Page 50 - Carbon Nanotubes

Electronic and structural properties of carbon nanotubes 41
Before we continue our description of electronic isfies eqn (3, and at the points the one-electron
structure methods for the carbon nanotubes using heli- wave functions transform under the generator of the
cal symmetry, let us reconsider the metallic and quasi- rotation group C,, with a phase factor given by kR.
metallic cases discussed in the previous section in more (R, + R2) = 0. This irreducible representation of the
detail. The graphene model suggests that a metallic C, group is split under reflection into the two irreduc-
state will occur where two bands cross, and that the ible representations a, and a2 of the C,, group that
Fermi level will be pinned to the band crossing. In terms are symmetric and antisymmetric, respectively, under
of band structure theory however, if these two bands the reflection plane; the states at K will belong to
belong to the same irreducible representation of a point these two separate irreducible representations. Thus,
group of the nuclear lattice that also leaves the point the serpentine nanotubes are always metallic because
in the Brillouin zone invariant, then rather than touch- of symmetry if the Hamiltonian allows sufficient band-
ing (and being degenerate in energy) these one-electron width for a crossing, as is normally the case[lO]. The
eigenfunctions will mix and lead to an avoided cross- sawtooth nanotubes, however, present a different pic-
ing. Only if the two eigenfunctions belong to differ- ture. The one-electron wave functions at K transform
ent irreducible representations of the point group can under the generator of the rotation group for this
they be degenerate. For graphene, the high symmetry nanotube with a phase factor given by LR-R, =
of the honeycomb lattice allows the degeneracy of the 2n/3. This phase factor will belong to one of the e rep-
highest-occupied and lowest-unoccupied states at the resentations of the C,,, group, and the states at If in
corners K of the hexagonal Brillouin zone in graph- the graphene Brillouin zone will therefore belong to
ene. Rolling up graphene into a nanotube breaks this the same symmetry group. This will lead to an avoided
symmetry, and we must ask what point group symme- crossing. Therefore, the band gaps of the non-
tries are left that can allow a degeneracy at the band serpentine nanotubes that satisfy eqn (5) are not truly
crossing rather than an avoided crossing. For the nano- metallic but only small band gap systems, with band
tubes, the appropriate symmetry operations that leave gaps we estimate from empirical and first-principles
an entire band in the Brillouin zone invariant are the calculation to be of the order of 0.1 eV or less.
C, rotation operations around the helical axis and re- Now, let us return to our discussion of carrying out
flection planes that contain the helical axis. We see an electronic structure calculation for a nanotube
from the graphene model that a reflection plane will using helical symmetry. The one-electron wavefunc-
generally be necessary to allow a degeneracy at the tions II;- can be constructed from a linear combination
Fermi level, because the highest-occupied and lowest- of Bloch functions ‘pi, which are in turn constructed
unoccupied states will share the same irreducible rep- from a linear combination of nuclear-centered func-
resentation of the rotation group. To demonstrate this, tions xj(r),
consider the irreducible representations of the rotation
group. The different irreducible representations trans-
form under 1 he generating rotation (of ~T/N radians)
with a phase factor an integer multiple 2am/N, where
m = 0, . . . , N - 1. Within the graphene model, each
allowed state at quasimomentum k will transform un-
der the rotation by the phase factor given by k.B/N,
and by eqn (5) we see that the phase factor at Kis just As the next step in including curvature effects beyond
2 ~rn/N. The eigenfunctions predicted using the the graphene model, we have used a Slater-Koster pa-
graphene model are therefore already members of the rameterization[31] of the carbon valence states- which
irreducible representations of the rotation point group. we have parameterized[32,33] to earlier LDF band
Furthermore, the eigenfunctions at a given Brillouin structure calculations[34] on polyacetylene-in the em-
zone point k in the graphene model must be members pirical tight-binding calculations. Within the notation
of the same irreducible representation of the rotation in ref. [31] our tight-binding parameters are given by
point group. V,, = -4.76 eV, V,, = 4.33 eV, Vpp, = 4.37 eV, and
For the nanotubes, then, the appropriate symmetries Vppa = -2.77 eV.[33] We choose the diagonal term
for an allowed band crossing are only present for the for the carbon p orbital, = 0 which results in the s
serpentine ([n, n]) and the sawtooth ([n,O]) conforma- diagonal term of E, = -6.0 eV. This tight-binding
tions, which will both have C,,, point group symme- model reproduces first-principles band structures qual-
tries that will allow band crossings, and with rotation itatively quite well. As an example, Fig. 4 depicts both
groups generated by the operations equivalent by con- §later-Koster tight-binding results and first-principles
formal mapping to the lattice translations R1 + R2 LDF results[l0,12] for the band structure of the [5,5]
and R1, respectively. However, examination of the serpentine nanotube within helical symmetry. AI1
graphene model shows that only the serpentine nano- bands have been labeled for the LDF results accord-
tubes will have states of the correct symmetry @e., dif- ing to the four irreducible representations of the C,,
ferent parities under the reflection operation) at the point group: the rotationally invariant a, and a, rep-
point where the bands can cross. Consider the K resentations, and the doubly-degenerate el and e2
point at (K, - K2)/3. The serpentine case always sat- representation. As noted in our discussion for the ser-