Figure 41: How to specify the kind of a nanotube.graphene out perpendicular to this line and role it up so that the end point falls on the origin we
defined.
Now we calculate the band structure, the dispersion relationship for carbon nanotubes. With carbon
nanotubes there are periodic boundary conditions around the short distance (around the diameter
of the tube). So there are only certainky-values, in the directionkxwe also use periodic boundary
conditions but the lengthLin this direction is much larger, hence there are more values forkxthen
forky. In fig. 42 there is a plot in k-space and on the left this issue is shown for the first brillouin zone
(which goes from−π/atoπ/a). If we have a (10,10) tube going around the diameter we only have
Figure 42: Metallic (10,10) armchair tube.10 values forky. That means there are fewer allowed k-vectors. But the tight binding problem does
not change with respect to the problem solved for graphene. For carbon nanotubes the calculation is
the same except there are only certain values forky. In fig. 42 on the right side the highest bonding