Implications of Time-Reversal Symmetry for the Band Structure of Single-Wall Carbon Nanotubes

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)) - -pi/T; K-1((3)) - (2N - M)/2R, K-2((3)) - pi/T; K-1((4)) - (M + N)/2R, K-2((4)) = -pi/T, and K-1((5)) = (N - M)/2R, K-2((5)) = pi/T, where N and M are the integers relating the chiral, C-h, symmetry, R, and translational, T, vectors of the nanotube by NR = C-h + MT, T = vertical bar T vertical bar, 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 = +/-pi/T, are shown to be degenerate due to the time-reversal symmetry.