Implications of time-reversal symmetry for the band structure of single-wall carbon nanotubes

Abstract

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: K1(1) = N/2R, K2(1)= 0; K1(2) = M/2R, K2(2)= −π/T; K1(3)= (2N −M)/2R, K2(3)= π/T; K1(4)= (M + N)/2R, K2(4)= -π/T, and K1(5)= (N − M)/2R, K2(5)= π/T, where M and N are the integers relating the chiral, Ch, symmetry, R, and translational, T, vectors of the nanotube by NR = Ch + MT, 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 K2 = ±π/T, are shown to be degenerate due to the time-reversal symmetry.