Page 37 - Carbon Nanotubes

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28 M. S. DRESSELHAUS al.

@ :metal :semiconductor armcha?’
Fig, 3. The 2D graphene sheet is shown along with the vec-
tor which specifies the chiral nanotube. The pairs of integers
(n,rn) in the figure specify chiral vectors Ch (see Table 1) for
carbon nanotubes, including Zigzag, armchair, and chiral tub-
ules. Below each pair of integers (n,rn) is listed the number
of distinct caps that can be joined continuously to the cylin-
drical carbon tubule denoted by (n,m)[6]. The circled dots
Fig. 1. The 2D graphene sheet is shown along with the vec- denote metallic tubules and the small dots are for semicon-
tor which specifies the chiral nanotube. The chiral vector OA ducting tubules.
or C, = nu, + ma, is defined on the honeycomb lattice by
unit vectors a, and u2 and the chiral angle 6 is defined with
respect to the zigzag axis. Along the zigzag axis 6 = 0”. Also
shown are the lattice vector OB = T of the 1D tubule unit the origin to the first lattice point B in the honeycomb
cell, and the rotation angle $ and the translation T which con- lattice. It is convenient to express T in terms of the in-
stitute the basic symmetry operation R = ($1 7). The diagram tegers (t, , f2) given in Table 1, where it is seen that the
is constructed for (n,rn) = (4,2).
length of T is &L/dR and dR is either equal to the
highest common divisor of (n,rn), denoted by d, or to
3d, depending on whether n - rn = 3dr, r being an in-
the axis of the tubule, and with a variety of hemispher- teger, or not (see Table 1). The number of carbon at-
ical caps. A representative chiral nanotube is shown oms per unit cell n, of the 1D tubule is 2N, as given
in Fig. 2(c). in Table 1, each hexagon (or unit cell) of the honey-
The unit cell of the carbon nanotube is shown in comb lattice containing two carbon atoms.
Fig. 1 as the rectangle bounded by the vectors Ch and Figure 3 shows the number of distinct caps that can
T, where T is the ID translation vector of the nano- be formed theoretically from pentagons and hexagons,
tube. The vector T is normal to Ch and extends from
such that each cap fits continuously on to the cylin-
ders of the tubule, specified by a given (n,m) pair.
Figure 3 shows that the hemispheres of C,, are the
smallest caps satisfying these requirements, so that the
diameter of the smallest carbon nanotube is expected
to be 7 A, in good agreement with experiment[4,5].
Figure 3 also shows that the number of possible caps
increases rapidly with increasing tubule diameter.
Corresponding to selected and representative (n, rn)
pairs, we list in Table 2 values for various parameters
enumerated in Table 1, including the tubule diameter
d,, the highest common divisors d and dR, the length
L of the chiral vector Ch in units of a (where a is the
length of the 2D lattice vector), the length of the 1D
translation vector T of the tubule in units of a, and
the number of carbon hexagons per 1D tubule unit
cell N. Also given in Table 2 are various symmetry
parameters discussed in section 3.
Fig. 2. By rolling up a graphene sheet (a single layer of car-
bon atoms from a 3D graphite crystal) as a cylinder and cap- 3. SYMMETRY OF CARBON NANOTUBFS
ping each end of the cylinder with half of a fullerene
molecule, a “fullerene-derived tubule,” one layer in thickness, In discussing the symmetry of the carbon nano-
is formed. Shown here is a schematic theoretical model for a tubes, it is assumed that the tubule length is much
single-wall carbon tubule with the tubule axis OB (see Fig. 1) larger than its diameter, so that the tubule caps can
normal to: (a) the 0 = 30” direction (an “armchair” tubule), be neglected when discussing the physical properties
(b) the 0 = 0” direction (a “zigzag” tubule), and (c) a gen- of the nanotubes.
eral direction B with 0 < 16 I < 30” (a “chiral” tubule). The
actual tubules shown in the figure correspond to (n,m) val- The symmetry groups for carbon nanotubes can be
ues of: (a) (5,5), (b) (9,0), and (c) (10,5). either symmorphic [such as armchair (n,n) and zigzag