and, correspondingly, the symmetry elements in group CNd/n include

and group CNd/a is of order N/d'. The irreducible representations of the groups Cd, and CNd/n in Eq. (19.15) are given in Table 19.6 and are appropriate roots of unity. For the two-dimensional E„ irreducible representations, the characters of the symmetry operations Cd, and CNd/n are given by

For the chiral nanotubes, a vector transforms according to the basis functions for the A and Ex irreducible representations, whereas quadratic terms in the coordinates form basis functions for the A, Ex, and E2 irreducible representations, as shown in the Table 19.5.

Referring to Fig. 19.2(a) for the (4,2) tubule, we have = 2tt(5/28) and il/d = 5, so that N rotations produce a total rotation of 27t(5). Since d = 2, N translations produce a distance 2T, so that NR is the diagonal of a rectangle that contains il = 10 times the area of the ID nanotube unit cell. The first lattice point is, however, reached at (N/d)R = [(5/2)Ch, T] of the ID tubule lattice. The number of elements in the Abelian group CNd/n is N/d, which is 14 in this case. The value of d' which is the highest common divisor of d and il/d is d' = 1, so that Cd, is the identity group containing only one symmetry element. This example shows that for (n,m) pairs having a common divisor d, a lattice point is reached after (N/d)R translations, so that il/d2 is not necessarily a multiple of 2ir.

We now give some examples to show that if either d> \ or n-m — 3r (so that dR = 3d) the ID tubule unit cell is reduced in size, and the number of R vectors to reach a lattice point is reduced. For example, tubule (7,5), for which caps are shown in Fig. 19.9, has a diameter of 8.18 A, N = 218 symmetry operations, and 91 translations R are needed to reach a lattice point. Since the integers (7,5) have no common divisor, and since n — m ^ 3 r, the unit cell is large. If we now consider a tubule with (n,m) = (8,5) with a diameter of 8.89 A, nearly 10% larger than the diameter for the (7,5) tubule, we note that since n — m is a multiple of 3, then N = 86 for the (8,5) tubule, where N is much smaller than for the (7,5) tubule. Likewise, for the (8,5) tubule, the length T = 16.1 A is much shorter than T — 44.5 A for the (7,5) tubule, and a lattice point is reached after 53 translations. An even smaller unit cell is obtained for the tubule corresponding to (n, m) = (10,5), despite the larger tubule diameter of 10.36 A. For the (10,5) tubule, N is only 70, and a lattice point is reached after only five translations R. Although the number of symmetry operations of the nonsymmorphic groups tends to increase with tubule diameter, those tubules for which either n — m — 3r, where r is an integer, or (n, m) contains a common divisor d, the size of the unit cell and therefore the number of symmetry operations is reduced by factors of 3 and d, respectively.

The symmorphic groups corresponding to the armchair and zigzag tubules have relatively small unit cells with T = a0 and T = y/3a0, respectively, where aQ — 2.46 A= \/3ac_c. The basic rotation angle if/ for both the (n, n) armchair tubule and the (n, 0) zigzag tubule is (// — 2ir/n.

To express the dispersion relations for electrons and phonons in carbon nanotubes, it is necessary to specify the basis vectors in reciprocal space K, which relate to those in real space Rj by

The lattice vectors and the unit cells in real space are discussed in §19.4.1. In Cartesian coordinates we can write the two real space lattice vectors as

where dR is defined by Eq. (19.8) and the length a0 is the lattice constant of the 2D graphene unit cell, a0 = ac^c\/3. From Eq. (19.21), it is easy to see that the lengths of Ch and T are in agreement with Eqs. (19.2) and (19.7). Then using Eqs. (19.20) and (19.21), we can write the corresponding reciprocal lattice vectors K, and K2 in Cartesian coordinates as

For the case of the armchair and zigzag tubules, it is convenient to choose a real space rectangular unit cell in accordance with Fig. 19.25. The area of each real space unit cell for both the armchair and zigzag tubules contains two hexagons or four carbon atoms. It should be noted that the real space unit cell defined by the vectors T and Ch is n times larger than the real space unit cells shown in Fig. 19.25. Also shown in this figure are the real space unit cells for a 2D graphene layer. It should be noted that the real space unit cell for the zigzag tubules follows from the definition given for the unit cells for chiral tubules [Fig. 19.2(a)], but the unit cell for the armchair tubule is specially selected for convenience.

Having specified the real space unit cells in Fig. 19.25, the corresponding unit cells in reciprocal space (or Brillouin zones) are determined by Eqs. (19.20) and (19.22) and are shown in Fig. 19.25 in comparison to the reciprocal space unit cell for a 2D graphene sheet. We note that for both the armchair and zigzag tubules, the ID reciprocal space unit cells shown in Fig. 19.25 are half as large as those for the 2D graphene sheet. On the graphene sheet, the real space lattice vectors Ch and T form a rectangular unit cell for the zigzag or armchair tubules which has an area N times larger than the area of the corresponding primitive cell in the graphene sheet, where N is given by Eq. (19.9) [19.84], and this large unit cell spans the circumference of the cylinder of the tubule.

The one-dimensional Brillouin zone of the chiral tubule is a segment along the vector K2 of Eq. (19.22). The extended Brillouin zone for the tubule is a collection of N wave vector segments of length |K2|, each separated from the next segment by the vector K,. Thus, by zone folding the N

wave vector segments of the 2D dispersion relations of the graphene layer back to the first Brillouin zone of the ID tubule, the ID dispersion relations for the N electron energy bands (or phonon branches) of the tubule are obtained. For the special cases of the armchair and zigzag tubules, the real space unit cells in Fig. 19.25 correspond to four carbon atoms and two hexagons, while the reciprocal space unit cells also contain two wave vectors. Therefore each dispersion relation for the (n, n) armchair tubules and the («,()) zigzag tubules is zone folded n = N/2 times in the extended Brillouin zone to match the ID Brillouin zone of the tubule. We present the results of this zone folding in discussing the electronic structure in §19.5 and the phonon dispersion relations in §19.7.

19.5. Electronic Structure: Theoretical Predictions

Fullerene-based carbon tubules are interesting as examples of a one-dimensional periodic structure along the tubule axis. In this section we summarize the remarkable electronic properties predicted for single-wall carbon nanotubes. Specifically, it is predicted that small-diameter nano-tubes will exhibit either metallic or semiconducting electrical conduction, depending on their diameter dt and chiral angle 6, and independent of the presence of dopants or defects. It is further shown that this phenomenon is not quenched by the Peierls distortion or by intertubule interactions. Because of the great difficulty in making measurements on individual single-wall tubules, few detailed experiments have thus far been reported, although this is a very active current research area. In this section, we review theoretical predictions for the electronic structure in zero magnetic field of single-wall symmorphic and nonsymmorphic carbon nanotubes. The effect of interlayer interaction for multiwall nanotubes is then considered along with intertubule interactions. Finally, theoretical predictions for the electronic structure of carbon nanotubes in a magnetic field are discussed. The status of experimental work on the electronic structure of carbon nanotubes is reviewed in §19.6, and the effect of magnetic fields is considered in §19.5.4.

Most of the calculations of the electronic structure of carbon nanotubes have been carried out for single-wall tubules. Confinement in the radial direction is provided by the monolayer thickness of the tubule. In the circumferential direction, periodic boundary conditions are applied to the enlarged unit cell that is formed in real space defined by T and Ch, and as a consequence, zone folding of the dispersion relations occurs in reciprocal space. Such zone-folding calculations lead to ID dispersion relations for electrons (described in this section) and phonons (see §19.7) in carbon tubules.

A number of methods have been used to calculate the ID electronic energy bands for fullerene-based single-wall tubules [19.25,47,86,88-93]. However, all of these methods relate to the 2D graphene honeycomb sheet used to form the tubule. The unit cells in real and reciprocal space used by most of these authors to calculate the energy bands for armchair and zigzag tubules are shown explicitly in Fig. 19.25. Furthermore, the armchair tubule can be described by a symmorphic space group (as is done in this section), but can also be described following the discussion for chiral tubules in §19.4.3 for a chiral angle of 0 = 30°.

We illustrate below the essence of the various calculations of the ID electronic band structure, using the simplest possible approach, which is a tight binding or Hiickel calculation that neglects curvature of the tubules. In making numerical evaluations of the energy levels and band gaps, it is assumed that the nearest-neighbor interaction energy yQ is the same as for crystalline graphite. Since the unit cells for the (n, m) armchair tubule and the (n, 0) zigzag tubules in real space are a factor of n smaller than the area enclosed by Ch and T in Fig. 19.2(a), the size of the Brillouin zone is a factor of n larger than the reciprocal space unit cell defined by the reciprocal lattice vectors given by Eq. (19.21). Thus, zone folding of the large unit cell in reciprocal space introduces discrete values of the wave vector in the direction perpendicular to the tubule axis [19.46]. Using this procedure, the 2D energy dispersion relations for a single graphene layer are folded into the ID Brillouin zone of the tubule. To illustrate this zone-folding technique, we start with the simplest form of the 2D dispersion relation for a single graphene sheet as expressed by the tight binding approximation [19.94]

where a0 = 1.42 x \/3 A is the lattice constant for a 2D graphene sheet and y0 is the nearest-neighbor C-C overlap integral [19.95], A set of ID energy dispersion relations is obtained from Eq. (19.23) by considering the small number of allowed wave vectors in the circumferential direction. The simplest cases to consider are the nanotubes having the highest symmetry. Referring to Fig. 19.25, we see the unit cells and Brillouin zones for the highly symmetric nanotubes, namely for (a) an armchair tubule and (b) a zigzag tubule. The appropriate periodic boundary conditions used to obtain energy eigenvalues for the (NX,NX) armchair tubule define the small number of allowed wave vectors kx in the circumferential direction

Substitution of the discrete allowed values for kxq given by Eq. (19.24) into Eq. (19.23) yields the energy dispersion relations Eq(k) for the armchair tubule [19.46]

in which the superscript a refers to armchair, k is a one-dimensional vector along the tubule axis, and Nx refers to the armchair index, i.e., (n, m) = (NX,NX). The resulting calculated ID dispersion relations Eq(k) for the (5,5) armchair nanotube (Nx = 5) are shown in Fig. 19.26(a), where we see six dispersion relations for the conduction bands and an equal number for the valence bands. In each case, two bands are nondegenerate (thin lines) and four are doubly degenerate (heavy lines), leading to 10 levels in each

Fig. 19.26. One-dimensional energy dispersion relations for (a) armchair (5,5) tubules, (b) zigzag (9,0) tubules, and (c) zigzag (10,0) tubules labeled by the irreducible representations of the point group £>(2,H1)l( at k = 0. The «-bands are nondegenerate and the e-bands are doubly degenerate at a general A:-point in the ID Brillouin zone [19.96],

Fig. 19.26. One-dimensional energy dispersion relations for (a) armchair (5,5) tubules, (b) zigzag (9,0) tubules, and (c) zigzag (10,0) tubules labeled by the irreducible representations of the point group £>(2,H1)l( at k = 0. The «-bands are nondegenerate and the e-bands are doubly degenerate at a general A:-point in the ID Brillouin zone [19.96], case, consistent with the 10 hexagons around the circumference of the (5,5) tubule. For all armchair tubules, the energy bands show a large degeneracy at the zone boundary, where ka0 = it, so that Eq. (19.23) becomes

for the 2D graphene sheet, independent of zone folding and independent of Nx. Although there are four carbon atoms in the unit cell for the real space lattice in Fig. 19.25(a), the two carbon atoms on the same sublattice of a graphene sheet are symmetrically equivalent, which causes a degeneracy of the energy bands at the boundary of the Brillouin zone. The valence and conduction bands in Fig. 19.26(a) cross at a A: point that is two thirds of the distance from k = 0 to the zone boundary at k = tr/a0. The crossing takes place at the Fermi level and the energy bands are symmetric for ±k values.

Because of the degeneracy point between the valence and conduction bands at the band crossing, the (5,5) tubule will exhibit metallic conduction at finite temperatures, because only infinitesimal excitations are needed to excite carriers into the conduction band. The (5,5) armchair tubule is thus a zero-gap semiconductor, just like a 2D graphene sheet.

Similar calculations, as given by Eqs. (19.23), (19.24), and (19.25), show that all (n, n) armchair tubules yield dispersion relations similar to Eq. (19.25) with 2n conduction and 2n valence bands, and of these 2n bands, two are nondegenerate and (n - 1) are doubly degenerate. All (n, n) armchair tubules have a band degeneracy between the highest valence band and the lowest conduction band at k — ±2tt/(3a0), where the bands cross the Fermi level. Thus, all armchair tubules are expected to exhibit metallic conduction, similar to the behavior of 2D graphene sheets [19.6,46,47,89,92,97,98],

The energy bands for the (iVy,0) zigzag tubule Ezq(k) can be obtained likewise from Eq. (19.23) by writing the periodic boundary condition on ky as:

to yield the ID dispersion relations for the 2Ny states for the (Ny, 0) zigzag tubule (denoted by the superscript z)

Referring to Fig. 19.26(b) we show for illustrative purposes the normalized energy dispersion relations for the zigzag (9,0) tubule, where Ny = 9, corresponding to a tubule with a diameter equal to that of the C60 molecule. In the case of the (9,0) zigzag tubule, the number of valence and conduction bands is 10, with 2 nondegenerate levels and 8 double degenerate levels yielding a total of 18 or 2Ny states, as expected from the number of hexagons for a circumferential ring in the 2D honeycomb lattice, which follows from Fig. 19.25(b). Most important is the band degeneracy that occurs at k — 0 between the doubly degenerate valence and conduction bands (symmetries e3u and ejg), giving rise to a fourfold degeneracy point at k = 0.

If we view the one-dimensional energy bands in the extended zone scheme, the dispersion for both the (5,5) and (9,0) tubules shown in Fig. 19.26(a) and (b) is just like "sliced" 2D energy dispersion relations, with slices taken along the directions kxq — (q/Nx)(2ir/y/3a) and kyq = (q/Ny)(2<jr/a) for the armchair and the zigzag tubules, respectively. For both types of tubules in Fig. 19.26(a) and (b), we have two ID energy bands which cross at the Fermi energy (E = 0), giving rise to metallic conduction. In both cases, the band crossing at E = 0 occurs because the corresponding 2D energy bands cross at the K point (corners of a hexagon) of the 2D Brillouin zone (see Fig. 19.25), where the 2D graphene energy bands for the conduction and valence bands are degenerate. Although the density of states at the K point is always zero in the 2D case, the density of states in the ID case can be finite, as shown below. Therefore, if there is no Peierls instability for this one-dimensional system (or if the Peierls gap is small compared to kT), carbon nanotubes can be metallic [19.46,89].

If we carry out calculations similar to those described by Eqs. (19.27) and (19.28) for the (10,0) zigzag tubule as shown in Fig. 19.26(c), we obtain ID dispersion relations with some features similar to Fig. 19.26(b), but also some major differences are observed. In the (10,0) case, the valence bands and the conduction bands each contain two nondegenerate levels and nine doubly degenerate levels to give a total of 20 states for each. Also, one degenerate level in the conduction band a~u) and another in the valence band show little dispersion. However, the noteworthy difference between the (9,0) and (10,0) zigzag tubules is the appearance of an energy gap between the valence and conduction bands at k = 0 for the case of the (10,0) tubule, in contrast to the degeneracy point that is observed at k = 0 for the (9,0) tubule. Thus, the (10,0) zigzag tubule is expected to exhibit semiconducting transport behavior, while the (9,0) tubule is predicted to show metallic behavior.

The physical reason for this difference in behavior is that for the (10,0) tubule, there are no allowed wave vectors ky q from Eq. (19.27) which go through the K point in the 2D Brillouin zone of the graphene sheet. Referring to Fig. 19.25 it is seen that for all (n, n) armchair tubules, there is an allowed k vector going through the 2D zone corner K point, since the K point is always on the ID zone boundary. However, for the (n, 0) zigzag tubule, the allowed ky q wave vectors only pass through the K point when n is divisible by 3, which is obeyed for the (9,0) tubule, but not for the (10,0) tubule. Thus, only one third of the (n, 0) zigzag tubules would be expected to show metallic conduction. The other two thirds of the zigzag tubules are expected to be semiconducting. This is a remarkable symmetry-imposed result.

It is surprising that the calculated electronic structure can be either metallic or semiconducting depending on the choice of (n, 0), although there is no difference in the local chemical bonding between the carbon atoms in the tubules, and no doping impurities are present [19.6]. These surprising results are of quantum origin and can be understood on the basis of the electronic structure of a 2D graphene sheet which is a zero-gap semiconductor [19.99] with bonding and antibonding tt bands that are degenerate at the K point (zone corner) of the hexagonal 2D Brillouin zone of the graphene sheet. The periodic boundary conditions for the ID tubules permit only a few wave vectors to exist in the circumferential direction. If one of these passes through the AT-point in the Brillouin zone (see Fig. 19.25), then metallic conduction results; otherwise the tubule is semiconducting and has a band gap.

The density of states per unit cell using a model based on zone folding a 2D graphene sheet is shown in Fig. 19.27 for metallic (9,0) and semiconducting (10,0) tubules [19.6], Here we see a finite density of states at

Fig. 19.27. Electronic ID density of states per unit cell for two (n, m) zigzag tubules based on zone folding of a 2D graphene sheet: (a) the (9,0) tubule which has metallic behavior,

(b) the (10,0) tubule which has semiconducting behavior. Also shown in the figure is the density of states for the 2D graphene sheet (dashed curves) [19.98].

(c) Plot of the energy gap for (n,0) zigzag nanotubes plotted in units of y0 as a function of n, where yn is the energy of the nearest-neighbor overlap integral for graphite [19.100].

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