410
14 Nanostructures
Fig. 14.17 A chiral nanotube (chiral vector is (10, 5), θ = −19.11 ◦ ) with hemispherical caps at both ends based on an
icosahedral C 140 fullerene. The tube diameter is 1.036 nm. Adapted from [1367]
The mechanical strength of carbon nanotubes is very large. For SWNT Young’s moduli of 10
3 GPa
have been found experimentally [1368] in agreement with theoretical predictions [1369].
14.3.2 Band Structure
In carbon nanotubes there is some mixing of the π (2p z ) and σ (2s and 2p z ) carbon orbitals due to the
radial curvature. This mixing is, however, small and can be neglected near the Fermi level [1370]. The
band structure of a nanotube is mainly determined by zone-folding of the graphene band structure.
The vector along the (infinitely extended) wire k z is continuous. The vector k ⊥ around the nanotube is
discrete with the periodic boundary condition
c h · g ⊥ = 2π m ,
(14.3)
where m is an integer. The distance of allowed k ⊥ -values is (5.5)
k ⊥ =
2π
π d
=
2
d
.
(14.4)
The character of the nanotube band structure depends on how the allowed k-values lie relative to the
graphene Brillouin zone and its band structure. This is visualized in Fig. 14.18. For the case of an
armchair tube (n, n), as shown in Fig. 14.18a, the K-point of the graphene band structure always lies
on an allowed k-point. Therefore, the nanotube is metallic, i.e. zero-gap, as seen in the bandstructure
in Fig. 14.18b. The Dirac point is between and X. For a zig-zag nanotube, the k-space is shown in
Fig. 14.18c for a (6, 0) nanotube. The corresponding band structure for a (6, 0) nanotube is also metallic
(Fig. 14.18d) with the Dirac point at the point.
In Fig. 14.19c the band structure of another metallic (12, 0) zig-zag nanotube is shown. However,
only for (3 m, 0) the K-point is on an allowed state and thus the tube metallic. For the other cases, as
shown for the k-space of a (8, 0) nanotube in Fig. 14.19b, this is not the case. The corresponding band
structure (Fig. 14.19c for (13, 0)) has a gap and thus the nanotube is a semiconductor. Generally, the
condition for a nanotube to be metallic is with an integer m
n 1 − n 2 = 3 m .
(14.5)
There are two semiconducting ‘branches’ with ν = (n 1 − n 2 ) mod 3 = ±1. The tubes with ν = +1
have a small band gap, those with ν = −1 have a larger band gap.
The density of states is a series of one-dimensional DOS, proportional to
√
E (6.79). It is compared
in Fig. 14.20 for a metallic and a semiconducting nanotube. Within 1 eV from the Fermi energy the
DOS can be expressed in an universal term [1373].
14 Nanostructures
Fig. 14.17 A chiral nanotube (chiral vector is (10, 5), θ = −19.11 ◦ ) with hemispherical caps at both ends based on an
icosahedral C 140 fullerene. The tube diameter is 1.036 nm. Adapted from [1367]
The mechanical strength of carbon nanotubes is very large. For SWNT Young’s moduli of 10
3 GPa
have been found experimentally [1368] in agreement with theoretical predictions [1369].
14.3.2 Band Structure
In carbon nanotubes there is some mixing of the π (2p z ) and σ (2s and 2p z ) carbon orbitals due to the
radial curvature. This mixing is, however, small and can be neglected near the Fermi level [1370]. The
band structure of a nanotube is mainly determined by zone-folding of the graphene band structure.
The vector along the (infinitely extended) wire k z is continuous. The vector k ⊥ around the nanotube is
discrete with the periodic boundary condition
c h · g ⊥ = 2π m ,
(14.3)
where m is an integer. The distance of allowed k ⊥ -values is (5.5)
k ⊥ =
2π
π d
=
2
d
.
(14.4)
The character of the nanotube band structure depends on how the allowed k-values lie relative to the
graphene Brillouin zone and its band structure. This is visualized in Fig. 14.18. For the case of an
armchair tube (n, n), as shown in Fig. 14.18a, the K-point of the graphene band structure always lies
on an allowed k-point. Therefore, the nanotube is metallic, i.e. zero-gap, as seen in the bandstructure
in Fig. 14.18b. The Dirac point is between and X. For a zig-zag nanotube, the k-space is shown in
Fig. 14.18c for a (6, 0) nanotube. The corresponding band structure for a (6, 0) nanotube is also metallic
(Fig. 14.18d) with the Dirac point at the point.
In Fig. 14.19c the band structure of another metallic (12, 0) zig-zag nanotube is shown. However,
only for (3 m, 0) the K-point is on an allowed state and thus the tube metallic. For the other cases, as
shown for the k-space of a (8, 0) nanotube in Fig. 14.19b, this is not the case. The corresponding band
structure (Fig. 14.19c for (13, 0)) has a gap and thus the nanotube is a semiconductor. Generally, the
condition for a nanotube to be metallic is with an integer m
n 1 − n 2 = 3 m .
(14.5)
There are two semiconducting ‘branches’ with ν = (n 1 − n 2 ) mod 3 = ±1. The tubes with ν = +1
have a small band gap, those with ν = −1 have a larger band gap.
The density of states is a series of one-dimensional DOS, proportional to
√
E (6.79). It is compared
in Fig. 14.20 for a metallic and a semiconducting nanotube. Within 1 eV from the Fermi energy the
DOS can be expressed in an universal term [1373].