Page 47 - Carbon Nanotubes

38 J. W. MINTMIRE and C. T. WHITE
sets, the nanotubes are chiral and have three inequiv-
alent helical operations.
Because real lattice vectors can be found that are
normal to the primitive real lattice vectors for the
graphite sheet, each nanotube thus generated can be
shown to be translationally periodic down the nano-
tube axis[12-14,23]. However, even for relatively small
diameter nanotubes, the minimum number of atoms
in a translational unit cell can be quite large. For ex-
ample, for the [4,3] nanotube (nl = 4 and nz = 3)
then the radius of the nanotube is less than 0.3 nm,
but the translational unit cell contains 148 carbon at-
oms as depicted in Fig. 2. The rapid growth in the
Fig. 1. Two-dimensional honeycomb lattice of graphene;
primitive lattice vectors R, and R, are depicted outlining number of atoms that can occur in the minimum
primitive unit cell. translational unit cell makes recourse to the helical and
any higher point group symmetry of these nanotubes
practically mandatory in any comprehensive study of
group symmetry) helical operations derived from the their properties as a function of radius and helicity.
primitive lattice vectors of the graphite sheet. Thus, These symmetries can be used to reduce to two the
while all nanotubes have a helical structure, nanotubes number of atoms necessary to generate any nano-
constructed by mapping directions equivalent to lat- tube[13,14]; for example, reducing the matrices that
tice translation indices of the form [n,O] and [n,n], to have to be diagonalized in a calculation of the nano-
the circumference of the nanotube will possess a re- tube’s electronic structure to a size no larger than that
flection plane. These high-symmetry nanotubes will, encountered in a corresponding electronic structure
therefore, be achiral[ 12-14,231. For convenience, we calculation of two-dimensional graphene.
have labeled these special structures based on the Before we can analyze the electronic structure of
shapes made by the most direct continuous path of a nanotube in terms of its helical symmetry, we need
bonds around the circumference of the nanotube[23]. to find an appropriate helical operator S (h, p) , rep-
Specifically, the [n,O]-type structures were labeled as resenting a screw operation with a translation h units
sawtooth and the [n, n]-type structures as serpentine. along the cylinder axis in conjunction with a rotation
For all other conformations inequivalent to these two p radians about this axis. We also wish to find the op-
erator S that requires the minimum unit cell size (Le.,
the smallest set of carbon atoms needed to generate
the entire nanotube using S) to minimize the compu-
tational complexity of calculating the electronic struc-
ture. We can find this helical operator S( h, p) by first
finding the real lattice vector H = mlR, + m2R2 in
the honeycomb lattice that will transform to S(h, p)
under conformal mapping, such that h = I H x BI /I BI
and p = 2.1r(H.B)/(BIZ. An example of H is depicted
in Fig. 2 for the [4,3] nanotube. If we construct the
cross product H x B of H and B, the magnitude I H x
BI will correspond to the area “swept out” by the he-
lical operator S. Expanding, we find

where I RI x Rz I is just the area per two-carbon unit
cell of the primitive graphene lattice. Thus (mln2 -
m2n1) gives the number of carbon pairs required for
the unit cell under helical symmetry with a helical op-
erator generated using H and the conformal mapping.
Given a choice for n, and n2, then (m~nz - mzn~)
must equal some integer value

Fig. 2. Depiction of conformal mapping of graphene lattice
to [4,3] nanotube. B denotes [4,3] lattice vector that trans-
forms to circumference of nanotube, and H transforms into Integer arithmetic then shows that the smallest mag-
the helical operator yielding the minimum unit cell size un- nitude nonzero value of N will be given by the great-
der helical symmetry. The numerals indicate the ordering of
the helical steps necessary to obtain one-dimensional trans- est common divisor of nl and n2. Thus, we can easily
lation periodicity. determine the helical operator H + S(h, p) that yields