Page 144 - Carbon Nanotubes

134 P. C. EKLUND et ai.
where p and q are integers that are uniquely deter- zag nanotubes, however, have other symmetry oper-
mined by the eqn ations, such as inversion and reflection in planes
parallel to the tube axis. Thus, the symmetry group,
rnp - nq = d, (8) assuming an infinitely long nanotube with no caps, is
given by
subject to the conditions q < m/d and p < n/d.
For the case d = 1, the symmetry group of a chi-
ral nanotube specified by (n, m) is a cyclic group of
order N given by
Thus e = 6)2,h in these cases. The choice of a,,, or
Dnh in eqn (16) is made to insure that inversion is a
symmetry operation of the nanotube. Even though we
where E is the identity element, and (RNjn = (2~ (WN), neglect the caps in calculating the vibrational frequen-
T/N)) . For the general case when d # 1, the cylinder cies, their existence, nevertheless, reduces the symme-
is divided into d equivalent sections. Consequently, it try to either Bnd or Bnh.
follows that the symmetry group of the nanotube is Of course, whether the symmetry groups for arm-
given by chair and zigzag tubules are taken to be d)& (or a,,,)
or a)2nh, the calculated vibrational frequencies will be
the same; the symmetry assignments for these modes,
however, will be different. It is, thus, expected that
where modes that are Raman or IR-active under and or TInh
but are optically silent under BZnh will only show a
weak activity resulting from the fact that the existence
of caps lowers the symmetry that would exist for a
and nanotube of infinite length.
3.2 Model calculations of phonon modes
Here the operation ed represents a rotation by 2n/d The BZ of a nanotube is a line segment along the
tube direction, of length 2a/T. The rectangle formed
about the tube axis; the angles of rotation in (!?hd/fi
are defined modulo 2?r/d, and the symmetry element by vectors C and T, in Fig. 3, has an area N times
(RNd/n = (2~(fi/Nd),Td/N)). larger than the area of the unit cell of a graphene sheet
The irreducible representations of the symmetry formed by vectors al and a2, and gives rise to a rect-
group C? are given by A, B, El, E2, . . . , EN/Z-, . The A angular BZ than is Ntimes smaller than the hexago-
representation is completely symmetric, while in the nal BZ of a graphene sheet. Approximate values for
B representation, the characters for the operations cd the vibrational frequencies of the nanotubes can be
and 6iNd/Q are obtained from those of a graphene sheet by the
method of zone folding, which in this case implies that

and

In the E, irreducible representation, the character of
any symmetry operation corresponding to a rotation In the above eqn, 1D refers to the nanotubes whereas
by an angle is given by 2D refers to the graphene sheet, k is the 1D wave vec-
tor, and ?and e are unit vectors along the tubule axis
and vector C, respectively, and p labels the tubule pho-
non branch.
The phonon frequencies of a 2D graphene sheet,
for carbon displacements both parallel and perpendic-
Equations (13-15) completely determine the character ular to the sheet, are obtained[l] using a Born-Von
table of the symmetry group e for a chiral nanotube. Karman model similar to that applied successfully to
Applying the above symmetry formulation to arm- 3D graphite. C-C interactions up to the fourth nearest
chair (n = m) and zigzag (m = 0) nanotubes, we find in-plane neighbors were included. For a 2D graphene
that such nanotubes have a symmetry group given by sheet, starting from the previously published force-
the product of the cyclic group e, and where constant model of 3D graphite, we set all the force
e;, consists of only two symmetry operations: the constants connecting atoms in adjacent layers to zero,
identity, and a rotation by 21r/2n about the tube axis and we modified the in-plane force constants slightly
followed by a translation by T/2. Armchair and zig- to describe accurately the results of electron energy loss