2.4.2 Electronic Band Structure of SWCNTs
The electronic structure of a SWCNT can be obtained from that of
graphene. Assuming periodic boundary conditions in the
circumferential direction characterized by the chiral vector , the wave
vector associated with the direction becomes quantized, while the
wavevector associated with the direction of the translational vector
(along the CNT axis) remains continuous for a CNT of infinite length. Thus
the energy bands consist of a set of onedimensional energy dispersion
relations which are cross sections of those of graphene.
Expressions for the reciprocal lattice vectors along the
CNT axis, (
), and in the circumferential direction, (
), are given by (see Table 2.2)

(2.5) 
The onedimensional energy dispersion relations of a SWCNT can be written as

(2.6) 
where
is a onedimensional wavevector along the CNT axis and
. The periodic boundary condition for a CNT gives
discrete values in the circumferential direction. The N pairs of energy
dispersion curves given by (2.6) correspond to the cross sections of
the twodimensional energy dispersion surface of graphene. In Fig. 2.7 several cutting
lines near one of the points are shown. The separation between two
adjacent lines and the length of the cutting lines are given by the
and
, respectively. If
the cutting line passes through a point of
the twodimensional BRILLOUIN zone (Fig. 2.7a), where the and
energy bands of graphene are degenerate by symmetry, then the
onedimensional energy bands have a
zero energy gap. When the point is located between to cutting lines,
is always located in a position onethird of the distance between
two adjacent lines (Fig. 2.7b) [29] and thus a
semiconducting CNT with a finite energy gap is formed. If for a CNT,
is exactly divisible by 3 the CNT is metallic. CNTs with
residuals 1 and 2 of the division by 3 are
semiconducting.
Figure 2.7:
The onedimensional wavevectors are
shown in the BRILLOUIN zone of graphene as bold lines for
(a) metallic and (b) semiconducting CNTs.

Figure 2.8:
Onedimensional energy dispersion relations of a)
the armchair CNT, b) the zigzag CNT, and c) the
zigzag CNT.
for armchair and zigzag CNTs correspond to
and
, respectively.
Solid lines denote degenerate bands and dashed lines nondegenerate bands.

Figure 2.8 shows the energy dispersion relations for the
armchair, the zigzag, and the zigzag CNTs. In general
armchair CNTs yield energy subbands with conduction and valence
bands. Of these bands, two are nondegenerate and are doubly
degenerate. The degeneracy comes from the two subbands with the same energy
dispersion, but different values. All armchair CNTs have a band
degeneracy between the highest valence and the lowest conduction band
(Fig. 2.8a). In zigzag CNTs the lowest conduction and the highest
valence bands are doubly degenerate (Fig. 2.8b and Fig. 2.8c).
In armchair and zigzag CNTs the bands are symmetric with respect to
. Since the band of an armchair CNT has a minimum at point , it
has a mirror minimum at point
and therefore two equivalent valleys
are present around the point
. The bands of zigzag and chiral
CNTs can have at most one valley (Fig. 2.8b and Fig. 2.8c).
In armchair CNTs the bands cross the FERMI level at
. Thus,
all they are expected to exhibit metallic conduction [12]. There
is no energy gap for the CNT at , whereas the CNT indeed
shows an energy gap.
Electrical conduction is determined by states around the FERMI energy. Therefore,
it is useful to develop an approximate relation that describes the dispersion
relations in the regions around the FERMI energy
. This can be
done by replacing the expression for
in (2.3) with a TAYLOR expansion around the point
where the energy gap is zero and . It is straightforward to
show that
,
with
. The corresponding energy dispersion
relation can be written as [33]

(2.7) 
The energy bands for zigzag CNTs can be obtained by imposing the
periodic boundary conditions, which define the number of allowed wavevectors in
the circumferential direction as
. This yields
the onedimensional dispersion relations for the states of the zigzag CNT

(2.8) 
Therefore, the energy gap for
subband can be written as difference between the energies of the
and branches at

(2.9) 
The energy gap has a minimum value of zero corresponding to .
If is not a multiple of three the minimum value of is equal to
. This means that the minimum energy gap is then given by

(2.10) 
where
is the diameter of the CNT in nanometers.
Based on (2.8) and (2.9), the DOS for semiconducting
zigzag CNTs is given by

(2.11) 
which is an approximation valid as long as
[34]. VAN HOVE singularities in the DOS appearing
at both the energy minima and maxima of the bands
(see Fig. 2.9) are important for determining various solidstate
properties of CNTs [35,36]
For all metallic CNTs, independent of their diameter and chirality, due to the
nearly linear dispersion relations around the FERMI energy the density of
states (DOS) per unit length along the CNT axis is a constant given by
[12].
Figure 2.9:
The density of states per unit cell of graphene
for a) the zigzag CNT which has metallic behavior and b)
the zigzag CNT which has semiconducting behavior. Dashed lines
show the density of states for the graphene sheet [21].

M. Pourfath: Numerical Study of Quantum Transport in Carbon NanotubeBased Transistors