Page 39 - Carbon Nanotubes

et
30 M. S. DRESSELHAUS ai.
Table 2. Values for characterization parameters for selected carbon nanotubes labeled by (n,rn)[7]
(n, m) d dR d, (A) L/a T/a N $/2n 7/a A4

5 15 6.78 475 1 10 1/10 1 /2 1
9 9 7.05 9 fi 18 1/18 A/2 1
1 1 7.47 m m 182 149/ 182 &ma 149
1 3 1.55 J93 m 62 11/62 1/,1124 17
1 1 7.72 m m 194 71/194 631388 11
10 10 7.83 10 43 20 1 /20 v3/2 1
6 18 8.14 m 1 12 1/12 1 /2 1
5 5 10.36 6175 m 70 1/14 63/28 5
5 15 17.95 6525 d7 70 3/70 l/(JZs) 3
15 15 31.09 m 621 210 1 /42 m 5

n 3n &a/n fin 1 2n 1/2n 1 /2 1
n n na/T n A 2n 1/2n fi/2 1

are given in Table 3 (for odd n = 2j + 1) and in Table 4 R = ($1 T) require translations T in addition to rota-
(for even n = 2j), wherej is an integer. Useful basis tions $. The irreducible representations for all Abe-
functions are listed in Table 5 for both the symmor- lian groups have a phase factor E, consistent with the
phic groups (D2j and DzJ+,) and non-symmorphic requirement that all h symmetry elements of the sym-
groups C,,, discussed by Eklund ef al. [8]. metry group commute. These symmetry elements of
Upon taking the direct product of group D, with the Abelian group are obtained by multiplication of
the inversion group which contains two elements the symmetry element R = ($ IT) by itself an appro-
(E, i), we can construct the character tables for Dnd = priate number of times, since Rh = E, where E is the
D, @ i from Table 3 to yield D,,, D,,, . . .for sym- identity element, and h is the number of elements in
morphic tubules with odd numbers of unit cells the Abelian group. We note that N, the number of
around the circumference [(5,5), (7,7), . . . armchair hexagons in the 1D unit cell of the nanotube, is not
tubules and (9,0), (ll,O), . . . zigzag tubules]. Like- always equal h, particularly when d # 1 and dR # d.
wise, the character table for Dnh = 0, @ ah can be To find the symmetry operations for the Abelian
obtained from Table 4 to yield D6h, Dsh, . . . for group for a carbon nanotube specified by the (n, rn)
even n. Table 4 shows two additional classes for group integer pair, we introduce the basic symmetry vector
D, relative to group D(zJ+l), because rotation by .rr R =pa, + qa,,. shown in Fig. 4, which has a very im-
about the main symmetry axis is in a class by itself for portant physical meaning. The projection of R on the
groups D2j. Also the n two-fold axes nC; form a class Ch axis specifies the angle of rotation $ in the basic
and represent two-fold rotations in a plane normal to symmetry operation R = (3 IT), while the projection
the main symmetry axis C,, , while the nCi dihedral of R on the T axis specifies the translation 7. In Fig. 4
axes, which are bisectors of the nC; axes, also form the rotation angle $ is shown as x = $L/2n. If we
a class for group D,, when n is an even integer. Corre- translate R by (N/d) times, we reach a lattice point
spondingly, there are two additional one-dimensional B" (see Fig. 4). This leads to the relationm =MCh +
representations B, and B2 in DZi corresponding to the dT where the integer M is interpreted as the integral
two additional classes cited above. number of 27r cycles of rotation which occur after N
The symmetry groups for the chiral tubules are rotations of $. Explicit relations for R, $, and T are
Abelian groups. The corresponding space groups are contained in Table 1. If d the largest common divisor
non-symmorphic and the basic symmetry operations of (n,rn) is an integer greater than I, than (N/d) trans-
lations of R will translate the origin 0 to a lattice point
B", and the projection (N/d)R.T = T2. The total ro-
tation angle $then becomes 2.rr(Mld) when (N/d)R
Table 3. Character table for group D(u+l, reaches a lattice point B". Listed in Table 2 are values
for several representative carbon nanotubes for the ro-
CR E 2C;j 2C:; ... 2Ci, (2j+ 1)C; tation angle $ in units of 27r, and the translation length

AI 1 1 1 ... 1 1 T in units of lattice constant a for the graphene layer,
as well as values for M.
'42 1 1 1 ... 1 -1
E, 2 2~0~6~ . . . 2co~j6~ 0 From the symmetry operations R = (4 IT) for tu-
2~0~26~
. . . 2c0s2j+~
E, 2 2c0s2+~ 2~0~44~ 0 bules (n, rn), the non-symmorphic symmetry group of
. .
the chiral tubule can be determined. Thus, from a
E, 2 2c0sjbj 2cos2j6, . . . 2cos j2c+5j 0 symmetry standpoint, a carbon tubule is a one-
dimensional crystal with a translation vector T along
where = 27/(2j + 1) and j is an integer the cylinder axis, and a small number N of carbon