be quantified by simultaneously measuring the Stokes (phonon creating) and anti-Stokes (phonon annihilation) Raman scattering of the G mode (also of the radial-breathing mode) as a function of current bias.^146,147 The Stokes/anti-Stokes ratio is well established as a means to measure local temperature using Raman spectroscopy,³⁶ although a fine care must be taken with respect to the resonance conditions that is not the same for Stokes and anti-Stokes Raman spectra, as discussed in Refs. 148–151, or the presence of non-linear Stokes–anti-Stokes correlated generation, which has been observed in graphene^152–154 but not yet addressed in carbon nanotubes.

Since the Raman frequencies are sensitive to the mass of atoms, they can be used to identify isotope substitution. The G band in SWNTs shows a considerably redshift when replacing the $12 C$ atoms by $13 C$⁠.^155–157 Interestingly, Otsuka et al.¹⁵⁸ showed a method for tracing the diverse growth profiles of individual SWNTs by embedding digitally coded isotope labels, with potential not only to sensitively detect $13 C$⁠, but contributing also to further understanding and control of SWNT chirality, length, and density during synthesis.

The intensity of the D band is related to the amount of defects in the structure and broadly utilized to monitor the quality of graphene-related materials.^166–168 A single substitutional impurity in the wall of a SWNT can be visualized using the D-band imaging.¹⁶⁹ In particular, a well-defined substitutional B impurity in graphene gives a stronger and sharper D band than the G band.¹⁷⁰ Besides, broadening of the other peaks is also observed. For instance, Fig. 9 shows the Raman spectra of a carbon nanotube fiber at two locations: 1 is at the body of the fiber, where most nanotubes are pristine-like; 2 is at the very edge of the fiber, where the defective structure of the nanotube ends is prominent. The spectrum changes from a two-peaks sharp G-band feature without a D band at 1 to a broad and multi-peak G structure with a D band at 2.

The intensity ratio between the G band and the disorder induced D band, $I D/ I G$⁠, has been a commonly used parameter for measuring the size $L a$ of a nano-crystal with ordered $sp 2$ C‒C bonds or the amount of localized defects in the $sp 2$ network.^65,171–175 This defect-quantity metric has been extended to study the annealing effect on disordered multi-wall carbon nanotubes,¹⁷⁶ Si and C ion-implanted double-wall carbon nanotubes,¹⁷⁷ the effect of $γ$ rays irradiated,¹⁷⁸ the boron and nitrogen doping levels in SWNTs^96,179 and MWNTs,¹⁸⁰ and to perform length characterization of DNA-wrapped carbon nanotubes.¹⁸¹ Thus, the D-band intensity analysis relative to the G band is widely used for monitoring either the pristine quality or the intentional modifications (e.g., chemical functionalization) of CNTs for applications.

The concepts introduced in this section are basic and generally applicable to carbon nanotube-related materials. A class of materials that deserves attention are the carbon nanotube composites, where the carbon nanotubes are commonly utilized as a composite to change the properties of a host, such as polymers. Several authors addressed these materials, and the interested reader can find the Raman spectroscopy concepts discussed here utilized to characterize carbon nanotube composites in several articles, for example, in Refs. 46, 47, 49, 54, and 182. In this literature, Raman spectroscopy has been used not only to identify the carbon nanotubes within the composites, but also to access their dispersion, to evaluate nanotube/matrix interactions, and to detect polymer phase transitions.⁴⁹ Strain or stress transferred to nanotubes from the surrounding environment can be quantified both globally and locally, enabling the characterization of the elastic properties of the composite material, via Raman band frequency shifts, thus addressing the load transfer effectiveness among nanotubes and host.⁴⁶ For example, a $G ′$ Raman band shift to a lower wavenumber upon application of a tensile stress indicates stress transfer from the matrix to the nanotubes and hence reinforcement by the nanotubes.¹⁸² Furthermore, polarized Raman can be used to access the degree of carbon nanotube alignment, since the Raman signal is stronger when the light is polarized along the carbon nanotube axis.^44,127,128

One of the useful information by Raman spectroscopy is Raman imaging, in which the Raman intensity of a phonon mode is plotted as a function of the two-dimensional position (X,Y) of the sample or device surface. Three-dimensional imaging is also possible by adding a Z-scan, but much less usual since selectness and resolution are not efficient along the laser beam propagation direction, depending on the focusing geometry. Usually, the hyper spectra are obtained by raster scanning the laser in the sample/device, and acquiring one spectrum per (X,Y) point. The hyper-spectral imaging is much more informative than simple microscopy imaging, because it carries all the spectral information at each point in the sample/device. In this section, we discuss how we can use Raman spectroscopy imaging for characterizing samples and the devices, addressing micro-Raman in Sec. IV A and nano-Raman in Sec. IV B and presenting arguments in favor of Raman imaging by comparing it with the outcomes from other imaging techniques in Sec. IV C.

A simple example of the use of micro-Raman spectroscopy imaging is when we put an isolated single-wall carbon nanotube on the $SiO 2$ surface and try to locate the SWNT at the specific position.³⁷ It is very hard to know the position of the nanotube on the surface since the diameter of SWNT is much smaller than the wavelength of the light, which means that it is not observable optically. Transmission electron microscope cannot be used for detecting the nanotube since the $SiO 2$ substrate is too thick to get the TEM images. By scanning the light focused by optical microscope over the device, we can get the Raman images with the spatial resolution in the order of half of the excitation wavelength (⁠ $λ/2$⁠), generally in the 250–500 nm range. This technique is known as micro-Raman imaging in which we need to observe the Raman spectra at each spot of the light. Further, by using tip-enhanced Raman spectroscopy (TERS), the resolution becomes comparable to the nanotube diameter, in the nano-Raman imaging, as discussed in Sec. IV B.

As a more advanced example, Raman mapping of a SWNT serpentine (straight tube segments connected by U-turns, see Fig. 10)¹⁸³ shows how the Raman spectroscopy based image goes beyond the wavelength of light. The SWNT serpentine is deposited on non-isotropic quartz and their Raman spectrum depends on the tube–substrate morphology, which are shown in Fig. 10. The top part of the figure shows a micro-Raman G-band image with green pointers showing locations numbered from 1 (bottom right) to 41 (top left). A Raman spectrum was collected at each of those positions, and the G band is depicted in the color map shown in the bottom graphic, showing an oscillatory behavior of the $G +$ peak near $1600 cm − 1$⁠, followed by the appearance and the disappearance of the broad metallic $G −$ signature around $1540 cm − 1$⁠. This means that the same SWNT-on-quartz system exhibits a mixture of semiconductor and metal behavior, depending on the orientation between the tube and the substrate. Therefore, Fig. 10 shows that different tube–substrate interactions for straight segments vs U-turns cause a modulation in the electronic properties along carbon nanotubes, forming an alternating semiconducting-metal-semiconducting-metal- $⋯$nano-device.¹⁰⁶ 

Another interesting device that has been proposed is the production of nanotube coil devices, including inductors, electromagnets, transformers, and dynamos. However, this type of construction commonly generates very defective nanotube coils. Nevertheless, such devices can be produced out of spontaneous self-coiling of the same SWNT into defect-free coils, going up to more than 70 turns with identical diameter and chirality, and these properties are corroborated by Raman imaging.^184,185 The left inset to panel (a) in Fig. 11 shows a Raman spectroscopy image of a SWNT coil formed by four complete turns. The image is based on the G-band Raman intensity [see device and characterization in panels (b)–(e)]. The black spectrum in (a) was taken at position 1 in the left inset, and the red spectrum at position 2 in the left inset. From the G-band line shape, the nanotube is identified to be semiconducting. The G-band signal in position 2 is significantly stronger than in position 1 because the coil is composed of the signal of several nanotube turns. The D band is completely absent, demonstrating the structural integrity. The right inset is the low-frequency region of the spectrum, showing the RBM peak from the single-wall carbon nanotube at around $90 cm − 1$⁠, obtained from the free end segment, as well as the RBM overtone peak at $∼180 cm − 1$⁠. The $90 cm − 1$ RBM is also present in the coil (position 2) but broadened by bundling.¹⁸⁴ 

While a regular microscope-based Raman system can be called a micro-Raman system, a tip-enhanced Raman spectroscopy (TERS) system can be called a nano-Raman system. The nano-Raman system surpasses the resolution dictated by the light diffraction limit by scanning a plasmonic nano-antenna on the sample and extracting information from the near-field of light. Surface-enhanced Raman spectroscopy (SERS) is one of the most common techniques by using the spatially localized near-field. In fact, SERS has been applied to carbon nanotubes.^186–191 However, because of the sharp resonance Raman effect in the one-dimensional structure, single-nanotube Raman spectroscopy by using the far-field can be achieved without any enhancement strategy of near-field.³⁷ Nevertheless, SWNTs became actually a prototype material for the advance of TERS, which is a variation of SERS where the plasmonic, local enhancement of the near-field is controlled by a scanning probe microscopy (SPM) system.^192,193 The merit of TERS is that TERS significantly improves the spatial resolution of Raman images. TERS provides a Raman imaging with a typical spatial resolution of $∼10$ nm, although resolution in the subnanometer scale has already been achieved.¹⁹⁴ 

The comparison between micro-Raman and nano-Raman imaging is shown in Fig. 12. Panel (a) shows a micro-Raman image of the G band by scanning the laser spot over a carbon nanotube serpentine.¹⁸³ Panel (b) is an expanded image from the squared area in (a). Panel (c) is the nano-Raman image of the G band at the same location as (b), obtained by the nano-Raman (TERS). The blurred image in (b), which is actually defined by the size of the laser spot, is replaced by a sharp image in (c), which clearly resolves all details on the carbon nanotube serpentine.

In the nano-Raman spectroscopy, the strong near-field is generated on the surface of the nano-antenna by exciting surface plasmon.¹⁹⁵ The behavior of the near-field is shown by the “tip-approach curve” [see Fig. 12(f)], which describes the TERS intensity as a function of the tip-sample distance (⁠ $Δ$⁠). The decaying nature of the near-field is well described with the classical electromagnetism theory by solving the Helmholtz equation for the electro-magnetic wave.^196–200 Except for tip-sample working distances below 1 nm, where quantum and atomistic effects on the plasmonic response have to be considered.²⁰¹ Nano-Raman in carbon nanotubes has been used to investigate local properties such as localized semiconductor-to-metal transition of a carbon nanotube²⁰² and electron and phonon renormalization near charged defects,¹⁶⁹ as shown in Fig. 13.

Figure 13 is an example of TERS for resolving local effects. It shows results from a (9,1) SWNT, where the Raman spectra in (a) were acquired at positions from R1 to R10 along the SWNT, as shown in (b), in steps of 10 nm. Figure 13(b) is a nano-Raman image based on the local intensity of the G band, and the yellow-dashed line is 100 nm in length, evidencing the high resolution of this Raman image. Notice in (a) a strong enhancement of the D peak at position R4, coincident with a localized photoluminescence (PL) emission shown in Fig. 13(c), indicating the presence of a defect acting as an exciton trapping state at R4. Analysis of the G $′$ peak (not shown in the figure) indicates that an exction trap state is formed by a negatively charged substitutional atom (nitrogen, in the case), which causes electron and phonon energies renormalization. Different TERS studies on SWNTs can be found in the literature,^{192–195,202} including the presence of linear carbon chains inside carbon nanotubes.²⁰³ A review of nano-Raman in carbon nanostructures, including carbon nanotubes, can be found in Ref. 204.

Raman spectroscopy imaging is not yet integrated into industrial protocols, but we propose it should, based on the merits by comparing it with the existing methods. As already pointed out in Sec. I, RS has access to the fine electric and vibrational properties of the materials with a $μ$eV energy resolution, superior to any other nanotechnology technique based on electron microscopy (e.g., electron energy-loss spectroscopy—EELS), scanning probe spectroscopies (SPSs), or even other optical techniques relying on less well-defined energy levels. Different optical techniques such as photoluminescence,^205–211 optical absorption,^212–214 and Rayleigh scattering^215–217 have provided a great contribution to the field of carbon nanotube characterization, including single nanotube $(n,m)$ identification, but when going to functional information, such as doping and strain, they are not as sensitive in energy as looking at the phonon energies.

Raman imaging can be compared with XPS (x-ray photo-electron spectroscopy) or ESCA (Electron Spectroscopy for Chemical Analysis) in which we can image the map of elements on the device or many other spectroscopies. Nowadays, the instrument that can measure several images at the same time is commercially available, which makes the characterization more precise and reliable information. Raman imaging has a lot of merits compared with other techniques. For example, Raman spectra can be observed in the ambient pressure while ESCA or TEM images requires the vacuum chamber. Further by changing the polarization direction of light in the polarized Raman spectroscopy, we can characterize the CNT alignment, the shape of the edge of graphene, strain direction, and other effects causing polarization selection. Raman imaging is also taken by applying the gate voltage on semiconductor devices, which can be directly compared with Kelvin force microscopy, in which the electro-static potential of the device as a function of the position is observed. In the Raman spectroscopy, the Kohn anomaly effect on the G band or RBM of carbon nanotubes can be observed as a function of the Fermi energy for each spot.

A further merit of Raman spectroscopy imaging is that, in principle, one can control the diameter of the incident light so that we first search the large area by a relatively large spot and then we focus on a specific region by much higher spatial resolution. Further, if we know the materials, we can focus on a narrow region of the Raman shift including the phonon frequency that we want to observe for saving time. Since the resonant Raman signal is sufficiently strong even for an isolated nanotube, we do not need to measure the Raman spectra for a long time if we just check if there are no signals on the specified point for several seconds.

Raman imaging can be used for characterizing (1) the edge structure, (2) the interface of the hetero-structure, (3) defects or impurity, (4) temperature gradient, (5) local strain, and (6) the Fermi energy. For (1)–(3), we frequently use the defect oriented Raman bands such as D and D $′$ bands^65,66 while for (4)–(6), the intrinsic Raman spectra such as the RBM, G, and G $′$ bands can be used. As for (4), it is known that the RBM frequency of nanotubes (or the G band of graphene) is monotonically decreased with increasing temperature for a wide region of temperature above the room temperature,²¹⁸ we can observe the temperature of the nanotube on the device as a function of the position. The local strain (5) is observed by the splitting of the G band and the Fermi energy is observed by the Kohn anomaly effect. In order to obtain these measurements, we do not need any further measurement but only micro-Raman setup.

Having established the most well-understood and broadly utilized aspects of Raman spectroscopy to characterize carbon nanotubes for applications, we now introduce the fundamental aspects of some complementary advanced techniques with potential to help generating novel Raman spectroscopy-based applications, including electro-chemical doping (Sec. V A), circular dichroism (Sec. V B) and conservation of angular momentum (Sec. V C), and coherent phonon spectroscopy (Sec. V D). Here, we focus on those three examples, but we also briefly comment that there are other innovative and recent aspects in the field of Raman microscopy applied to CNT samples, for example: (i) CARS microscopy;²¹⁹ (ii) improved spatial resolution by structured illumination;²²⁰ and (iii) the use of statistical spectrum processing methods to improve spectral analysis and the robustness of the information obtained,²²¹ among others that are not discussed here.

One of the fundamental techniques in semiconductor physics is doping of carriers in which the Fermi energy of the nanotubes and graphene-related materials is shifted from the Dirac point at the K points in the hexagonal Brillouin zone. In the field-effect transistor (FET), the carriers are doped in the channel region between the source and drain electrodes by applying the voltage at the gate electrode to enable electric current. When we apply the ionic gel in which the ion surrounded by a molecule is accumulated by the gate potential, we can change the Fermi energy more than 1 eV, which we call electro-chemical doping. In the electro-chemical doping, since we do not need insulator layers for the accumulated ions since the ions are surrounded by insulating molecules, the capacitance at the gate electrode becomes large, which is the reason for getting a large Fermi energy. A large change of the Fermi energy is useful for finding many physics such as superconducting phases and the Kohn anomaly effect though the speed of accumulation of charge in the gel is not so fast.

In electronic devices, it is important to evaluate the Fermi energy for a given gate voltage to obtain the carrier concentration or mobility. The Fermi energy is observed by many experimental methods such as scanning tunneling spectroscopy, the Hall conductivity, and angle-resolved photo-electron spectroscopy. Here, we show that non-contact Raman measurement can be used for evaluating the Fermi energy.

The Raman measurement by applying the gate voltage is called gate-modulated Raman spectroscopy,²²² in which the Fermi energy of graphene or nanotubes are given as a function of the gate voltage of either the top or the bottom gate. However, since the function depends on parameters of the gate electrode such as the area of electrodes, thickness, and dielectric constant of the dielectric materials as a capacitor, we cannot get a general form of the function. Thus, direct measurement of the Fermi energy is required for each device.

In the case of first-order Raman scattering, since the wavevector of the phonon is $q=0$⁠, the excitation of the e–h pair occurs vertically in the k space (⁠ $k= k ′$⁠) near the K point in the Brillouin zone as shown in Fig. 14(a). When the Fermi energy, $E F$ is a positive (negative) value, the excitation with the $E eh$ up to 2 $E F$ is suppressed because the final state is occupied (the initial state is unoccupied). When the $E F$ increases from zero to $ℏ ω q ( 0 )/2$⁠, the phonon frequency becomes soft since the hardening contribution is partially suppressed. When the $E F$ further increases more than $ℏ ω q ( 0 )/2$⁠, the phonon frequency becomes hard since the softening contribution is partially suppressed.

In Fig. 14(b), we show the Kohn anomaly for G $′$ phonon in which phonon $q$ vector exists around the K point. In this case, the electron is excited from the K to K $′$ valleys leaving a hole behind at the K point. For a specific phonon frequency $ω q$⁠, the correction term of $ℏ ω q−ℏ ω q ( 0 )≡Π( ω q E F)$ is called the self-energy. The self-energy, $Π( ω q, E F)$⁠, is a complex function of $q$⁠, in which the real part gives a shift of the phonon energy $ℏ ω q−ℏ ω q ( 0 )$⁠, and the imaginary part gives the broadening of the spectra. Thus, $γ q$ in Eq. (6) can be calculated self-consistently by the self-energy. It is noted in Figs. 14(c) and 14(d) that both (c) intraband and (d) interband excitations of the e–h pair are possible for the Kohn anomaly of $q=0$ phonon. As for $q=K$ phonon, we can then imagine a virtual vertical transition of electron and hole by translating a vector $−( K+ q ′$⁠) in Figs. 14(c) and 14(d).²²⁴ 

Circular dichroism (CD) is defined by the nonequivalent optical absorption for left-handed circularly polarized light (LCP) and right-handed circularly polarized light (RCP). Since a chiral nanotube does not have a mirror symmetry for a mirror parallel to the nanotube,⁴² we expect CD for chiral nanotubes if we can separate the enantiomer of left-handed and right-handed chiral nanotubes. CD is a relevant property for application since it can be used to generate optically active devices, with potential application to information doubling by left- and right-handed circularly polarized light in the optical fibers as a single-photon emitter.^15,16 Although CD is not a Raman spectroscopy, the CD spectrum is an important characterization tool for characterizing the enantiomer (either left-handed or right-handed nanotube) purity of the nanotubes from application because of bio-compatibility in which only left-handed molecules appears in bio-materials. It is somewhat confusing to show here that though the left-handed and right-handed nanotubes for an enantiomer pair gives the opposite sign of CD, all left-handed nanotubes do not give the same sign as CD.

By using the gel sorting of carbon nanotubes, Wei et al. observed the CD spectra of separated nanotube in solution since the gel is a natural product with left handed and thus not only single chirality $(n,m)$ but also enantiomer is separated.²²⁶ It is noted that CD is observed for a chiral fullerene, such as $C 76$⁠.

Since a photon of LCP (RCP) has an angular momentum $+ℏ$ (⁠ $−ℏ$⁠) in the propagation direction, optical absorption of SWNT occurs from the $μ$ to $μ+1$ (⁠ $μ−1$⁠) cutting lines when the incident light propagates in the direction perpendicular to the nanotube axis. The cutting lines are defined by a set of one-dimensional Brillouin zones of a SWNT in the two-dimensional Brillouin zone of graphene.^33,227 It is noted that the Kataura plot in Figs. 4(b) and 5 is from the $μ$ to the $μ$ cutting lines (⁠ $E μ μ$⁠), which is optically allowed for linearly polarized light in the direction parallel to the nanotube axis.

The optical transition occurs by the conservation of angular momentum.²²⁸ Since the final states for a given initial state at the $μ$th cutting line are not the same for LCP and RCP, we expect that the optical absorption probabilities for LCP and RCP may not be the same. However, since we have a time-reversal symmetry in the electronic structure of carbon nanotubes, the optical absorption probability for LCP from $μ$ to $μ+1$ near the K point becomes the same as the optical absorption probability for RCP from $−μ$ to $−μ−1$ near the K $′$ point, which gives zero value of CD. This means that the conventional theory for CD does not work for a chiral carbon nanotube.

Sato et al. calculated CD spectra,²²⁸ which is compared with the experimental results²²⁶ as shown in Fig. 15. Sato et al. have discussed the origin of the CD in a nanotube, in which the phase difference of the light at each carbon atom is essential for the appearance of CD.²²⁸ The phase difference effect gives a shift of $k$ vector in the optical transition in the direction of the $K 2$ (the direction of the cutting line), which breaks the symmetry of optical absorption probabilities between the K and K $′$ points for the light propagating in the direction parallel to the nanotube axis. As for the propagating direction perpendicular to the nanotube axis, the phase difference effect appears as a pre-factor (⁠ $β$⁠) of the optical absorption.²²⁸ By considering the enhancement of optical absorption by the exciton effect, the calculated CD values of undoped carbon nanotubes for the geometry of perpendicular incidence reproduce the experimentally observed CD spectra as a function of the wavelength as shown in Fig. 15.

Since the nanotube directions are random in the solution, we need to integrate CD values for the angle relative to the propagation direction of light. However, in the case of parallel incidence of light to the nanotube axis, since the depolarization effect²²⁹ suppresses the electric field of LCP or RCP in the nanotube, we do not expect the CD values for this geometry. It is the reason why the calculated results of perpendicular incidence reproduce experimental results. The CD spectra for perpendicular incidence show the peak at $E 12$⁠, $E 13$⁠, $E 24$⁠, $E 35$⁠, and $E 46$⁠, from the lower energy, which can be calculated by the calculation of the Kataura plot.^116,230

By comparing with the peak positions of $E i j$⁠, we can assign the $(n,m)$ indices of a nanotube. The enantiomer pair of $(n,m)$ is $(n+m,−m)$⁠, which gives an opposite sign of CD for $(n,m)$⁠, and thus we can assign $(n+m,−m)$ from the sign of CD. The sign of CD for $E 12$ is given by the relative position of the nearest and second-nearest cutting lines near the K point. The relative position of the cutting lines is classified to be type-I and type-II semiconductor SWNTs, which are determined by the value of mod $(2n+m,3)$=1 or 2, respectively^108,230 Thus, the sign of CD is opposite for type-I and type-II within the same handedness. Thus, the assignment of the enantiomer is uniquely determined by CD spectra by comparing the calculated results of $E i j$⁠.

Since a photon of LCP (RCP) has a spin-angular momentum of $ℏ$ (⁠ $−ℏ$⁠), in order to get the helicity-changing Raman scattering, we need a change of angular momentum by $2ℏ$ in a Raman scattering process. The angular momentum of an incident photon is transferred to the photo-excited electrons by electron–photon interaction.

In Table I, we list the symmetry of the phonon, $ν$⁠, and the values of $N$⁠, $N ν$⁠, $m ν ph$⁠, and $p$ that give the helicity-changing Raman spectra for the $D 2 h$⁠, $D 3 h$⁠, $D 4 h$⁠, and $D 6 h$ symmetries.^41,42 For example, the G band of graphene corresponds to the $E 2 g$ mode of $D 6 h$⁠. The in-plane Raman mode of TMDs corresponds to the $E ′$ mode of $D 3 h$⁠. From Table I, we can predict many unknown helicity-changing Raman signals for a twofold or fourfold rotational symmetry in Table I. Although the point group symmetry shows only $D n h$ in Table I, conservation law appears for both the higher and lower symmetries. For a lower symmetry, we can use Table I, too, since $N ν$ in the lower symmetry is either 1 or 2. For a higher symmetry such as $T d$⁠, and $O h$⁠, $I h$⁠, the angular momentum can be discussed for each high symmetry axis if the symmetry axis has a unique direction relative to the propagation direction of the light.

The helicity changing Raman spectra can assign the symmetry of the phonon, which is independent of characterization by polarized Raman spectra.^126,128,233 Thus, it will be useful for low-dimensional materials.

When we irradiate a material with a single-pulsed laser light (or pump pulse) with a time width of several femtoseconds (fs), the valence electrons excite almost at the same time. The photo-excited electrons start to oscillate phonons at the same time by electron–phonon interaction. Such phonons are called coherent phonons (CPs) since the phonons oscillate with the same phase. Since the CP is a macroscopic oscillation of the materials, the transmission (or reflection) probability of light for the material oscillates as a function of time delay from the pump pulse with the frequency of the phonon as shown in Fig. 16(a).^234,235 In Fig. 16(a), the oscillations of the transmission (or reflection) probability $ΔT/T$ are measured by irradiating the second pulse of a laser (probe pulse) with changing the time delay. By taking a Fourier transformation of $ΔT/T$⁠, we get CP spectra that have peaks at the phonon frequencies as shown in Fig. 16(b).

CP spectroscopy is similar to Raman spectroscopy since the CP spectra are given as a function of phonon frequency. CP spectroscopy requires a pump–probe measurement of laser optics, which is not easy for preparing the facility. Nevertheless, the CP method has many merits that Raman spectroscopy does not have. Let us list the unique points of CP: (1) CP spectra are free from the Rayleigh scattering; (2) the resonant condition is relaxed; (3) we can change the energy of the probe pulse; (3) we can change the repetition rate of the pumped pulse; (4) the initial phase of the phonon can be measured; and (5) the mixed sample of different SWNTs can be measured separately.

Since we do not measure the scattered light in CP, but the time modulation of transmission (or reflection) probability, the effect of Rayleigh scatting at the low frequency does not appear. In order to get the same phase of the oscillation of CP, the pulse width of the pump pulse should be much smaller than the period of the phonon oscillation, $T$⁠. For example, since the period of the RBM modes at $100 cm − 1$ is 333 fs (⁠ $T$ is given by $T=h/E$⁠, where $h$ is the Planck constant; $1 eV=8065c m − 1=1.6× 10 − 19$ J), 50 fs is sufficient for pump pulse. However, the period of the oscillation for the G band (⁠ $∼1600 cm − 1$⁠) is 20 fs, less than 10 fs is necessary to observe the G band by CP. For such a short pulse width, we know the uncertainty principles of quantum mechanics, $ΔtΔE≥ℏ$ in which $ΔE$ for $Δt=T$ becomes 66 meV. The value of $ΔE$ becomes large when we consider the short lifetime of the photo-excited electron. A large $ΔE$ relaxes the resonant condition for the CP spectra, which makes many RBM peaks for a given laser energy.

One of the advantages of CP spectroscopy is that we can measure the CP signal of one $(n,m)$ SWNT even in the mixed sample. In the case of Raman spectroscopy, we can measure the RBM of one $(n,m)$ SWNT by using a sharp resonant effect by one-dimensional Van Hove singularity. However, in the case of CP, since the pump pulse is short, the resonant windows become large so that we cannot specify $(n,m)$ by the resonant effect.

In CP spectroscopy, on the other hand, we use the so-called “pulse-train excitation,” in which the incident pulses are injected repeatedly in the frequency of a RBM. When the repeated pulses generate CP for the specific $(n,m)$ SWNT, the CP has the same phase as the phase of the repeated pulses, and thus the intensity of the CP becomes much larger than other CP intensities whose phases are random. In Fig. 17, we show the CP of pulse train excitation.^236,237 When we use a single pulse excitation as shown in Fig. 17(a), we show many beats in $ΔT/T$ as a function of the time delay whose Fourier transform shows many peaks of RBM modes. When we set the repeated frequency of the incident pulses to 6.96 THz as shown in Fig. 17(b), we get only one peak of the RBM for (11,3) from the same mixed sample. Similarly by changing the repeated frequencies in Figs. 17(c) and 17(d), we get the corresponding CP peaks. This technique is useful for focusing the CP spectra in any mixed samples.

The pulse-train technique can be used for the G band. However, since the repeated frequency for the G band is much larger than that for RBM, no experiments adopt the technique for the G band. Nugraha et al. discussed the possibility of the pulse-train technique for the repeated pulse for several-integer times the period of the vibrational frequency.²³⁸ It is also interesting to discuss the question of the RBM mode in CP spectra, whether the RBM mode starts by shrinking the diameter or by expanding the diameter. Since the initial force to a carbon atom by photo-excited electron is determined by the sign of electron–phonon interaction, the answer is that the direction depends on type-I or type-II SWNTs that has an opposite sign of electron–phonon interaction.²³⁹ The initial phase of SWNT is measured by some experiments.

Raman spectroscopy in carbon nanotubes has been vastly studied in the last 20 years and it has reached the maturity to be utilized in applications. In this paper, we highlighted the use of Raman spectroscopy to help the development of applications based on carbon nanotubes, including several references from the literature, which runs from materials science to biotechnology. We focused initially on the well-established relations between the most prominent Raman features, namely, the RBM, D, G, and G $′$ bands, on carbon nanotube structure (diameter and helicity) and other external factors, such as strain, doping, the presence of defects, and environmental interactions. We provided equations, fitting parameters, and observed ranges of parameters that compose ready-to-use protocols. We then described how Raman spectroscopy has been used to characterize synthesis, selective sorting, doping, strain, and defects at the end of carbon fibers. Finally, we discussed micro- and nano-Raman imaging as a technique superior to standard microscopy, since spectroscopic imaging carries functional information, as well as other complementary advanced techniques that can be used to further improve the ability of Raman spectroscopy to characterize materials and devices.

A challenge to this field is the use of Raman spectroscopy not only as a method to characterize the materials and devices that will then be utilized independently of the Raman effect, but also using Raman response as the device working protocol itself. The carbon nanotubes can be designed as optical devices that will be read using Raman spectroscopy, and the Raman spectrum itself will be the functional indicator, for example, for sensors of strain, doping, and environmental conditions. The technique is mature for such an endeavor, while lasers and detectors are also becoming more powerful and cheaper, opening a route for novel disruptive innovations.

A.J. and R.S. sincerely acknowledge the late Professor Mildred S. Dresselhaus, Professor Gene Dresselhaus, Professor Marcos A. Pimenta, and all collaborators for 20 years of collaboration on Raman spectroscopy. A.J. acknowledges CNPq (No. 302775/2018-8). R.S. acknowledges JSPS KAKENHI (No. JP18H01810).

The data that support the findings of this study are available from the corresponding author upon reasonable request.