INTRODUCTION
The interaction between light and solids gives rise to many optical properties in solid-state systems that can be observed by a variety of experimental techniques such as photo absorption and emission, infrared absorption, Raman scattering, photoelectron and Auger spectroscopy, and ultraviolet and X-ray photoelectron spectroscopy. Especially in one-dimensional (1-D) solids, such as carbon nanotubes,[1-3] much useful information on the optical properties can be obtained by exploiting low-dimensional effects, such as the direction of the polarization of the light relative to the nanotube axis, the resonance between the laser excitation energy and a singularity in the joint density of states, and the variety of crystal structures of nanotubes stemming from their chirality. Thus, by rotating the orientation angle of the nanotube relative to the incident and scattered light polarization directions, by changing the laser excitation energies, or by selecting different geometries for the nanotubes, we can systematically investigate the optical properties for many different types of carbon nanotubes in a consistent way. Especially fruitful has been the study of the resonance Raman spectra from only one nanotube by using micro-Raman measurements of a nanotube on a Si/ SiO2 substrate. Because of the large amount of detailed information that can be obtained from this technique, micro-Raman spectroscopy is considered to be a standard nanotechnique for probing the optical properties of a nanotube in a quick, nondestructive way at room temperature and atmospheric pressure.
Since the reported observation of carbon nanotubes in 1991, high-resolution transmission electron microscopy (HRTEM) and scanning probe microscopy [SPM, such as scanning tunneling microscopy/spectroscopy (STM/STS), atomic force microscopy (AFM), Kelvin force microscopy (KFM), magnetic force microscopy (MFM), and so on] have provided definitive tools for observing tiny samples on nanometer length scales. However, for single-wall carbon nanotubes (SWNTs), the high quality required of the measurement instruments and of the personnel to observe structure at the atomic level restricts the number of instruments and researchers who can make such measurements.
The finding of so many new phenomena and concepts in the nanotube field in the past decade reminds us of the good old times of the adventurers on sailing ships who ventured out to find the new world. In the next decade, nanotube research will enter a new period in which nano-tube samples are easily obtained, in which many nanotube-based industrial technologies and applications will be developed, and in which fundamental physics and chemistry discoveries will continue to be made, but more and more the research will be stimulated by the needs of the applications and by the availability of new measurement tools. In this sense, a standard technique to characterize nanotubes is highly desirable. Here it is proposed that resonance Raman spectroscopy should fill this role.
OVERVIEW
One unique feature of a SWNT is the fact that the electronic structure is either metallic or semiconducting, depending only on the chirality of the nanotube and not on the impurities that it contains.[1] Here the chirality of a SWNT denotes the spiral geometries of a nanotube structure, which is specified by two integers (n, m) as discussed below. Thus, the interaction of the conduction electrons in the metallic nanotube with light can be investigated, e.g., by comparing the Raman spectra of metallic nanotubes with that of semiconducting nanotubes that have no doping with noncarbon atoms. Thus, carbon nanotubes provide an ideal material for understanding the physical properties of a low-dimensional solid. Another important fact about nanotubes is that the diameter of a nanotube is much smaller than the wavelength of light, but the length of the nanotube is usually comparable to or greater than the optical wavelength. Thus microscopic and macroscopic views of nanotubes can coexist within a single nanotube, and it is expected that these unique properties will lead to new optical applications in the future.
Fig. 1 Single-wall carbon nanotubes. The ends of nanotubes are capped by fullerene hemispheres. (a) Armchair nanotube, (b) zigzag nanotube, and (c) chiral nanotube.
Among the various optical techniques, resonance Raman spectroscopy of nanotubes has been investigated in depth during the last few years. The resonance Raman effect is a strong enhancement 103) of the Raman intensity when a real optical absorption takes place in the excitation process. The resonance condition for each nanotube is very sharp in energy, known as van Hove singularities, with a width of less than 10 meV. Because the resonant energies of each (n, m) nanotube are different from one another, the energies of the van Hove singularities can be used for identification (something like a fingerprint) of the nanotube atomic structure. Thus, van Hove singularity is a key word to understand optical properties of nanotubes. We can select a particular (n, m) SWNT for investigation by selecting the appropriate laser energy Elaser, or we can investigate metallic or semiconducting nanotubes separately by selecting Elaser, even for samples consisting of SWNT bundles containing a mixture of metallic and semiconducting SWNTs. Recently, using a chemical vapor deposition (CVD) growth technique for preparing isolated SWNTs on a Si/SiO2 substrate, we have observed resonance Raman spectroscopy from one nanotube. Compared with the spectra from SWNT bundle samples, single nanotube spectroscopy gives much more definitive information on the electronic and phonon properties of SWNTs.
In this article, we first discuss the singular nature of the electronic energy states and then discuss optical absorption and the resonance Raman spectra. Finally, we present the double resonance theory of Raman nanotubes, which is necessary for describing the experimentally observed dispersive phonon modes in SWNTs.
VAN HOVE SINGULARITIES
A single-wall carbon nanotube is a graphene (honeycomb) sheet rolled up into a cylinder. In Fig. 1, we show the cylindrical structures of SWNTs. Each end of the nanotube is terminated by a hemisphere of fullerene containing six pentagonal carbon rings. Because it is considered that nanotubes grow with a cap, the cap structure is essential for generating different kinds of geometries for nanotubes. The relative positions of the pentagonal rings with respect to one another is almost arbitrary except for the requirement that two pentagonal rings do not touch each other (isolated pentagon rule), thus giving rise to many possible geometries for generating nanotube structures. Among the various nanotube structures, there are only two kinds of nanotubes that have mirror symmetry along the nanotube axis, namely, the armchair and zigzag nanotubes, as shown in Fig. 1a and b, respectively. The names armchair and zigzag are taken from the shape of the edge cuts shown on the right side of Fig. 1. All other nanotubes (Fig. 1c) exhibit axial chirality and are called chiral nanotubes.
Fig. 2 The unrolled honeycomb lattice of a nanotube. OA is the equator of the nanotube and OB corresponds to the translation vector of this 1-D material. By connecting OB to AB’, we can make a seamless cylindrical shape. OAB’B is a unit cell of the nanotube. The figure corresponds to the (4,2) chiral nanotube and there are N =28 hexagons in the unit cell.
The geometry of a nanotube is uniquely expressed by two integers, (n, m). In Fig. 2, the rectangle OAB’B is shown, and by connecting OB to AB’, we can make the cylindrical shape of the nanotube. In this case, the vector OA (hereafter we call OA the chiral vector) corresponds to the equator of the nanotube. OA can be expressed by a linear combination of the two unit vectors, a1 and a2, of the honeycomb lattice, so that
The length of OA divided by p gives the diameter of the nanotube, dt. Two lattice vectors OB and AB’, which are perpendicular to OA, correspond to the translational vector T in the 1-D lattice of the nanotube, and T is a function of n and m, as shown in Table 1. The rectangle OAB’B denotes the unit cell of the nanotube containing N hexagons and 2N carbon atoms, where N is expressed by
The integer dR is the common multiple divisor of (2n+m) and (2m+n). Further details of the mathematics describing these variables are given in Table 1 and are further explained in Ref. [1]. (This introductory textbook of nanotube can be cited for the ”Introduction” and ”van Hove Singularity” sections.)
Table 1 Parameters for carbon nanotubes
Fig. 3 The energy dispersion relations for 2-D graphite are shown throughout the whole region of the Brillouin zone. The inset shows the energy dispersion along the high symmetry lines.
The valence electrons of sp2 carbons, such as graphite and nanotubes, consist of p (2p) electrons. Because a carbon atom in a nanotube has one p electron (and because there are 2N carbon atoms in the nanotube unit cell), 2N 1-D electronic p bands are obtained for a SWNT by applying periodic boundary conditions around the circumferential direction OA. Such a treatment for the electronic structure is called ”zone folding.” Along the direction OA, the wave vectors (2p/wavelength), which are perpendicular to the nanotube axis direction k?, are discrete and are given by the condition:
Thus, one obtains N inequivalent k? = 2pp/IChI = 2p/dt discrete wave vectors for bonding and antibonding p bands and the 1-D wave vectors k||, parallel to the nanotube axis, are continuous (— p/T< k|< p/T) for an infinitely long SWNT.
One-dimensional energy dispersion relations for a nanotube Ei_d(p,k|) are obtained by cutting the 2-D energy dispersion relations of graphite E2_D(k) along the k| direction with wave vectors k?p placed at equal distances of 2/dt with a length of 2p/T,
where, by a tight binding calculation, E2_D(k) is given by:
in which the coordinates of kx and ky are given in Table 1. The plus and minus signs in Eq. 5 correspond to unoccupied (antibonding) and occupied (bonding) p energy bands, respectively. go>0 is the tight-binding nearest-neighbor overlap energy parameter. When we plot the Eg2-D in the hexagonal 2-D Brillouin zone (BZ) (Fig. 3), the bonding and antibonding energy dispersion relations touch each other at the hexagonal corners, the K and K points of the 2-D Brillouin zone. Because of the periodicity of Eg2-D (kx, ky) in k space, the K and K points are inequivalent to each other. The asymmetry between the shapes of the bonding and antibonding p bands comes not directly from Eq. 5 but from the higher-order correction to the tight-binding parameters known as the overlap parameter s. In the following discussion, however, for simplicity, we use s=0.
Fig. 4 The energy dispersion relations for carbon nanotubes. (a) (5,5) Armchair, (b) (9,0) zigzag, (c) (10,0) zigzag nanotubes. Bold and thin lines are doubly or singly degenerate energy bands, respectively. The Fermi energy is located at E(k)/t= o. (5,5) and (9,o) are metallic nanotubes while (10,0) is semiconducting.
Cutting these 2-D energy dispersion relations by equidistant lines parallel to the nanotube axis (hereafter we call these lines ”cutting lines”) corresponds to obtaining the 1-D energy dispersion of nanotubes, as shown in Fig. 4. In this formulation the Fermi energy is located at E(k)/t=0, and thus depending on whether there are energy bands that cross the Fermi energy or not, the nanotube is either metallic or semiconducting, respectively. The condition to get either metallic or semiconducting nanotubes is whether the cutting lines cross the K or K points where the bonding and antibonding p bands of 2-D graphite touch each other (Fig. 3).
Fig. 5 Electronic density of states for the (a) (10,0) semiconducting zigzag nanotube, (b) (9,0) metallic zigzag nanotube. Dotted lines denote the density of states for 2-D graphite.
Fig. 6 Cutting lines around the K point in the Brillouin zone for (a) metallic and (b) semiconducting nanotubes.
In Fig. 5, we show by solid lines the electronic density of states (DOS) for (a) (10,0) and (b) (9,0) zigzag nanotubes. Dotted lines denote the DOS for 2-D graphite for comparison. For metallic nanotubes, the density of states at the Fermi energy is constant as a function of energy, while there is an energy gap for semiconducting nanotubes. The value of the density of states for metallic SWNTs at the Fermi energy, D(EF) is 8/V3nay0 in units of states per unit length along the nanotube axis per electron volt, which is independent of diameter. If D(EF) is given by per gram per electron volt, the D(EF) is relatively large for small-diameter nanotubes. The many spikes in the DOS for nanotubes correspond to the energy positions of the minima (or maxima) of the energy dispersion curves of Fig. 4. Each spike exhibits a singularity of 1/y/E — E0 (where E0 is the energy extremum) that is characteristic of 1-D materials and is known as a van Hove singularity.
The energy position of the van Hove singularity near the Fermi energy is determined by how the cutting lines are oriented near the K point of the 2-D BZ. In Fig. 6, we show the cutting lines around the K point for (a) metallic and (b) semiconducting nanotubes. In the case of metallic nanotubes, the central cutting line just goes through the K point, and the two nearest cutting lines are located at the same distance of K2 from the K point. The corresponding energy dispersions for the central line and for the two neighboring lines are, respectively, metallic linear energy dispersions, and are the first subbands (Fig. 4a), which give the van Hove singularities nearest to the Fermi energy. An important fact about the energy dispersion of 2-D graphite around the K point is that the energy dispersion of Eq. 4 is linear in k, when we measure k from the K point, i.e.,
Fig. 7 Equienergy contours for the electronic p bands of 2-D graphite.
Within the linear approximation of the energy dispersion, the equienergy contour of 2-D graphite is a circle around the K point (Fig. 7). Thus, only the distance of the two neighboring cutting lines from the K point is essential for determining the energy position of a van Hove singularity. Because the separation between two cutting lines, K2, is inversely proportional to the diameter of the nanotube, dt, the van Hove singular energies relative to the Fermi energy are inversely proportional to dt, too.
In the case of a semiconducting nanotube, on the other hand, the K point is always located at a position one-third of the distance between the two lines as shown in Fig. 6b. Thus, the first and the second van Hove singular energies from the Fermi energy correspond to the 1-D energy dispersion for the nearest and the second nearest cutting lines, respectively, whose energies are one-third and two-thirds of the smallest energy separation of the van Hove singularity of metallic nanotubes with a similar diameter.
The linear energy approximation, however, does not work well for k points far from the K point. In this case, the equienergy contours deform with a deformation that increases with increasing k, and eventually become a triangle that connects three M points around the K point (Fig. 7) when we consider the periodicity of the electron dispersion relations in k space. This effect is known as the trigonal warping effect of the energy dispersion. When the trigonal warping effect is included, the direction of the cutting lines, which depends on the chiral angle of the nanotube (or simply on its chirality), is essential to determine the precise positions of the van Hove energies. For example, the two neighboring cutting lines in the metallic nanotube in Fig. 6a are not equivalent to each other, which gives rise to a splitting of the van Hove peaks in the DOS. This splitting of van Hove peaks is a maximum for zigzag nanotubes and monotonically decreases to zero for armchair nanotubes for which the two neighboring cutting lines are equivalent to each other. For both metallic and semiconducting nanotubes, the trigonal warping effect appears as a modification of the energy position of the van Hove singularities that depends on chirality. This value of the splitting of the van Hove singularities (0.1 eV at most) is large enough to be easily detected by the energy accuracy of resonance Raman spectroscopy (10 meV). It is therefore possible to specify the (n, m) values for SWNTs from a detailed analysis of the resonance Raman spectra of each SWNT using a tunable laser.
