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VOLUME 92 (2010) | ISSUE 8 | PAGE 559
Implications of time-reversal symmetry for band structure of single-wall carbon nanotubes
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: K_1^{(1)}={\cal N}/2R, K2(1)=0; K_1^{(2)}={\cal M}/2R, K2(2)=-π/T; K_1^{(3)}=(2 {\cal N} -{\cal M})/2R, K2(3)=π/T; K_1^{(4)}=({\cal M} +{\cal N})/2R, K2(4)=-π/T, and K_1^{(5)}=({\cal N} -{\cal M})/2R, K2(5)=π/T, where {\cal N} and {\cal M} are the integers relating the chiral, Ch, symmetry, R, and translational, T, vectors of the nanotube by {\cal N} {\bf R}={\bf C}_h + {\cal M} {\bf T}, 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 K_2=\pm \pi/T are degenerate due to the time-reversal symmetry.