Implications of time-reversal symmetry for the band structure of single-wall carbon nanotubes |
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Authors: | S V Goupalov |
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Institution: | 1.Department of Physics,Jackson State University,Jackson,USA;2.Ioffe Physicotechnical Institute,St. Petersburg,Russia |
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Abstract: | When electron states in carbon nanotubes are characterized by two-dimensional wave vectors with the components K
1 and K
2 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: K
1(1) = N/2R, K
2(1)= 0; K
1(2) = M/2R, K
2(2)= −π/T; K
1(3)= (2N −M)/2R, K
2(3)= π/T; K
1(4)= (M + N)/2R, K
2(4)= -π/T, and K
1(5)= (N − M)/2R, K
2(5)= π/T, where M and N are the integers relating the chiral, C
h
, symmetry, R, and translational, T, vectors of the nanotube by N
R = C
h
+ M
T, T = |T|, and R is the nanotube radius. The states at the edges of the one-dimensional Brillouin zone, which are symmetric about the M-points with K
2 = ±π/T, are shown to be degenerate due to the time-reversal symmetry. |
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Keywords: | |
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