Page 48 - Carbon Nanotubes

Electronic and structural properties of carbon nanotubes 39

the minimum unit cell (primitive helical motif) size of posed by the rolling up of the nanotube. We then con-
2Natoms, where N is the greatest common divisor of tinue by analyzing a structurally correct nanotube
n, and n,. within a Slater-Koster sp tight-binding model using
Next, consider the possible rotational symmetry the helical symmetry of the nanotube. We then finish
around the helical axis. This operation is of special in- by discussing results for both geometry optimization
terest because all the rotation operations around the and band structure using first-principles local-density
chain axis will commute with the helical operator functional methods.
S (, h, p) , allowing solutions of any one-electron Ham-
iltonian model to be block-diagonalized into the ir- 3.1 Graphene model
reducible representations of both the helical operation One of the simplest models for the electronic struc-
and the rotation operator simultaneously. We observe ture of the states near the Fermi level in the nanotubes
that the highest order rotational symmetry (and small- is that of a single sheet of graphite (graphene) with pe-
est possible rotation angle for such an operator) will riodic boundary conditions[lO-161. Let us consider a
be given by a C, rotation operator, where N remains Slater-Koster tight-binding model[31] and assume that
the greatest common divisor of n, and nz[13]. This the nearest-neighbor matrix elements are the same as
can be seen straightforwardly by considering the real those in the planar graphene (Le., we neglect curva-
lattice vector B,, ture effects). The electronic structure of the nanotube
will then be basically that of graphene, with the ad-
ditional imposition of Born-von Karman boundary
conditions on the electronic states, so that they are pe-
The conformal mapping will transform this lattice op- riodic under translations of the circumference vector
eration B, to a rotation of 2a/N radians around the B. The electronic structure of the a-bands near the
nanotube axis, thus generating a C, subgroup. Fermi level then reduces to a Huckel model with one
The rotational and helical symmetries of a nano- parameter, the Vppn hopping matrix element. This
tube defined by B can then be seen by using the cor- model can be easily solved, and the one-electron eigen-
responding helical and rotational symmetry operators values can be given as a function of the two-
S(h,pp) and C, to generate the nanotube[l3,14]. This dimensional wavevector k in the standard hexagonal
is done by first introducing a cylinder of radius Brillouin zone of graphene in terms of the primitive
lattice vectors R1 and R,:
E(k) = I/,,J~+~cos~.R~ +~cos~.R~+~cos~.(R~ -R2) (4)

The Born-von Karman boundary conditions then
1 Bi/27r. The two carbon atoms located at d = (R, + restrict the allowed electronic states to those in the
R2)/3 and 2d in the [O,O] unit cell of Fig. 1 are then graphene Brillouin zone that satisfy
mapped to the surface of this cylinder. The first atom
is mapped to an arbitrary point on the cylinder sur-
face, which requires that the position of the second be
found by rotating this point by 27r(d.B)/IBI2 radi- that rn an integer[l0,14,15]. In terms of the two-
ans about the cylinder axis in conjunction with a trans- dimensional Brillouin zone of graphene, the allowed
lation Id x IBI /IBI units along this axis. The positions states will lie along parallel lines separated by a spac-
BI
of these first two atoms can then be used to generate ing of 2~/l , Fig. 3a depicts a set of these lines for
2(N- 1) additional atoms on the cylinder surface by the [4,3] nanotube. The reduced dimensionality of
(N- 1) successive 2a/N rotations about the cylinder these one-electron states is analogous to those found
axis. Altogether, these 2N atoms complete the speci- in quantum confinement systems; we might expect
fication of the helical motif which corresponds to an that the multiple crossings of the Brillouin zone edge
area on the cylinder surface given by NJR, x Rzl. would lead to the creation of multiple gaps in the elec-
This helical motif can then be used to tile the remain- tronic density-of-states (DOS) as we conceptually
der of the nanotube by repeated operation of the he- transform graphene into a nanotube. However, as de-
lical operation defined by S (h, p) generated by the H picted in Fig. 3b, these one-electron states are actually
defined using eqn (2). continuous in an extended Brillouin zone picture. A
direct consequence of our conclusion in the previous
section-that the helical unit cell for the [4,3] nano-
3. ELECTRONIC STRUCTURE OF tube would have only two carbons, the same as in
CARBONNANOTUBES graphene itself-is that the a-band in graphene will
We will now discuss the electronic structure of transform into a single band, rather than several.
single-shell carbon nanotubes in a progression of more Initially, we showed that the set of serpentine nano-
sophisticated models. We shall begin with perhaps the tubes [n,n] are metallic. Soon thereafter, Hamada
simplest model for the electronic structure of the nano- et al.[15] and Saito et a1.[16,17] pointed out that the
tubes: a Huckel model for a single graphite sheet with periodicity condition in eqn (5) further groups the re-
periodic boundary conditions analogous to those im- maining nanotubes (those that cannot be constructed