Optoelectronics with Carbon Nanotubes

(a) (b)
Figure I-2 (a) Energy dispersion of graphene showing inequivalent K and K’ points as
blue and red dots, and (b) details at low energy, showing the linear dispersion. K, K’,
M, and Γ are high-symmetry points in graphene’s Brillouin zone. The plane of the
hexagons denotes the Fermi level.
Figure I-3 further illustrates the difference between these two types of SWNTs in
reciprocal space. If the quantized momentum vector in the circumferential direction of an
SWNT coincides with a Dirac point, it is metallic. If it misses the Dirac point, it creates an
energy gap (i.e., the cone is cut away from the point) and the SWNT is semiconducting. Using
the (n, m) indices, it can be shown that if (n - m) / 3 is an integer, the SWNT is metallic;
otherwise it is semiconducting. Therefore, if SWNTs are produced randomly in bulk, two-thirds
should be semiconducting on the average. It should be pointed out that this is a simplified
picture in the first order approximation, and the linear dispersion approximation may be less
valid in certain situations such as in small-diameter tubes (i.e., large unit wave vector in the
circumferential direction). For example, the so-called trigonal warping effect (a distortion of
equi-energy contours away from the circular shape that connect M points in a triangle) can create
a small energy gap in the order of meVs in metallic nanotubes 4, 13 .
6