Optoelectronics with Carbon Nanotubes

Ck 2 i;
i1,2,3...
(Eq. I.3)
determines whether the SWNT is metallic or semiconducting; if the momentum vector coincides
with the location in graphene k-space where there is no bandgap, it is metallic, otherwise, it is
semiconducting. This is a direct consequence of the energy dispersion relation in graphene
which is now discussed in greater detail.
The energy dispersion of graphene considering only the π and π* orbitals was calculated
as early as 1947 by Wallace 12 using the tight-binding approximation. This model shows that
graphene is a zero-bandgap semiconductor (or semimetal) with a linear dispersion near the Fermi
level (Figure I-2). At low energy, it can be approximated as cone shapes meeting at K and K’
symmetry points of the Brillouin zone, called the Dirac points (Figure I-2 (b)). Taking the low-
energy approximation in terms of k = |k-kF| based on the graphene energy eigenvalues from Ref.
12, we get
E( k) EF
3
0ka
2
E vk
F F
5
(Eq. I.4)
where γ0 is the C-C transfer energy and a is the unit vector length (Figure I-1). Using known
values for γ0 and a gives the Fermi velocity v F ≈ 8.7 × 10 5 m/s. Note that there is only one
electron per atom contributing to these orbitals; in an updoped graphene at zero temperature, the
valence band is completely filled and the conduction band is empty, making graphene a semi-
metal (or a zero-bandgap semiconductor). How these cones are effectively “cut” by an SWNT’s
circumferential momentum determines whether the SWNT is metallic or semiconducting.