Page 91 - Carbon Nanotubes

80 S. IHARA and S. ITOH
3.1.4 Recent results of electronic calcula- 3.2.2 Elongated tori. The experiments, at the
tions. Total energy calculations or molecular orbital present time, suggest that the torus of type (D) with
calculations are necessary to explore electronic, opti- parallel fringes at a separation of 3.7 A, such as Cm,
cal, and chemical properties of toroidal forms. From is likely to exist. Thus, the type (C) structures having
the ab initio self-consistent field (SCF) calculation height of 3.7 A could exist. See Fig. 6.
[22] for the torus C120, the HOMO-LUMO (highest- If we consider the l/k part of the chain of the cir-
occupied-lowest-unoccupied molecular orbitals) gap, cle, the number of hexagons can be put nl and n2 for
which is responsible for the chemical stability, is 7.5 the outer and inner circle of the upper (or lower) hex-
eV. This is close to that of SCF calculation for C60 of agonal chain (see Fig. 6), respectively. Each upper and
7.4 eV. (If the value of the HOMO-LUMO gap is zero, lower hexagonal chain contains n: + n: + 2(nl+ n2)
the molecule is chemically active, thus unstable.) In atoms. The number of the hexagons along the height
SCF, the HOMO-LUMO is different from local den- is put L, where L is a positive integer. For torus C240,
sity approximation. For stability, ours is consistent nl = n2 = 3, and L = 1 and k = 5. If we elongate (by
with the result of the all-electron local density approx- putting hexagons for allowed locations) the thickness
imation calculation where the value is 1.0 eV for the of the tube, then ro - r,, nl - n2 increases. On the
HOMO-LUMO gap[23]. Recent tight-binding calcu- other hand, if we elongate the height of the torus, L
lation of the same author[24], indicates that the increases.
HOMO-LUMO gap for C360 is 0.3 eV. These values By inserting a cylindrical tube of hexagons, we
indicate that toroidal structures are chemically stable. stretch the length of the toroidal forms whose heights
The tight-binding calculation of the HOMO-LUMO are larger than the radii, by putting n, = n2 = 3, k =
gap for tori CSw and C576 gives 0.04 eV and 0.02 eV, 5 and increasing L. The stretched toroidal forms we
respectively[ 101. thus obtained[l7], type (D), areCm, C3M), c4809 CW,
Our Huckel-type calculation for isomers of C,[ 161 C7m, CSm. . . (See Fig. 7). These forms are links be-
indicates that the positions and directions of the poly- tween toroidal forms and short (nanometer-scale)
gons change the electronic structures substantially for length turn-over tubes. The values of the cohesive en-
C240 or CzsO. Because of the geometrical complexity ergies for tori CXO, c3607 C4g0, c600, C720, and c840
of the torus, any simple systematics, as have been are -7.338, -7.339, -7.409, -7.415, -7.419, and
found for the band gaps of the carbon nanotubes[6], -7.420 eV/atom, respectively. Note that their cohe-
could not be derived from our calculations. But, the sive energies decrease with increasing height of the tori
common characteristics of the isomers for C240 with (or L) (i.e., number of hexagons). Simulations showed
large HOMO-LUMO gaps are that their inner and that these stretched toroidal forms are thermodynam-
outer tubes have the same helicities or that the penta- ically stable.
gons and heptagons are radially aligned. Note that the Using the torus c2gg of D6h which is derived from
HOMO-LUMO gap of the torus C240, which is shown the torus c240 of , shallow tori, type (D), are gen-
in Fig. 3 (b), is 0.497 eV. erated by putting L = I, k = 6, and n2 = 3, with vary-
ing nl (= 3,4, 5,6,7,8,9). Tori having D6h symmetry
3.2. Results of the experiments are shown in Fig. 8.
and elongated tori In Table 1, cohesive energies for the tori (of L =
3.2.1 Results of the experiments. Several ex- 1, k = 6) for various n, and n2 are given. The cohe-
perimental groups try to offer support for the exis- sive energy is the lowest for nl - n2 = 0, and also has
tence of the toroidal form of graphitic carbon[25].
Transmission electron microscopy (TEM) images
taken by Iijima, Ajayan, and Ichihashi[26] provided
experimental evidence for the existence of pairs of
pentagons (outer rim) and heptagons (inner rim),
which are essential in creating the toroidal struc-
ture[lO-171, in the turn-over edge (or turn-around
edge[26]) of carbon nanometer-sized tubes. They sug-
gested that the pentagon-heptagon pairs appearing in
the turn-over edge of carbon nanotubes have some
symmetry along the tube axis. They used a six-fold
symmetric case where the number of pentagon-
heptagon pairs is six. This accords with the theoretical
consideration that the five-, six-, seven-fold rotational
symmetric tori are most stable.
Iijima et al. also showed that the parallel fringes
appearing in the turn-over edge of carbon nanotubes
have a separation of 3.4 A[26]. (This value of sepa-
ration in nested tubes is also supported by other au- Fig. 6. Part of the elongated torus: here, n,, n2, and L are
the number of the hexagons along the inner circle, outer cir-
thors[27].) It is quite close to that of the “elongated” cle, and height of the torus, respectively; this figure is for the
toroidal form of C,, proposed by us[15]. case of n, = 12, n2 = 6, and L = 1.