132 Solving the Schrödinger equation in periodic solids
M Γ K(^) Γ
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MFigure 6.5. Tight-binding band structure of graphene. The points of the Brillouin
zone are shown on the right hand side.We speak now of an(n,m)nanotube wherenandmare the integer coefficients. We
restrict ourselves here to the case wherem=0. Such tubes are called ‘zigzag’ tubes
because a circle running around the tube consists of a zigzag structure of nearest
neighbour bonds.
A carbon nanotube is a one-dimensional object – therefore the states are labelled
by a Bloch vectoralongthe tube. We neglect effects due to the curvature of the
sheet, which alter the interactions. For large tubes (i.e. tubes with a large diameter)
this is a good approximation. Across the tube, the wave functionmust be periodic.
The difference from a periodic cell in a periodic crystal must be emphasised here.
In a crystal, the potential and the density are periodic, but the wave function (in
general) is not. In the present case the wave function must match onto itself across
the tube – hence it is really periodic. This implies that the transverse component of
the wave vector must be 2πj/Lfor a tube circumferenceLand integerj. For each
nwe find an energy value, that is, for each fixed longitudinalk-vector we find a
discrete energy spectrum.
For this case, the period along the tube isa
√
- This means that the longitudinal
Brillouin zone runs up tok=π/(a
√
3 ). This point is denoted asX. The transverse
period is given asL=na. In order to calculate the band structure, we perform a
loop over the longitudinalk-vector. For each such vector we run over the possible
transversek-vectors (values 2πj/Lwherejlies between 0 andn). We calculate the
two energies in (6.20), and plot these as a function of the longitudinalk. The result
is shown in Figure 6.6 for a tube with an odd and an even number of orbitals. Note
the difference between the two: one is a metal, the other an insulator. In reality, the
even tube has a small gap due to the curvature with respect to the graphene case.
Tubes of another type, the so-called arm-chair tubes, characterised bym=n, are
always metallic. The reader is invited to investigate that case.