|VOLUME 92 | ISSUE 8 |
|Implications of time-reversal symmetry for band structure of single-wall carbon nanotubes
S. V. Goupalov
Department of Physics, Jackson State University, Jackson, MS 39217 USA
A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
When electron states in carbon
nanotubes are characterized by two-dimensional wave vectors with the components
K1 and K2 along the nanotube circumference and cylindrical axis,
respectively, then two such vectors
symmetric about a M-point in the reciprocal space of graphene are
shown to be related by the time-reversal operation.
To each carbon nanotube there
correspond five relevant M-points with the following coordinates:
, K2(1)=0; ,
, K2(4)=-π/T, and
where and are the integers relating the chiral,
Ch, symmetry, R, and translational, T, vectors
of the nanotube by ,
T=| T|, and R is the nanotube radius. We show that the states
at the edges of the one-dimensional Brillouin zone which are symmetric
about the M-points with are degenerate due to
the time-reversal symmetry.
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