Abstract Single-wall carbon nanotubes provide ideal model 1-D condensed matter systems in which to address fundamental questions in many-body physics, while, at the same time, they are leading candidates for building blocks in nanoscale optoelectronic circuits. Much attention has been recently paid to their optical properties, arising from 1-D excitons and phonons, which have been revealed via photoluminescence, Raman scattering, and ultrafast optical spectroscopy of semiconducting carbon nanotubes. On the contrary, dynamical properties of metallic nanotubes have been poorly explored, although they are expected to provide a novel setting for the study of electron-hole pairs in the presence of degenerate 1-D electrons. In particular, (n,n)-chirality, or armchair, metallic nanotubes are truly gapless with massless carriers, ideally suited for dynamical studies of Tomonaga-Luttinger liquids. Unfortunately, progress towards such studies has been slowed by the inherent problem of nanotube synthesis whereby both semiconducting and metallic nanotubes are produced. Here, we use post-synthesis separation methods based on density gradient ultracentrifugation and DNAbased ion-exchange chromatography to produce aqueous suspensions strongly enriched in armchair nanotubes. Through resonant Raman spectroscopy of the radial breathing mode phonons, we provide macroscopic and unambiguous evidence that density gradient ultracentrifugation can enrich armchair nanotubes. Furthermore, using conventional, optical absorption spectroscopy in the nearinfrared and visible range, we show that interband absorption in armchair nanotubes is strongly excitonic. Lastly, by examining the G-band mode in Raman spectra, we determine that observation of the broad, lower frequency (G − ) feature is a result of resonance with non-armchair "metallic" nanotubes. These findings regarding the fundamental optical absorption and scattering processes in metallic carbon nanotubes lay the foundation for further spectroscopic studies to probe many-body physical phenomena in one dimension. Single-wall carbon nanotubes provide ideal model 1-D condensed matter systems in which to address fundamental questions in many-body physics, while, at the same time, they are leading candidates for building blocks in nanoscale optoelectronic circuits. Much attention has been recently paid to their optical properties, arising from 1-D excitons and phonons, which have been revealed via photoluminescence, Raman scattering, and ultrafast optical spectroscopy of semiconducting carbon nanotubes. On the contrary, dynamical properties of metallic nanotubes have been poorly explored, although they are expected to provide a novel setting for the study of electron-hole pairs in the presence of degenerate 1-D electrons. In particular, (n,n)-chirality, or armchair, metallic nanotubes are truly gapless with massless carriers, ideally suited for dynamical studies of Tomonaga-Luttinger liquids. Unfortunately, progress towards such studies has been slowed by the inherent problem of nanotube synthesis whereby both semiconducting and metallic nanotubes are produced. Here, we use post-synthesis separation methods based on density gradient ultracentrifugation and DNAbased ion-exchange chromatography to produce aqueous suspensions strongly enriched in armchair nanotubes. Through resonant Raman spectroscopy of the radial breathing mode phonons, we provide macroscopic and unambiguous evidence that density gradient ultracentrifugation can enrich armchair nanotubes. Furthermore, using conventional, optical absorption spectroscopy in the nearinfrared and visible range, we show that interband absorption in armchair nanotubes is strongly excitonic. Lastly, by examining the G-band mode in Raman spectra, we determine that observation of the broad, lower frequency (G − ) feature is a result of resonance with non-armchair "metallic" nanotubes. These findings regarding the fundamental optical absorption and scattering processes in metallic carbon nanotubes lay the foundation for further spectroscopic studies to probe many-body physical phenomena in one dimension. I. INTRODUCTION Armchair carbon nanotubes have a distinguished status among all the members, or species, of the singlewall carbon nanotube (SWCNT) family. Being the only truly gapless species in the family, they are defined by the simple relation n = m, where n and m are the chiral indices, (n, m), which define the diameter and chiral angle of a given nanotube. As a result of their unique one-dimensional (1-D) electronic dispersions with zero effective mass and zero band gap, they are expected to exhibit some of the unusual properties characteristic of 1-D metals. I. INTRODUCTION 1 While strong electron-electron interactions can lead to the formation of Tomonaga-Luttinger liquid states, 1-10 strong electron-phonon interactions can renormalize phonon frequencies and lifetimes (Kohn anomalies) and induce Peierls lattice instabilities, especially in small-diameter nanotubes. I. INTRODUCTION Although a number of theoretical proposals exist for probing many-body effects in armchair SWCNTs, [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] experimental realizations have been limited to DC transport measurements. [22] [23] [24] [25] [26] Recent theories, 13, 21 in particular, have addressed the issue of Fermi edge singularities, specifically for SWCNTs. Balents 13 predicts that an orthogonality catastrophe strongly reduces the singleparticle van Hove singularity to the form (ε − ∆) β−1/2 , with β ≈ 0.3 for carbon nanotubes, where ε is energy and ∆ is the energy gap from the Fermi energy to the first subband, which can be probed via tunneling measurements. Mishchenko and Starykh 21 predict that optical absorption between the linear band and the first massive band (the so-called E M 01 or E M 10 transition) is enhanced, compared to the non-interacting case, and develops a power-law frequency dependence ∝ (ω−∆) −γ , where γ ≈ 0.2 for typical nanotubes. [27] [28] [29] [30] Such predictions provide the theoretical impetus for experimentally probing nonequilibrium many-body responses driven through optical excitations and other quantum effects in armchair SWCNTs. At the same time, as exceptionally conductive wires, exhibiting ballistic conduction even at room temperature, [31] [32] [33] they are promising for a variety of electronic applications. I. INTRODUCTION However, specific and systematic studies of armchair SWCNTs aimed at resolving the aforementioned issues have been severely hampered by sample issues. Although experimental measurements are typically more facile to implement using macroscopic ensembles, because of the heterogeneity inherent to nanotube synthesis, as-produced, bulk samples normally contain a wide assortment of nanotube species of different diameters, chiral angles, lengths, optical enantiomers and electronic types. This heterogeneity in structural properties often leads to overlap in electronic and optical properties, resulting in a response that is a superposition of signals from both metallic and semiconducting nanotubes. Often when such overlap exists, the response from semiconducting SWCNTs can obscure that from metallic nanotubes, leading to only limited and incomplete pictures of optical behavior in these gapless materials as demonstrated by numerous optical studies on ensemble-type samples (e.g., references 34, 35 ). The alternative to such macroscopic studies is to perform micro-spectroscopy on isolated, individual SWCNTs. These studies, however, are typically more elaborate to implement experimentally as the signal tends to be small from a single nanotube. Additionally, study of a particular species of nanotube, like an armchair SWCNT, is almost dependent on chance as nanotube synthesis cannot be sufficiently controlled to grow specific chiralities. As a result, researchers are limited to the laborious process of searching many isolated tubes to find one that fits the experimental needs. Lastly, even in cases where a single example of an armchair SWCNT can be located, constituting a single measurement, the resulting data cannot necessarily be generalized to be representative of all armchair SWCNTs due to possible defects and environmental effects. I. INTRODUCTION Here, we present a route to study armchair SWCNTs on a macroscopic scale using various forms of optical spectroscopy. Using post-synthesis separation techniques such as density gradient ultracentrifugation (DGU) and DNA-based chromatography, researchers have been able to create macroscopic ensembles strongly enriched in armchair species, which can then be probed optically to reveal the intrinsic optical response of metallic nanotubes. In Section II, we briefly review the electronic band structure and optical selection rules for armchair carbon nanotubes and the related narrow-gap semiconducting nanotubes which constitute the class of "metallic" SWCNTs. We then in Section III review the previous experimental studies done on "metallic" SWCNTs using optical absorption, resonant Raman scattering, and resonant Rayleigh scattering spectroscopies. In Section IV, we discuss how to prepare macroscopic ensembles using post-growth separation techniques. DGU (Section IV A) provides colored aqueous suspensions consisting of ensembles of multiple armchair species. The DNA-based method (Section IV B) goes further to produce suspensions of a single-chirality armchair species. I. INTRODUCTION Such highly enriched and specialized samples are perfect playgrounds for studying the optical properties of armchair SWCNTs. Through optical absorption (Section V), we assess the optical structure including line shape, line width and line position of optical transitions in armchair SWCNTs. Based on analysis of both DGU and DNAbased armchair samples, we comment on the importance of excitons to the optical properties of metallic SWCNTs and how the role of excitons explains the rainbow of visible colors exhibited by armchair species of different diameters. Through resonant Raman spectroscopy, we used the excitation profiles of the radial breathing modes (RBMs) to determine optical transition energies and the relative abundances of all "metallic" nanotube species both before and after DGU enrichment (Section VI). While DGU-based separation revealed strong enrichment in armchair species, even "unsorted" SWCNT material revealed a preference for armchair species. Extending our studies to higher energy phonons, armchair ensembles revealed the intrinsic line shape of the G-band phonon as originating only from the G + peak, a transverse optical phonon in armchair SWCNTs (Section VII). Lastly, in Section VIII we summarize our results and discuss future avenues of investigation. II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS The so-called "metallic" SWCNTs are defined as any (n, m) species where the quantity (n − m) is an integer multiple of 3, or in other words, ν = (n − m) mod 3 = 0, II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS where n and m are the chiral indices. This leads us to the more accurate classification (as will be shown shortly) of "metallic" species as ν = 0 species, which can be further subdivided into the armchair (n = m) and the nonarmchair (n = m) species. The diversity in physical crystal structure between different ν = 0 species results in a rich mixture of electronic band structure effects, crystal symmetries, and quasiparticle interactions that strongly define and affect the optical properties (and their related processes) of armchair and non-armchair SWCNTs. II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS Here, we briefly summarize the most important and relevant effects as they pertain to optical spectra of ν = 0 nanotubes. II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS A. Band structure of ν = 0 SWCNTs II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS In 1992, with the initial discovery of carbon nanotubes having just occurred by Iijima et al., 36 the electronic band structure of SWCNTs and specifically "metallic" tubes was calculated by multiple groups [37] [38] [39] following different calculation approaches but yielding similar conclusions. It was shown that SWCNTs could possess ei-ther metallic or semiconducting electronic conduction, depending on the crystal structure of the particular type of tube and specified by the diameter and chiral angle of the nanotube (determined by the chiral indices (n, m)). Specifically, the so-called (n, n) armchair species were shown to be metals with no energy band gap and a finite electronic density-of-states at the Fermi energy. [37] [38] [39] This is a consequence of one of the allowed k-states of an armchair nanotube passing through the K (K')-point of the two-dimensional graphene Brillouin zone, where the so-called Dirac cones meet. The zone-folding approach of Saito et al. 37 yielded analytical expressions for calculating the electronic dispersion of the various bands resulting from the quantization of the allowed kstates of nanotubes (i.e., cutting lines) due to their onedimensionality. Using a k · p formalism near the K-point of graphene, Ajiki and Ando 40, 41 derived an analytical expression for the dispersion of the conduction (+) and valence (−) bands: II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS Here, κ ϕ (q) = (2/d t )(q + ϕ + ν/3) where γ 0 is the nearest neighbor hopping energy, a c−c (= 1.44Å) is the carboncarbon bond length, d t is the nanotube diameter, q is the band index, ν is the type number as defined earlier (which is 0 for "metallic" species), and ϕ = φ/φ 0 is the magnetic flux threading a nanotube in units of the magnetic flux quantum, φ 0 = ch/e. By considering the effect of nanotube curvature through lattice optimization before calculating band structure, Hamada et al. correctly predicted that such non-armchair "metals" (ν = 0) were actually narrow-gap semiconductors due to a shift in momentum of the cutting line that would normally pass through the K-point, resulting in the opening of a small band gap. This shift is a result of enhanced electronic transfer in the circumferential direction of the nanotube due to its curvature. Kane and Mele 42 later derived an analytical expression for this narrow band gap II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS where E g is the curvature-induced gap, α is the chiral angle and all other quantities are defined as before. This curvature-induced band gap is reflected in the electronic band dispersions of "metallic" species in Fig. 1 . In Fig. 1a , which represents an example of an armchair nanotube, no band gap is observed because the momentum shift is parallel and along the already existing cutting line. In the case of a non-armchair, ν = 0 species, a band gap has opened in the linear dispersion bands (q = 0) due to the curvature effect. II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS In the vicinity of the K-point of the first Brillouin zone of graphene, all equienergy lines around the Kpoint are circular around the K-point. While valid for small-energy excitations, this approximation fails at − bands due to trigonal warping. Also, the massless q = 0 bands become massive in the vicinity of k = 0 due to the curvature-induced band gap. Valence (conduction) bands are indicated in blue (red). Solid black (green) arrows indicate allowed optical transitions for electric polarization parallel (perpendicular) to the nanotube axis. Black dashed arrows indicate forbidden optical transitions. II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS larger energies when one approaches the M-point where the equienergy lines become triangular. This distortion of the equienergy contours in the two-dimensional (2D) graphene Brillouin zone from circular to triangular is known as trigonal warping. As a result, the allowed electronic transitions develop a dependence on chiral angle and electronic type with increasing energy. This is particularly relevant for ν = 0 species where initially equidistant cutting lines from the K-point, which intersected the same equienergy line, now become inequivalent due to trigonal warping, resulting in different energy electronic states. The overall effect is that a single "metallic" electronic state becomes split into two by a quantity known as the trigonal-warping splitting. 43 Using a higher-order k · p calculation, [44] [45] [46] the trigonal-warping effect can be approximated by II. BAND STRUCTURE AND SELECTION RULES FOR OPTICAL TRANSITIONS IN METALLIC SWCNTS where ϕ trigonal eff is the effective magnetic flux threading a nanotube equal in energy to the trigonal-warping splitting, β is a constant of order unity, and all other quantities are defined as before. As a note, the higher-order k ·p formalism has been shown to be equivalent with tightbinding approach with regards to both the curvatureinduced band gap and trigonal-warping effects. 44-46 The trigonal-warping effect is most clearly shown in a comparison of the first massive bands (|q| = 1) for armchair and non-armchair species in Fig. 1 . In Fig. 1a , a single parabolic band exists for each of the valence and conduction bands for the armchair case. However, for the non-armchair case as shown in Fig. 1b Fig. 2 and Tables I-III provide theoretical estimates of the curvature-induced gap and trigonal-warping splitting for various ν = 0 species of SWCNTs. Both effects were experimentally realized by scanning tunneling spectroscopy measurements on zigzag tubes, where a splitting of the van Hove singularities and gap opening at the Fermi energy were observed in the electronic density-of-states. 47 B. Selection Rules for Optical Transitions To properly predict, model, and interpret the optical properties of armchair and non-armchair ν = 0 SWCNTs in their respective spectra through such processes as optical absorption and Raman scattering, we must know which optical transitions are allowed between which energy states (i.e., optical selection rules). Using the k · p method, Ajiki transitions are labeled in red. The energy splitting between upper and lower branches is caused by trigonal warping. Both curvature-induced gap and trigonal-warping splitting are zero for armchair species and maximum for (3n, 0) zigzag species. the incident light is parallel to the nanotube axis. However, interband optical transitions between the massless, linear bands are not allowed for any light polarization due to symmetry. In the case where the polarization of the excitation is perpendicular to the nanotube axis, interband transitions are allowed only between bands where the band index changes by 1, i.e., ∆q = ±1 for both massive and massless bands. However, due to the depolarization effect, these perpendicular transitions are expected to be heavily suppressed, 41 although excitonic effects are predicted to retain this transition as a well-defined peak in absorption at a renormalized energy. 49 Subsequently, Jiang et al., 50 using the tight-binding method, calculated the electric dipole matrix elements (whose square is proportional to optical absorption), yielding analytical expressions as a function of chiral index and wavevector, k. As Ajiki and Ando, they found that parallel interband transitions are allowed only between massive bands of the same index and k-vector and not at all allowed between massless linear bands, for armchair species. Furthermore, by examining the kdependence of the matrix element, they found that the dipole matrix element reaches a maximum for k-values that coincide with the positions of the van Hove singularities (VHS) in the electronic density-of-states for each band. As a result, the strong optical absorption observed in nanotubes is a result of the combination not only of the singularity in the density-of-states but also the coinciding maxima in dipole matrix elements. Additionally for armchair SWCNTs, it was shown that the dipole matrix element is zero for all bands at the k = 0 point, indicating a node in optical absorption for armchair species due to their high symmetry. For polarization perpendicular to the tube axis, Jiang et al. 50 reached similar conclusions to that of Ajiki and Ando 41 with the further refinement that only transitions between "linear" and parabolic bands are significant in intensity as they are allowed everywhere in k-space. Transitions between massive bands around k-points near the VHSs, however, are heavily suppressed due to the appearance of nodes in the dipole matrix elements at the positions of the VHS. B. Selection Rules for Optical Transitions With the application of a magnetic field, a band gap opens, proportional in size to the magnetic flux, φ. 40, 41, 51, 52 The band gap reaches a maximum when φ = φ 0 /2 (where φ 0 = h/e is the magnetic flux quantum) and then decreases to zero at φ = φ 0 . This gap opening allows optical transitions between the previously massless linear bands (which are now massive in the vicinity of the band gap) for light polarized parallel to the tube axis. Effectively, the magnetic field has changed a metallic armchair nanotube into a narrow-gap semiconductor. All other selection rules previously mentioned remain the same. As a result, the optical selection rules for narrowgap semiconductors (non-armchair, ν = 0 species) in the absence of a magnetic field are the same as for armchair species in the presence of a magnetic field. B. Selection Rules for Optical Transitions With regards to resonant Raman processes, the optical selection rules are similar to those of optical absorption with the added considerations of the inclusion of the emission or absorption of a phonon of a given symmetry and the emission of the scattered photon. For first-order Raman processes, corresponding to vibrational modes such as the radial breathing mode (RBM) (A-symmetry) and the G-band (A-, E 1 -and E 2 -symmetry components), A-and E 1 -symmetry phonons can be observed between states E q → E q (parallel interband transition E ii ) and E q → E q±1 (perpendicular interband transition E ij where j = i ± 1), where q is the index identifying a particular VHS in the valence [before (after) arrow] and conduction [after (before) arrow] bands. B. Selection Rules for Optical Transitions 53 E 2 -symmetry phonons can only be observed via the E q → E q±1 excitation process. Because of the strong suppression of perpendicular transitions due to the depolarization effect, 41 phonons excited via E ii typically dominate Raman spectra of ensemble samples. C. Excitonic Effects An important additional factor to consider in the optical spectra of metallic carbon nanotubes is the role excitons play. While excitons are commonly associated with the optical transitions and spectra of semiconductors, recent experiments [54] [55] [56] [57] and theoretical studies 46, [58] [59] [60] [61] have reported the strong influence of excitonic effects in the optical spectra and band structure of metallic SWCNTs. While the optical selection rules governing absorption and resonant Raman processes remain unaffected, 46 the energy positions and line shapes of the major optical features are modified as a result of the formation of excitons. Allowed optical transitions for excitons associated with a particular electronic state still obey the earlier optical selection rules with the additional constraint that only 1 of the 16 possible excitonic states (after taking into account valley and spin degeneracies) is optically active. 62 Therefore, optical transitions are only allowed for such optically active excitonic states in the absence of additional perturbations. 62 The inclusion of excitonic effects into the optical spectra of ν = 0 SWCNTs decreases the optical transition energy by the exciton binding energy, estimated to be ∼50-100 meV for the first optical transition in "metallic" species 46, 59, 60 with diameters ranging from 0.8 to 2 nm, with binding energy decreasing with increasing nanotube diameter. Furthermore, the electron-hole Coulomb interaction causes a significant suppression of the electronic continuum in 1-D systems. 63, 64 The result of the continuum suppression on optical spectra is a significant decrease in the high-energy tail in absorption due to the VHS. This causes the line-shape of absorption features to become highly symmetric and almost Lorentzian-like, but a small high-energy tail still remains. 46, 59, 60 This is due to the incomplete transfer of oscillator strength from the continuum to the excitonic states. 60 Interestingly, predictions by Malic et al. 60 show that the magnitude of the trigonal-warping-induced splitting of the E is not significantly affected by excitonic effects. This should produce observable absorption features in non-armchair, ν = 0 species of sufficiently small diameter where the trigonal-warping splitting can exceed the line-widths. Low-energy excitation of excitons across the curvature-induced band gap in non-armchair ν = 0 SWCNTs or the magnetic-fieldopened band gap in armchair SWCNTs should produce observable optical absorption in the terahertz frequency range only in the local k-space vicinity of the gap. Both lowest lying excitons (one optically inactive and one optically active) will be equally populated at experimentally attainable temperatures removing any suppression of optical emission. 61 Excitations away from the band gap, however, will remain forbidden. 46, 61 No excitonic phases are predicted to exist at any value of band gap, since the exciton binding energy monotonically decreases with decreasing band gap. Optical studies of metallic carbon nanotubes have evolved in richness with the significant progress in sample preparation and availability. Initial studies on powders and films of as-produced SWCNT material, first produced in the 1990s, confirmed general predictions of the optical and electronic properties of SWCNTs. Later, in the early 2000s, with the development of aqueous surfactant suspensions containing highly individualized tubes using ultrasonication and ultracentrifugation, 65 our ability to study optical and electronic properties of specific (n, m) species grew, allowing researchers to refine early models and theories. Most recently, with the advent of highly selective separation techniques such as density gradient ultracentrifugation (DGU) and DNA-based ion-exchange chromatography, optical studies of singlechirality SWCNT samples on macroscopic scales were possible, which are devoid of the influence of other neighboring and often interfering species. At the same time, advancements in single-molecule, micro-spectroscopies and low-density, long-length SWCNT growth have allowed researchers to also probe properties of single (n, m) nanotube species. Presented here is a brief summary of the important contributions to the current literature surrounding the properties of ν = 0 SWCNTs across the techniques of optical absorption, resonant Raman scattering, and resonant Rayleigh scattering spectroscopies. A. Optical Absorption Kataura et al. 66 performed one of the earlier optical absorption studies on thin films that were produced from ethanol dispersions of SWCNTs synthesized by the electric arc-discharge method. The ethanol dispersion was spray-painted onto a quartz substrate and optical absorption was measured in the ≈2500-200 nm range. Although the sample was composed of aggregates (bundles) of SWCNTs, three distinct, broad absorption features were observed corresponding to the first and second optical transitions of semiconducting SWCNTs and the first optical transition of "metallic" or ν = 0 SWCNTs, confirming predictions originally made by nearest-neighbor tight-binding approximation calculations. However, absorption features could not be attributed to specific chiral structures. More importantly, this study marks the first appearance of what would later be named the socalled Kataura plot, a plot of optical interband transitions versus nanotube diameter. Such a plot has widely been used by subsequent researchers to aid in the assignment of measured optical transitions to specific (n, m) species and has served a predictive tool as well. A. Optical Absorption After this initial work, further information obtained from optical absorption spectroscopy progressed slowly due to limitations imposed by sample quality. Although liquid suspensions of nanotubes could be produced via ultrasonication and/or chemical functionalization, absorption spectra remained poorly defined due to the reaggregation of SWCNTs in suspension or modification of band structure due to functionalization, producing results similar to the initial work by Kataura et al. In 2002, O'Connell et al., 65 however, succeeded in suspending SWCNTs in water using surfactant micelles and a combination of ultrasonication and ultracentrifugation using nanotubes produced by the high-pressure carbon monoxide (HiPco) method. This breakthrough resulted in aqueous suspensions containing individualized SWCNTs that were not significantly modified electronically by the suspension process. Optical absorption spectra of such samples revealed a multitude of optical features corresponding to the distinct optical transitions of semiconducting and metallic species. Furthermore, direct band gap fluorescence was observed for semiconducting species in the near-infrared region in this study. Although such a photoluminescence process does not occur for metallic nanotubes, scanning excitation studies did reveal a region in the visible excitation (∼450-550 nm) where virtually no photoluminescence was observed, corresponding to the optical transitions of metallic nanotubes. Most importantly, these types of aqueous surfactant suspensions of SWCNTs opened the door to many other optical studies via absorption, resonant Raman scattering, and ultrafast optical studies. As related to optical absorption studies, these solution-phase samples along with subsequent (n, m) identification of optical features by other techniques, allowed studies of chemical doping and functionalization (for example, diazonium functionalization chemistry by Strano et al. 67 ). Such studies revealed that, in many cases, metallic nanotubes were more susceptible to such chemical species presumably because of the facile availability of delocalized π-electrons. A. Optical Absorption With the rapid progress in separation techniques such as DGU that resulted in samples that were separated by electronic type and/or diameter, clear delineation of semiconducting and metallic transitions could be observed. 68, 69 In particular, with the knowledge of the general excitation regions where metallic and semiconducting transitions occur, optical absorption was used to estimate the electronic type purity of enriched samples by integrating the area under metallic and semiconducting species. Even more useful was the observation by Miyata et al. 70 and Green et al. 71 that thin films produced from metal-enriched nanotube material exhibit little broadening and red-shifting as compared to the individualized enriched suspension. This is unlike the case of semiconductor-enriched suspensions, which exhibit significant peak broadening and red-shifting upon aggregation of SWCNTs into films. Using length-sorted samples produced via DGU from CoMoCAT SWCNT material, Searles et al. 72 performed magneto-optical-absorption studies in magnetic fields up to 35 T, where they determined the orbital magnetic susceptibility anisotropy, ∆χ, to be 2-4 times greater for ν = 0 species (5,5), (6, 6) and (7, 4) than for semiconducting species of comparable diameter. A. Optical Absorption At the same time that sample preparation improved, absorption spectroscopy techniques improved in sensitivity to allow the measurement of absorption spectra of individual SWCNTs. Berciaud et al. 73 succeeded in measuring absorption spectra, using photothermal heterodyne imaging (PHI) and a cw tunable laser excitation source, of individual HiPco-derived SWCNTs, deposited from aqueous suspension onto a microscope coverslip. Spots observed in PHI images were considered to be individual tubes and had subsequent absorption and photoluminescence spectra measured. In cases where both absorption and photoluminescence were observed, the tubes were attributed to semiconducting species. Such nanotubes also exhibited exciton-phonon sidebands in their spectra, additionally confirming their semiconducting nature. In cases where PHI spots only produced an absorption spectrum and no excition-phonon sidebands, such tubes were attributed to metallic species although assignment to a specific (n, m) species was not possible. Wang et al., 54 using SWCNTs grown by the chemical vapor deposition (CVD) method across a trench produced in a support substrate, measured optical absorption spectra using a modulation technique and broadband white light source produced by supercontinuum generation. Individual SWCNTs were independently (n, m) identified by electron beam diffraction before absorption measurement. In particular, Wang et al. measured the absorption spectrum of a (21,21) armchair nanotube and observed a highly symmetric absorption line shape followed by a very weak continuum at slightly higher energy. This highly symmetric feature was attributed to excitonic absorption and constituted the first experimental evidence that excitons can exist in one-dimensional carbon nanotube metals (first predicted by Deslippe et al ). 59 Additionally, Wang et al. and later Zeng et al., 74 using Rayleigh scattering in a backscattering geometry on an individual (13, 10) , claimed to observe a weak exciton-phonon sideband due to an optical phonon, which supports the claim of exciton involvement in metallic SWCNT optics. This feature, however, was not clearly observed in either case. B. Resonant Raman Scattering Resonant Raman scattering (RRS) was one of the earliest optical techniques to gather information specifically surrounding metallic carbon nanotubes. Because of Raman scattering's resonant nature and the unique band structure of carbon nanotubes, only phonons corresponding to species electronically resonant with the excitation energy should appear. Combined with the frequency dependence of certain vibrational modes for nanotube crystal structure, (n, m)-specific information can be extracted. B. Resonant Raman Scattering The first RRS spectra measured on carbon nanotubes were reported by Rao et al. 75 on laser-ablationsynthesized SWCNT powder. By examining excitation energy dependence using multiple discrete laser wavelengths, multiple Raman-active modes predicted by theory were observed including the radial breathing mode (RBM) and G-band mode. The RBM and G-band modes were in particular observed to change with excitation energy, illustrating the resonance effect in nanotube Raman scattering. At one particular excitation wavelength, the G-band was observed to become anomalously broad, which would later be shown to be due to a unique feature of non-armchair, ν = 0 nanotubes. [76] [77] [78] To examine this resonant effect more closely, Sugano et al. 79 performed more detailed excitation-wavelength-dependent RRS studies using a tunable excitation source (cw dye laser) on SWCNT powder produced by the electric arcdischarge method. There they observed resonant enhancement of certain RBM features that seemed to agree with theoretical predictions of the locations of electronic transitions for certain tubes. Furthermore, the predicted diameter dependence of electronic transitions was observed, with transition energy decreasing with decreasing RBM frequency (i.e., increasing nanotube diameter). B. Resonant Raman Scattering Moving focus to the G-band phonon, Pimenta et al. 80 made the association of the broad G-band feature observed by Rao et al. to "metallic" SWCNTs using laser-ablation-produced SWCNTs. Brown et al. 81 subsequently assigned the broad, lower frequency component of the G-band, the so-called G − peak, to a Breit-WignerFano resonance between the discrete circumferential vibrational mode and the continuum of electronic states. As such, the broad G − peak was used as an indicator of the presence of metallic nanotubes for many years until further refinements were made (see Section VII). In another work by Brown et al., 82 it was shown that anti-Stokes and Stokes Raman spectra taken at the same excitation wavelengths demonstrated very different line shapes in their respective G-bands, one corresponding to metallic species and the other to semiconducting species. 80 This demonstrated the discrete nature of the electronic density-of-states of a given species and the ability to selectively excite one species over another by careful selection of laser energy. 80 However, in mixed powder samples like those employed above, very little (n, m)-specific data can be obtained. To resolve this issue, researchers resorted to the use of RRS micro-spectroscopy to investigate individual SWCNTs on a supporting substrate. Using such an approach, Jorio et al. 83 measured Raman spectra of multiple individual tubes, grown via CVD onto silicon substrates, focusing on the RBM frequency. Combined with predictions based on a theoretical Kataura plot, RBMs were assigned to possible (n, m) indices, including several metallic species, providing definitive assignments of Raman spectral features to specific metallic chiralities. Jorio et al. 84 followed up their earlier work with a study of the G-band frequencies of 62 different individual tubes. With that array of data on different (n, m)-identified species, they attempted to extract an empirical relationship describing the frequency dependence of the lower energy G − peak relative to the G + peak, which was claimed to have no frequency depen- 80 , where C is a constant dependent on electronic type). 80 Examining yet another phonon mode, the dispersive G'-mode, Souza Filho et al. 85 measured the intensity of that mode as a function of excitation energy for several individual metallic SWCNTs grown via CVD on SiO 2 . The observed dispersion was claimed to correlate with the then-recently-predicted trigonal warping effect in the electronic density-of-states. Furthering investigations on the trigonal warping effect via Raman, Son et al. 86 observed both lower-(E M− ii ) and upper-branch (E M+ ii ) metallic optical transitions, due to trigonal warping splitting (see Figs. 1b and 2b), in individual metallic tubes via measurement of the intensity dependence of a tube's RBM as a function of excitation energy (Raman excitation profile). There, it was observed that the higher-energy, upper-branch metallic transition resulted in lower Raman intensity at its resonance maximum as compared to the lower energy, lower-branch transition for the same tube. However, this only occurred for nanotubes of large diameter (>1.5 nm) causing questions as to why its appearance had not previously been observed in smaller diameter tubes. 80 Returning to the G-band mode, experimental studies by Wu et al., 87 Telg et al., 88 Fouquet et al., 89 Michel et al., 77 and Park et al. 90 attempted to correlate G-band line shape with specific (n, m), especially in light of a new explanation for the origin of the broad G − feature in metallic nanotubes as arising from the Kohn anomaly 91 (discussed later in Section VII). Raman measurements were made on individual, (n, m)-identified tubes. In all the studies except Park et al., the appearance of the broad G − peak was attributed to a ν = 0, non-armchair species with the largest G − intensity observed for zigzaglike species. In the case of armchair nanotubes, the G − peak was absent. Park et al.'s observations agreed with that of the other reports mentioned prior except in the case of the armchair SWCNT, where they claimed a broad G − peak was present as well. In a completely new avenue of investigation, Farhat et al. 92 observed a broad Raman feature not corresponding to previously observed vibrational Raman features for individual ν = 0 tubes on a substrate. The broad Raman feature, which moved in scattered photon energy with changing excitation energy, was explained to be a result of inelastic scattering of photoexcited carriers with the electronic continuum present in the linear bands of metallic band structure. However, it should be pointed out that the species examined in this study were not truly metallic as they were non-armchair, ν = 0 nanotubes. 80 One particular issue with single-tube Raman measurements is the inability to select which (n, m) are investigated due to the lack of control in the growth process. Finding a particular type of chirality of nanotube is a matter of chance. With the development of aqueous surfactant suspensions, however, all species present in the as-produced nanotube material can be examined if the appropriate excitation conditions are available. Strano et al., 93 using a surfactant-suspended dispersion of HiPco SWCNTs, attempted to use a tunable excitation source (a combination of cw dye and solid-state lasers) to map out most of the metallic species via excitation dependence of their RBM features and assign them to specific (n, m) indices. While not all the assignments were correct due to the unknown knowledge that upper branch metallic transitions were not present in nanotubes with a diameter <1.3 nm, the basic methodology to assigning them was sound. Subsequent studies by Doorn et al., 94 Telg et al., 95 Fantini et al., 96 and Maultzsch et al. 97 on surfactant-suspended HiPco nanotubes and later surfactant-suspended CoMoCAT nanotubes, 98 correctly assigned RBM features using optical transition energies obtained from Raman excitation profiles to (n, m) species. The combined data sets of tabulated (n, m) with RBM frequencies and optical transition energies allowed Maultzsch et al. 97 and also Jorio et al. 99 to develop empirical equations to predict both RBM frequencies and optical transition energies for nanotubes, which has served as an experimental predictive tool for locating previously unobserved spectral features. 100 Jorio et al., 99 in particular, attempted to take existing expressions for tight-binding calculations predicting SWCNT optical transitions and add additional chiral-angle-dependent and logarithmic terms to account for chiral-dependent electron and hole effective masses and many-body corrections, respectively, with parameters tuned to fit existing optical data. 99 Looking at another type of ensemble nanotube sample, arrays of vertically aligned carbon nanotubes grown via CVD and containing a broad range of diameters and chiralities, Doorn et al. 55 performed similar excitationwavelength-dependent studies to that of the tunable excitation suspension work previously mentioned. Due to the broad diameter distribution (0.7-4 nm), 77 different metallic optical transitions were observed including 15 clear, upper-branch metallic transitions (the rest were lower-branch metallic transitions), spanning the first and second optical transitions. The data show that upperbranch metallic optical transitions were not observed for SWCNTs with a diameter <1.3 nm, yet for diameters >2 nm, significant Raman intensity due to resonance with upper-branch transitions was observed, especially for zigzag and near-zigzag species, in contrast to the work of Son et al. 86 By examining scaling behavior of the optical transitions to a model previously created for examining semiconducting nanotube transitions 101 where excitons are known to play an important role, Doorn et al. suggested that excitons are also important in the optical transitions of metallic nanotubes, in agreement with single-tube absorption measurements. 102 To further investigate this connection between excitons and ν = 0 nanotubes, May et al., 103 using similar vertically aligned carbon nanotube arrays, examined the optical transition energy dependence of the metallic species (13,1) via its RBM as a function of temperature. At moderate temperatures above room temperature, the optical transition energy was observed to decrease with increasing temperature. This monotonic decrease in transition energy up to 570 K was explained as a softening of the optical transition due to lattice expansion and electron-phonon coupling upon heating. However, above 570 K, a sudden discontinuity and increase occurred in the optical transition energy with temperature. This was attributed to thermal dissociation of the exciton bound state (exciton binding energies for metallic nanotubes have been estimated to be ≈ 50 meV, 59 ,102 which corresponds to 580 K) with the optical transition state now consisting of band-to-band transitions. This work and the previous one 55 highlight the importance of excitons in metallic nanotubes, as will be discussed later in Section V. 99 Tables I-III summarize many of the basic (n, m)-specific nanotube parameters and experimentally determined data from optical absorption and resonant Raman experiments reviewed briefly here and presented later on in Sections V-VI, along with theoretical predictions of curvature-induced band gap and trigonal warping splitting. C. Resonant Rayleigh Scattering Resonant Rayleigh scattering or elastic light scattering has recently become a powerful tool for the optical study and rapid characterization of nanotube samples. 105-107 Applicable to individual tubes deposited or grown on a substrate over a trench, the technique can rapidly identify optical transitions due to its large scattering intensity when using a broadband, tunable excitation source such as supercontinuum-generated white light. Sfeir et al. 105 made the observation of excitation spectra consistent with "metallic" species exhibiting trigonal warping splitting. However, the individual tubes measured were not (n, m)-identified. In a subsequent study by Sfeir et al., 106 where individual tubes were (n, m)-identified using electron beam diffraction, measurements on metallic species revealed a single optical transition feature for armchair species and a two-peak feature for non-armchair, ν = 0 species due to trigonal warping. Most recently, using the approach of Sfeir et al., Liu et al. 107 combined electron beam diffraction with in-situ Rayleigh scattering to measure 206 optical transitions of (n, m)-identified nanotubes. In particular, for ν = 0 species, optical transitions were measured ranging from E M 11 to E M 33 , observing trigonal warping splitting for nonarmchair tube. From this broad range of data, covering nanotubes as small in diameter as (10, 10) to as large as (31, 31) , an empirical equation was produced to predict optical transition energies with appropriate (n, m)-and E ii -dependent parameters. However, it should be noted that small-diameter metals with optical transitions >2.5 eV were not studied, which is where most deviations from graphene-like behavior occur (small-diameter metals are the focus of our work presented later). IV. METHODS FOR ENRICHMENT OF ARMCHAIR CARBON NANOTUBES Optical investigations of armchair and other ν = 0 nanotubes have been often impaired by multiple issues related to sample quality and signal detection. Due to an inability at this time for researchers to exactly control and tune the chirality distribution of species grown in nanotube syntheses, as-produced nanotube materials are plagued with issues of sample heterogeneity with respect to physical parameters such as nanotube length, diameter, chiral angle, electronic type, and aggregation state. Optical measurements on metallic SWCNTs contained within macroscopic ensembles composed of heterogeneous material, such as colloidal suspensions, powders, and thin films, typically suffer from problems such as spectral overlap wherein the response of a particular optical mode of a particular (n, m) species is obscured by overlap with the optical response from other nanotube species. Furthermore, even when spectral overlap is not an issue and spectral features are well-defined, optical spectra containing contributions from both metallic and semiconducting species can be dominated by signal from semiconductor components due to their often larger optical response. This is particularly true for resonant Raman scattering. IV. METHODS FOR ENRICHMENT OF ARMCHAIR CARBON NANOTUBES One approach to circumvent these sample heterogeneity issues is in the use of single-nanotube microspectroscopy where a single nanotube is deposited or grown on a substrate or host medium and its optical response is subsequently measured. While this allows for clear measurement of the optical properties of a particular SWCNT species, such measurements typically produce small signal and still suffer from the problem of lack of chirality control (i.e., one does not have fine control of TABLE I: Basic parameters, optical transition energies, and phonon frequencies of ν = 0 (or mod 0), single-wall carbon nanotubes, suspended in aqueous surfactant with a diameter range from 0.73 to 1.65 nm. Here, (n, m) are the chiral indices, 2n+m is the family number, dt is the nanotube diameter, RBM is the radial breathing mode frequency, α is the chiral angle, Eg is the curvature-induced band gap, ∆E could not be determined. what species can be studied since the growth process cannot be controlled). This results in the need to do many measurements on many different tubes to find suitable metallic tubes for study, which can be time-consuming. Additionally, because of the one-by-one nature of singlenanotube spectroscopy, studies are not necessarily statistically representative of a particular nanotube material. Another approach in the optical study of armchair SWCNTs that avoids sample issues while simultaneously avoiding the experimental difficulties of single-nanotube micro-spectroscopies is to produce macroscopic ensembles of SWCNTs enriched in armchair nanotube species by post-synthesis separation methods. Researchers have attempted such post-production separations since the early days of nanotube research via methods such as selective chemical functionalization, 67 complexation, 108 dielectrophoresis, 109 etc. However, these methods often modify the properties of the nanotubes or do not produce sufficient material for ensemble optical measurements. Additionally, the specificity of separation to one specific chirality of such methods is often low. IV. METHODS FOR ENRICHMENT OF ARMCHAIR CARBON NANOTUBES Recently, researchers have appealed to techniques normally reserved for biochemistry to separate SWCNTs due to the large volume of work demonstrating highpurity and specificity in separation of biological macromolecules, such as DNA and proteins, based upon molec- dient ultracentrifugation and DNA-based ion-exchange chromatography, both of which lead to the production of armchair-enriched macroscopic ensembles suitable for optical measurements. A. Density Gradient Ultracentrifugation Density gradient ultracentrifugation (DGU) separates different molecular species by exploiting differences in their respective mass densities as they travel through a viscous medium under a large applied centrifugal field. Briefly, an aqueous suspension of surfactant-suspended SWCNTs travel through a mass density gradient via ultracentrifugation (∼200,000g for several hours). Due to the variation in density of the gradient, different density nanotube species will migrate towards different regions of the gradient, under the applied force of the centrifuge versus the buoyant force of each nanotube due to its mass density, until an equilibrium is reached. At this point, the nanotubes are now sufficiently separated in vertical distance in the gradient so that they may be extracted using simple fractionation techniques (see diagram in Fig. 3 ). As the physical species being separated is an individual SWCNT surrounded by a surfactant micelle and several layers of hydration, the intrinsic differences in mass density of different chirality nanotubes (and hence the chiral-specific buoyant forces) are primarily due to the varying dimensions of different chiralities (diameter and length) as well as the size and surface coverage of the surfactant micelle around the nanotube, which can be a complex function of nanotube diameter, surfactant type, ionic concentration, and sterics. However, this complexity can also lead to high specificity of separation as well. A. Density Gradient Ultracentrifugation DGU was first applied to SWCNTs by Arnold et al. CoMoCAT-produced SWCNT materials. There, they demonstrated the efficacy of DGU for separating SWCNTs by observing diameter-based sorting as measured by optical absorption spectroscopy. Arnold et al. subsequently demonstrated DGU using anionic surfactants commonly used to suspend SWCNTs by ultrasonication and centrifugation, namely sodium dodecyl sulfate (SDS) and sodium cholate (SC). 68 In performing DGU with the single surfactant SC, it was shown that not only could diameter-sorting be achieved with greater efficiency than their previous DNA-suspended work, 110 but that with multiple iterations of DGU, single-chirality semiconducting samples, namely (6, 5) and (7, 5) , could be produced with CoMoCAT SG65 grade nanotube material, as demonstrated by optical absorption and photoluminescence excitation (PLE) spectroscopy measurements. In the same work, it was shown that the addition of a second surfactant, SDS, to the density gradient, resulted in sorting by electronic type. Specifically, laser-ablationproduced SWCNT material could be separated into thin, concentrated fractions of nearly pure metallic and semiconducting nanotubes as shown by optical absorption. This opened the door for many different types of measurements on metallic nanotubes, ranging from DC electrical transport to optical spectroscopy. To confirm their type separation, thin films of purely metallic and semiconducting nanotubes were fabricated by vacuum filtration. Significantly lower sheet resistivity was observed for the metal-enriched film as opposed to the semiconducting film, as would be expected. 68 On the heels of this breakthrough in SWCNT separation science utilizing commonly employed surfactants as opposed to DNA, a flood of research followed, improving and expanding the DGU technique further. Yanagi et al. 69 added a third surfactant to the combination of SDS and SC in their electronic-type sorting with the addition of sodium deoxycholate (DOC), resulting in an expansion in vertical separation distance of the thin typeseparated fractions observed by Arnold et al.. This resulted in significantly more facile fractionation of the separated products, which they demonstrated on CoMo-CAT, HiPco, and laser ablation carbon nanotube material. In addition to optical absorption and sheet resistance measurements on their
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