Page 49 - Carbon Nanotubes

40 J. W. MINTMIRE and C. T. WHITE
from the condition n, = n2) into one set that has mod-
erate band gaps and a second set with small band gaps.
The graphene model predicts that the second set of
nanotubes would have zero band gaps, but the symme-
try breaking introduced by curvature effects results in
small, but nonzero, band gaps. To demonstrate this
point, consider the standard reciprocal lattice vectors
K, and K2 for the graphene lattice given by Ki.Rj =
The
2~6~~. band structure given by eqn (4) will have a
band crossing (i.e., the occupied band and the unoccu-
pied band will touch at zero energy) at the corners K
of the hexagonal Brillouin zone, as depicted in Fig. 3.
These six corners K of the central Brillouin zone are
given by the vectors kK = f (K, - K2)/3, f (2K1 +
K2)/3, and k(Kl + 2K2)/3. Given our earlier defini-
tion of B, B = nlRl + n2R2, a nanotube will have zero
band gap (within the graphene model) if and only if
k, satisfies eqn (5), leading to the condition n, - n2 =
3m, where m is an integer. As we shall see later, when
we include curvature effects in the electronic Hamil- Fig. 3. (a) Depiction of central Brillouin zone and allowed
tonian, these "metallic" nanotubes will actually fall graphene sheet states for a [4,3] nanotu_be conformation.
Note Fermi level for graphene occurs at K points at vertices
into two categories: the serpentine nanotubes that are of hexagonal Brillouin zone. (b) Extended Brillouin zone pic-
truly metallic by symmetry[lO, 141, and quasimetallic ture of [4,3] nanotube. Note that top left hexagon is equiv-
with small band gaps[15-18]. alent to bottom right hexagon.
In addition to the zero band gap condition, we have
examined the behavior of the electron states in the vi-
cinity of the gap to estimate the band gap for the mod- graphite sheet with periodic boundary conditions im-
erate band gap nanotubes[l3,14]. Consider a wave posed analogous to those in a quantum confinement
vector k in the vicinity of the band crossing point K problem. This level approximation allowed us to ana-
and define Ak = k - kx, with Ak = IAkl. The func- lyze the electronic structure in terms of the two-
tion ~(k) defined in eqn (4) has a cusp in the vicinity dimensional band structure of graphene. Going beyond
of R, but &'(k) is well-behaved and can be expanded this level of approximation will require explicit treat-
in a Taylor expansion in Ak. Expanding ETk), we ment of the helical symmetry of the nanotube for prac-
find tical computational treatment of the electronic structure
problem. We have already discussed the ability to de-
EZ(Ak) = V;p,azAk2, (6) termine the optimum choice of helical operator S in
terms of a graphene lattice translation H. We now ex-
where a = lRll = JR2J the lattice spacing of the amine some of the consequences of using helical sym-
is
honeycomb lattice. The allowed nanotube states sat- metry and its parallels with standard band structure
isfying eqn (5) lie along parallel lines as depicted in theory. Because the symmetry group generated by the
Fig. 2 with a spacing of 2a/B, where B = I B I . For the screw operation S is isomorphic with the one-dimen-
nonmetallic case, the smallest band gap for the nano- sional translation group, Bloch's theorem can be gen-
tube will occur at the nearest allowed point to I?, eralized so that the one-electron wavefunctions will
which will lie one third of the line spacing from K. transform under S according to
Thus, using Ak = 2?r/3B, we find that the band gap
equals[ 13,141

The quantity K is a dimensionless quantity which is
conventionally restricted to a range of --P < K 5 T, a
central Brillouin zone. For the case cp = 0 (i.e., S a
pure translation), K corresponds to a normalized quasi-
where rcc is the carbon-carbon bond distance (rcc = momentum for a system with one-dimensional trans-
a/& - 1.4 A) and RT is the nanotube radius (RT = lational periodicity (i.e., K = kh, where k is the
B/2a). Similar results were also obtained by Ajiki and traditional wavevector from Bloch's theorem in solid-
Ando[ 1 Sj . state band-structure theory). In the previous analysis
of helical symmetry, with H the lattice vector in the
3.2 Using helical symmetry graphene sheet defining the helical symmetry genera-
The previous analysis of the electronic structure of tor, K in the graphene model corresponds similarly to
the carbon nanotubes assumed that we could neglect the product K = k.H where k is the two-dimensional
curvature effects, treating the nanotube as a single quasimomentum vector of graphene.