Fig. 19.20. Diagram showing a sequence of C2 dimer additions which result in the addition of one row of hexagons to a carbon tubule, (a) A projection mapping of the cap atoms and the atoms on the cylinder of a (6,5) carbon tubule, where each pentagon defect is denoted by a light gray shaded hexagon containing a number (see §3.3). (b) The same projection mapping as in (a), except that the cap atoms do not appear explicitly, (c) The projection mapping of (b) after a C2 dimer has been absorbed, resulting in the movement of the pentagonal defect 3. (d) The same as (c) except that 4 is the active pentagon which absorbs the C2 dimer. (e) The same as (d) except that 5 is the active pentagon, (f) The same as (e) except that 1 is the active pentagon, (g) The final state implied by (f) which is geometrically identical to (b) except that one row of hexagons (10 carbon atoms) has been added to the tubule.

Fig. 19.21. Proposed growth mechanism of carbon tubules at an open end by the absorption of C2 (dimers) and C3 (trimers). (a) Absorption of Q dimers at the most active edge site of a chiral carbon tubule resulting in the addition of one hexagon. Also shown is an out-of-sequence absorption of a C3 trimer, (b) Absorption of C2 dimers at the open end of an armchair carbon tubule, (c) Absorption of a C3 trimer at the open end of a zigzag carbon tubule and subsequent C2 dimer absorption.

Fig. 19.21. Proposed growth mechanism of carbon tubules at an open end by the absorption of C2 (dimers) and C3 (trimers). (a) Absorption of Q dimers at the most active edge site of a chiral carbon tubule resulting in the addition of one hexagon. Also shown is an out-of-sequence absorption of a C3 trimer, (b) Absorption of C2 dimers at the open end of an armchair carbon tubule, (c) Absorption of a C3 trimer at the open end of a zigzag carbon tubule and subsequent C2 dimer absorption.

method for relieving strain in the cap region. The introduction of pentagons leads to positive curvature, while capping with heptagons lead to changes in tubule size (see Fig. 19.12) and orientation. One explanation for the semi-toroidal tubule termination (Fig. 19.13) is provided by the introduction of six heptagons at the periphery of an open tube. The subsequent addition of hexagons keeps the tubule diameter constant so that the toroidal surface can form. Thus the introduction of heptagon-pentagon pairs can produce a variety of tubule shapes. However, regarding this explanation of the semi-toroidal termination, it is hard to understand why suddenly six heptagons are introduced along the circumference at once, soon to be replaced by six pentagons which likely are strongly correlated with the prior introductions of the heptagons and the strain field associated with each heptagon.

At present the growth model for carbon nanotubes remains incomplete with regard to the role of temperature and helium gas. Since the vapor phase growth occurs at only 1100°C, any dangling bonds that might participate in the open tubule growth mechanism would be unstable, so that the

Fig. 19.22. Schematic diagram for open tube growth of nanotubes from a carbon supply. The figure shows the addition of carbon atoms to the open ends, the capping of the longest open end, and the initiation of new tubules, (a) Some shells are terminated at the positions indicated by arrowheads as illustrated in the circle, (b) Terminated shells carry left-handed or right-handed kink sites, owing to the helical tube structure, (c) Similar termination of the shells near the top of the tip, forming steps, one atom in height, as indicated by the arrowheads [19.37].

Fig. 19.22. Schematic diagram for open tube growth of nanotubes from a carbon supply. The figure shows the addition of carbon atoms to the open ends, the capping of the longest open end, and the initiation of new tubules, (a) Some shells are terminated at the positions indicated by arrowheads as illustrated in the circle, (b) Terminated shells carry left-handed or right-handed kink sites, owing to the helical tube structure, (c) Similar termination of the shells near the top of the tip, forming steps, one atom in height, as indicated by the arrowheads [19.37].

closed tube approach would be favored. In this lower temperature regime, the growth of the tubule core and the thickening process occur separately (see §2.5). In contrast, for the arc discharge synthesis method, the temperature where tubule growth occurs has been estimated to be about 3400°C [19.77], so that, in this case, the carbon is close to the melting point. At these high temperatures, tubule growth and the graphitization of the thickening deposits occur simultaneously, so that all the coaxial tubules grow at once at these elevated temperatures [19.35], and the open tubule growth may be favored.

An unexplained issue for the growth mechanism is the role of the helium gas. In most experiments helium gas and some other gases as well are used at about 100 torr for cooling the carbon system, since the tubule structure is not stable but is only a quasi-stable structure. Although the growth appears to be sensitive to the gas pressure, it is not clear how the He gas cooling of the carbon system causes growth of the quasi-stable tubule or

fullerene phases. It is expected that future studies will provide a more detailed explanation of the growth mechanism, especially regarding the role of temperature, buffer gases, and electric fields.

One interesting growth feature, reported by several groups [19.37,38, 78], is the containment of a small-diameter carbon tubule inside a larger-diameter tubule, as in Fig. 19.23, where it is seen that the inner tubule has no access to a carbon source. Such a feature seems to require growth by an open tube mechanism, although the nucleation of the cap from the terminated cap may be necessary for the growth of this feature.

Smalley [19.67] has suggested that the open ends of the tubules are stabilized by the electric fields that can be generated near the graphite surface in the arc discharge. Because of the high temperature of the particles in the arc discharge (up to 3400°C [19.77]), many of the species in the gas phase are expected to be charged, thereby screening the electrodes. Thus the potential energy drop associated with the electrodes is expected to occur over a distance of ~ 1/xm or less, thereby causing very high electric fields. It is these high electric fields which may stabilize the open-ended tubes [19.67,79], which ordinarily would represent a very high energy state because of the large number of dangling bonds.

It has been suggested that the catalyzed growth mechanism for the singlewall tubule is different from that for the multiwall tubule, due to the role of the catalyst. It is believed that nanosize iron particles act as catalysts for the formation of single-shell tubules, which are shown in Fig. 19.6, as they bridge carbide particles. In this growth process, the cementite particles eventually become coated with graphitic material, but the filaments themselves do not seem to contain any transition metal species after the growth is completed [19.79].

It is known from laser pyrolysis studies that transition metal carbide nanoscale particles such as cementite (FeC3) catalyze the growth of graphene layers on their periphery [19.80], by a growth process similar to the formation of vapor-grown carbon fibers (see §2.5), whose growth is also catalyzed by transition metal nanoparticles [19.32]. It has also been shown [19.79,81] using an arc discharge between two graphite electrodes that nanocrystalline LaC2 particles surrounded by graphitic layers are formed when one of the two electrodes is packed with La-oxide (La203) powders in the center of the rod. The carbons formed on the surface of LaC2 particles were found to be well graphitized, serving as a protection to the encapsulated pyrophoric LaC2 particles. As another example, carbon encapsulated Co-carbide nanoparticles have also been formed using the arc discharge method [19.31],

Although the symmetry of the 2D graphene layer is greatly lowered in the ID nanotube, the single-wall nanotubes have interesting symmetry properties that lead to nontrivial physical effects, namely a necessary degeneracy at the Fermi level for certain geometries.

19.4.1. Specification of Lattice Vectors in Real Space

To study the properties of carbon nanotubes as ID systems, it is necessary to define the lattice vector T along the tubule axis and normal to the chiral vector Ch defined by Eq. (19.1) and Fig. 19.2(a). The vector T thus defines the unit cell of the ID carbon nanotube. The length T of the translation vector T corresponds to the first lattice point of the 2D graphene sheet through which the vector T passes. From Fig. 19.2(a) and these definitions, we see that the translation vector T of a general chiral tubule as a function of n and m, can be written as [19.82]:

with a length

where the length Ch is given by Eq. (19.2), d is the highest common divisor of («, m), and

11 if n-mis not a multiple of 3d g.

Thus for the (5,5) armchair tubule dR = 3d = 15, while for the (9,0) zigzag tubule dR = d = 9. The relation between the translation vector T and the symmetry operations on carbon tubules is discussed below and in §19.4.2. As a simple example, T — ~j3ac_c for a (5,5) armchair nanotube and

T = 3ac_c for a (9,0) zigzag nanotube, where ac_c is the nearest-neighbor carbon-carbon distance. We note that the length T is greatly reduced when (n, m) have a common divisor and when (n — m) is a multiple of 3.

Having specified the length T of the smallest translation vector for the ID carbon nanotube, it is useful to determine the number of hexagons, N, per unit cell of a chiral tubule specified by integers (n, m). From the size of the unit cell of the ID carbon nanotube defined by the orthogonal vectors T and Ch, the number N is given by

"R

where dR is given by Eq. (19.8) and we note that each hexagon contains two carbon atoms. As an example, application of Eq. (19.9) to the (5,5) and (9,0) tubules yields values of 10 and 18, respectively, for N. We will see below that these unit cells of the ID tubule contain, respectively, five and nine unit cells of the 2D graphene lattice, each 2D unit cell containing two hexagons of the honeycomb lattice. This multiplicity is used in the application of zone-folding techniques to obtain the electronic and phonon dispersion relations in §19.5 and §19.7, respectively.

Referring to Fig. 19.2(a), we see that the basic space group symmetry operation of a chiral tubule consists of a rotation by an angle tp combined with a translation r, and this space group symmetry operation is denoted by R = (i//1 t ) and corresponds to the vector R = pax+qa2 shown in Fig. 19.24. The physical significance of the vector R is that the projection of R on the chiral vector Ch gives the angle ip scaled by Ch/2ir, while the projection of R on T gives the translation vector r of the basic symmetry operation of the ID space group. The integer pair (p, q) which determines R is found using the relation mp — nq = d (19.10)

subject to the conditions q < m/d and p < n/d. Taking the indicated scalar product R • T in Fig. 19.24 we obtain the expressions for the length of T

where d is the highest common divisor of (n, m), T is the magnitude of the lattice vector T, and N is the number of hexagons per ID unit cell, given by Eq. (19.9). For the armchair and zigzag tubules, Eq. (19.11) yields t = T/2 and r = V3772, respectively.

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