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4.5. VARIATIONAL MONTE-CARLO APPLIED TO THE CARBON NANOTUBES97
a)
b) c)
( 2,1)
( 2,2)
(3,0)
Figure 4.9: Reciprocal space of a) the (2,1) tube , b) the armchair (2,2) tube,
and c) the zig-zag (3,0) tube. The armchair and zig-zag tubes have a metallic
dispersion.
4.5 Variational Monte-Carlo applied to the Carbon
nanotubes
The study of carbon nanotubes is also an interesting event, since their discovery
by Iijima in 1991 [116], has become a full research field with significant contributions
from all areas of research in solid-state. Single-walled carbon nanotube
(SWCNT) are constituted by a single graphene plane wrapped into a cylinder.
The Fermi surface of graphene is very particular: it is reduced to six discrete
points at the corners of the 2D lattice Brillouin zone [117] as shown in Fig.4.1.
As a result, depending on their diameter and their chirality which determine
the boundary conditions of the electronic wave functions around the tube, the
nanotube can be either semiconducting or metallic, depending on the lines of k
points in the Brillouin Zone crosses the 6 points of the Fermi surface (see Fig.
4.9). It has been shown that only the zero-chirality armchair nanotubes have zero
electronic band gap, the others having a small charge gap and being therefore
semi-conductors or insulators depending on their radius and chirality. Let us note
that the special zig-zag geometry has also a zero charge gap in the free electronic
band theory.
While superconductivity is well established in graphene, this phenomenon was
only recently investigated for the nanotubes : intrinsic superconductivity was
experimentally observed in ropes of nanotubes [118]. Data show the existence of
intrinsic superconductivity in ropes of carbon nanotubes in which the number of
tubes varies between 30 and 400. The question of the existence of superconducting
correlations in the limit of the individual tube cannot however be answered yet.
Whereas graphene is doped with alkali-dopants, there are no chemical dopants
in the ropes of carbon nanotubes. As shown in previous works there is some