Page 40 - Carbon Nanotubes

Physics of carbon nanotubes 31

AI 1 1 1 1 ... 1 1 1
A2 1 1 1 1 ... 1 -1 -1
Bl 1 -1 1 1 ... 1 I -1
B2 1 -1 1 1 ... 1 -1 1
El 2 -2 2 cos Gj 2 cos 24j . . . 2cos(j - l)+j 0 0
E2 2 2 2 cos 2Gj 2 cos 4+j . . . 2COS2(j - l)+j 0 0

Ej-! 2 (-1)j-’2 2cos(j- l)bj 2cos2(j- 1)4 ... 2cos(j- l)2+j 0 0

where $, = 27r/(2j) and j is an integer.

hexagons associated with the 1D unit cell. The phase We illustrate some typical results below for elec-
factor E for the nanotube Abelian group becomes E = trons and phonons. Closely related results are given
exp(27riM/N for the case where (n,m) have no com- elsewhere in this volume[8,1 I].
mon divisors (i-e., d = 1). If M = 1, as for the case The phonon dispersion relations for (n,O) zigzag tu-
of zigzag tubules as in Fig. 2(b) NR reach a lattice bules have 4 x 3n = 12n degrees of freedom with 60
point after a 2n rotation. phonon branches, having the symmetry types (for n
As seen in Table 2, many of the chiral tubules with odd, and Dnd symmetry):
d = 1 have large values for M; for example, for the
(6J) tubule, M = 149, while for the (7,4) tubule,
M = 17. Thus, many 2~ rotations around the tubule
axis are needed in some cases to reach a lattice point
of the 1D lattice. A more detailed discussion of the
symmetry properties of the non-symmorphic chiral
groups is given elsewhere in this volume[8].
Because the 1D unit cells for the symmorphic Of these many modes there are only 7 nonvanishing
groups are relatively small in area, the number of pho- modes which are infrared-active (2A2, + 5E1,) and
non branches or the number of electronic energy 15 modes that are Raman-active. Thus, by increasing
bands associated with the 1D dispersion relations is the diameter of the zigzag tubules, modes with differ-
relatively small. Of course, for the chiral tubules the ent symmetries are added, though the number and
1D unit cells are very large, so that the number of pho- symmetry of the optically active modes remain the
non branches and electronic energy bands is aIso large.
Using the transformation properties of the atoms
within the unit cell (xatom ’IfeS ) and the transformation
properties of the 1D unit cells that form an Abelian
group, the symmetries for the dispersion relations for
phonon are obtained[9,10]. In the case of n energy
bands, the number and symmetries of the distinct en-
ergy bands can be obtained by the decomposition of
the equivalence transformation (xatom sites ) for the at-
oms for the ID unit cell using the irreducible repre-
sentations of the symmetry group.

- x ch
Table 5. Basis functions for groups D(2,) and Do,+,,

Fig. 4. The relation between the fundamental symmetry vec-
tor R =pa, + qaz and the two vectors of the tubule unit cell
for a carbon nanotube specified by (n,m) which, in turn, de-
termine the chiral vector C, and the translation vector T.
The projection of R on the C,, and T axes, respectively, yield
(or x) and T (see text). After (N/d) translations, R reaches
a lattice point B”. The dashed vertical lines denote normals
to the vector C, at distances of L/d, X/d, 3L/d,. . . , L from
the origin.