Hu f sf

where e2p is the site energy of the 2p atomic orbital and f{k) =eiWv5 + 2e-ifcW2v'3cos*|£ (5)

where a = |ai| = |a.21 = \/3ac-c- Solution of the secular equation det(7i — ES) = 0 implied by (4) leads to the eigenvalues

for the C-C transfer energy 70 > 0, where s denotes the overlap of the electronic wave function on adjacent sites, and E+ and E- correspond to the n* and the n energy bands, respectively. Here we conventionally use y0 as a positive value. The function w(k) in (6) is given by w(k) = y/\f(k)F= \l l+4cos^^cos^ + 4cos2^. (7)

In Fig. 2 we plot the electronic energy dispersion relations for 2D graphite as a function of the two-dimensional wave vector k in the hexagonal Brillouin zone in which we adopt the parameters y0 = 3.013 eV, s = 0.129 and e2p = 0

Fig. 2. The energy dispersion relations for 2D graphite with 70 = 3.013 eV, s = 0.129 and e2p = 0 in (6) are shown throughout the whole region of the Brillouin zone. The inset shows the energy dispersion along the high symmetry lines between the r, M, and K points. The valence n band (lower part) and the conduction n* band (upper part) are degenerate at the K points in the hexagonal Brillouin zone which corresponds to the Fermi energy [2]

Fig. 2. The energy dispersion relations for 2D graphite with 70 = 3.013 eV, s = 0.129 and e2p = 0 in (6) are shown throughout the whole region of the Brillouin zone. The inset shows the energy dispersion along the high symmetry lines between the r, M, and K points. The valence n band (lower part) and the conduction n* band (upper part) are degenerate at the K points in the hexagonal Brillouin zone which corresponds to the Fermi energy [2]

so as to fit both the first principles calculation of the energy bands of 2D turbostratic graphite [8,9] and experimental data [2,10]. The corresponding energy contour plot of the 2D energy bands of graphite with s = 0 and e2p = 0 is shown in Fig. 3. The Fermi energy corresponds to E = 0 at the K points.

Near the K-point at the corner of the hexagonal Brillouin zone of graphite, w(k) has a linear dependence on k = |k| measured from the K point as a/3

Thus, the expansion of (6) for small k yields

so that in this approximation, the valence and conduction bands are symmetric near the K point, independent of the value of s. When we adopt e2p = 0 and take s = 0 for (6), and assume a linear k approximation for w(k), we get the linear dispersion relations for graphite given by [12,13]

If the physical phenomena under consideration only involve small k vectors, it is convenient to use (10) for interpreting experimental results relevant to such phenomena.

The 1D energy dispersion relations of a SWNT are given by

K

Fig. 3. Contour plot of the 2D electronic energy of graphite with s = 0 and e2p = 0 in (6). The equi-energy lines are circles near the K point and near the center of the hexagonal Brillouin zone, but are straight lines which connect nearest M points. Adjacent lines correspond to changes in height (energy) of O.I70 and the energy value for the K, M and r points are 0, 70 and 370, respectively. It is useful to note the coordinates of high symmetry points: K = (0,47t/3a), M = (2n/\/3a, 0) and r = (0, 0), where a is the lattice constant of the 2D sheet of graphite [11]

where T is the magnitude of the translational vector T, k is a 1D wave vector along the nanotube axis, and N denotes the number of hexagons of the graphite honeycomb lattice that lie within the nanotube unit cell (see Fig. 1). T and N are given, respectively, by

_ V3Ch y/3ndt AT 2(n2+m2 + nm) 1 = —-— = —--, and TV = ---. (12)

Here dR is the greatest common divisor of (2n + m) and (2m + n) for a (n,m) nanotube [2,14]. Further Ki and K2 denote, respectively, a discrete unit wave vector along the circumferential direction, and a reciprocal lattice vector along the nanotube axis direction, which for a (n, m) nanotube are given by

where bi and b2 are the reciprocal lattice vectors of 2D graphite and are given in x, y coordinates by

The periodic boundary condition for a carbon nanotube (n, m) gives N discrete k values in the circumferential direction. The N pairs of energy dispersion curves given by (11) correspond to the cross sections of the two-dimensional energy dispersion surface shown in Fig. 2, where cuts are made on the lines of fcK2/|K2| + ^Ki. In Fig. 4 several cutting lines near one of the K points are shown. The separation between two adjacent lines and the length of the cutting lines are given by the K1 and K2 vectors of (13), respectively, whose lengths are given by

If, for a particular (n, m) nanotube, the cutting line passes through a K point of the 2D Brillouin zone (Fig. 4a), where the n and n* energy bands of 2D graphite are degenerate (Fig. 2) by symmetry, then the 1D energy bands have a zero energy gap. Since the degenerate point corresponds to the Fermi energy, and the density of states are finite as shown below, SWNTs with a zero band gap are metallic. When the K point is located between two cutting lines, the K point is always located in a position one-third of the distance between two adjacent K1 lines (Fig. 4b) [14] and thus a semiconducting nanotube with a finite energy gap appears. The rule for being either a metallic or a semiconducting carbon nanotube is, respectively, that n — m = 3q or n — m = 3q, where q is an integer [2,8,15,16,17].

Fig. 4. The wave vector k for one-dimensional carbon nanotubes is shown in the two-dimensional Brillouin zone of graphite (hexagon) as bold lines for (a) metallic and (b) semiconducting carbon nanotubes. In the direction of Ki, discrete k values are obtained by periodic boundary conditions for the circumferential direction of the carbon nanotubes, while in the direction of the K2 vector, continuous k vectors are shown in the one-dimensional Brillouin zone. (a) For metallic nanotubes, the bold line intersects a K point (corner of the hexagon) at the Fermi energy of graphite. (b) For the semiconductor nanotubes, the K point always appears one-third of the distance between two bold lines. It is noted that only a few of the N bold lines are shown near the indicated K point. For each bold line, there is an energy minimum (or maximum) in the valence and conduction energy subbands, giving rise to the energy differences Epp (dt)

Optical Properties and Raman Spectroscopy of Carbon Nanotubes 219 The 1D density of states (DOS) in units of states/C-atom/eV is calculated by t ^ " r 1

dk where the summation is taken for the N conduction (+) and valence ( —) 1D bands. Since the energy dispersion near the Fermi energy (10) is linear, the density of states of metallic nanotubes is constant at the Fermi energy: D(EF) = a/(2n27odt), and is inversely proportional to the diameter of the nanotube. It is noted that we always have two cutting lines (1D energy bands) at the two equivalent symmetry points K and K' in the 2D Brillouin zone in Fig. 3. The integrated value of D(E) for the energy region of EM (k) is 2 for any (n, m) nanotube, which includes the plus and minus signs of Eg2D and the spin degeneracy.

It is clear from (16) that the density of states becomes large when the energy dispersion relation becomes flat as a function of k. One-dimensional van Hove singularities (vHs) in the DOS, which are known to be proportional to (E2 — E2)-1/2 at both the energy minima and maxima (±E0) of the dispersion relations for carbon nanotubes, are important for determining many solid state properties of carbon nanotubes, such as the spectra observed by scanning tunneling spectroscopy (STS), [18,19,20,21,22], optical absorption [4,23,24], and resonant Raman spectroscopy [25,26,27,28,29].

The one-dimensional vHs of SWNTs near the Fermi energy come from the energy dispersion along the bold lines in Fig. 4 near the K point of the Bril-louin zone of 2D graphite. Within the linear k approximation for the energy dispersion relations of graphite given by (10), the energy contour as shown in Fig. 3 around the K point is circular and thus the energy minima of the 1D energy dispersion relations are located at the closest positions to the K point. Using the small k approximation of (10), the energy differences EM (dt) and ES1(dt) for metallic and semiconducting nanotubes between the highest-lying valence band singularity and the lowest-lying conduction band singularity in the 1D electronic density of states curves are expressed by substituting for k the values of |K1| of (15) for metallic nanotubes and of |K1/3| and |2K1/3| for semiconducting nanotubes, respectively [30,31], as follows:

When we use the number p (p = 1, 2,...) to denote the order of the valence n and conduction n* energy bands symmetrically located with respect to the Fermi energy, optical transitions Eppr from the p-th valence band to the p'-th conduction band occur in accordance with the selection rules of 5p = 0 and 5p = ±1 for parallel and perpendicular polarizations of the electric field with respect to the nanotube axis, respectively [23]. However, in the case of perpendicular polarization, the optical transition is suppressed by the depolarization effect [23], and thus hereafter we only consider the optical absorption of 5p = 0. For mixed samples containing both metallic and semiconducting carbon nanotubes with similar diameters, optical transitions may appear with the following energies, starting from the lowest energy, ESn(dt), 2E1S1(dt), EH(dt), 4E1S1(dt),

In Fig. 5, both Epp(d t) and Epp(dt) are plotted as a function of nanotube diameter dt for all chiral angles at a given dt value. [3,4,11]. This plot is very useful for determining the resonant energy in the resonant Raman spectra corresponding to a particular nanotube diameter. In this figure, we use the values of 70 = 2.9e^ and s = 0, which explain the experimental observations discussed in the experimental section.

Fig. 5. Calculation of the energy separations Epp(dt) for all (n,m) values as a function of the nanotube diameter between 0.7 < dt < 3.0 nm (based on the work of Kataura et al. [3]). The results are based on the tight binding model of Eqs. (6) and (7), with 70 = 2.9 eV and s = 0. The open and solid circles denote the peaks of semiconducting and metallic nanotubes, respectively. Squares denote the Epp (dt) values for zigzag nanotubes which determine the width of each Epp (dt) curve. Note the points for zero gap metallic nanotubes along the abscissa [11]

Fig. 5. Calculation of the energy separations Epp(dt) for all (n,m) values as a function of the nanotube diameter between 0.7 < dt < 3.0 nm (based on the work of Kataura et al. [3]). The results are based on the tight binding model of Eqs. (6) and (7), with 70 = 2.9 eV and s = 0. The open and solid circles denote the peaks of semiconducting and metallic nanotubes, respectively. Squares denote the Epp (dt) values for zigzag nanotubes which determine the width of each Epp (dt) curve. Note the points for zero gap metallic nanotubes along the abscissa [11]

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