Optoelectronics with Carbon Nanotubes

(a)
Figure I-3. The Brillouin zones of (a) metallic and (b) semiconducting SWNTs
mapped onto the hexagonal Brillouin zones of graphene. The blue (K) and red
(K’) dots indicate the Dirac points where conduction and valence bands meet, as
in Figure I-2. Note that in (b), the “cutting line” of the quantized momentum
misses the K point, thus creating an energy gap. Reciprocal lattice vectors k1
corresponds to the chiral vector C in Figure I-1, and k2 to the translational vector
in the direction of the long axis of the CNT (not shown in Figure I-1).
Based on Equation (I.4) and the above argument, the one-dimensional energy-dispersion
can be expressed as
2 2
E( ) E v k j
; j=0, 1, 2, ... (Eq. I.5)
F F
(b)
where Δ is one-half of the energy gap, which depends on the diameter of the tube. Figure I-4
shows the first three bands of metallic and semiconducting SWNTs calculated for when ν is (a)
0, 3, and 6 (metallic), and (b) 1, 2, and 4 (semiconducting). Once the one-dimensional energy
dispersion is thus known, the corresponding density of states (DOS) can be calculated (shown in
green in Figure I-4). As seen in Figure I-4, the DOS shows singularities named van Hove
singularities because of the SWNT’s reduced dimensionality.
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