| The single-wall carbon nanotubes(SWCNTs) with various bands gaps may be an insulator, a semiconductor, or a metal. The orbicular geometrical characteristics of the cylinder make quantization of angular momentum perpendicular to circumferences, and lead a band gap, which is called primary gap. With the further study of Kane and Mele, if the curvature effect is considered, SWCNTs may have secondary gap, which is inversely proportional to the square of the tubes'radius. In experiments, it is observed that only armchair tube is true metal, while the zigzag metallic nanotubes had the secondary gap and were in fact small-gap semiconductors. In references, the armchair tubes are usually classified to the first class metallic tube, while the other metallic tubes are classified to the second class metallic tube. From the analysis of geometry, the curvature effect makes the bond length and angle to change. In this paper, we have make use of the lattice model on cylinder to study the relation of secondary gap and the radius as well as chirality of SWNTs. The main works are in the following:1.We make use of the corresponding relation of the cylindrical lattice model and flat lattice and the condition which three bonds are equivalent to construct reciprocal lattice, which is still a triangular lattice on cylinder.. In addition, we have given the Brillouin zone of the cylindrical lattice and have discussed the corners of the first Brillouin zone in relation to the corners of first Brillouin zone of graphene. In detail, the corners of the first Brillouin zone of armchair tubes move along the circumferences as well as the ones of zigzag tubes move along the axial direction of the tube, while the ones of the chiral tube move along the both directions.2.Basic on the model of the cylinder lattice, we consider the tight-binding Hamiltonian ofπelectrons in the approximation of the nearest-hopping., while the Hamiltonian can be diagonalized. Here, we have mainly studied the causes of the secondary gap, and then analyzed the relationship between the secondary gap and the chirality as well as radius. The concrete results in the following: 1) We have obtained the band structure of SWCNTs, while considering the curvature effect modification to the nearest-hopping amplitudes and using the Fourier transformation in cylindrical coordinate. The band concludes valence band and conduction band, which are described as the azimuthal quantum number along the circumferences and the wave vector along the axial direction. 2) To analyze the reason secondary gap, we have discussed the transition when the increase does not equal three, in momentum space under a continuous zero-energy gap conditions for the existence of three transitions that increase the formation of a triangle, given the valence band and I. with the location of degenerate points and come to sub-energy gap appears due to curvature effects lead to the results of band-edge movement. 3) We have discussed the band structures and energy gaps of the armchair tube and the zigzag tube, and obtained the armchair tube is real metal as well as the metallic zigzag tube appears secondary gap, which are accordance with the theatrical results Kane and Ding. In experiments, Ouyang measured the secondary gap 0.080±0.005, 0.042±0.004 and 0.029±0.004 eV for (9, 0), (12, 0) and (15, 0), respectively, while in our theatrical results are 0.0792, 0.0446 and 0.0285 eV. 4) We have studied the secondary class metallic tubes'secondary gap. In this case, the results can't be analytically obtained, so we get the numerical values. We find when l = 1, 2, 3, 4(l satisfy the condition m ? n = 3l) and R=6.4804, 6.8924, 7.3439, 7.8281A, the corresponding secondary gap are 0.0068, 0.0110, 0.0131, 0.0137eV, so the secondary gap is relate to the radius and l . Finally, we have fitted the expressions as the radius and l .
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