【作者】 王锐；

【导师】 王少峰；

【作者基本信息】 重庆大学 ， 凝聚态物理， 2009， 硕士

【摘要】 单壁碳纳米管可以为绝缘体、半导体或者金属,这是因为不同手性和半径的管具有不同的能隙。由于柱面的环状几何特征,使得垂直于轴线的环周上动量量子化,进而导致能隙的产生,这个能隙称为碳纳米管的主能隙。Kane和Mele进一步研究发现,在考虑曲率效应后,可能出现与管半径的平方成反比的次级能隙。实验上也发现,只有扶手型管的能隙严格保持为零,而锯齿型和手性金属碳纳米管都会出现次级能隙,变为小能隙半导体。文献中也把扶手型管称为第I类金属管,把其它的金属管划归第II类金属管。从几何分析知道,曲率效应使得碳原子间的键长和键角等因素发生变化。本文利用了碳纳米管的柱面晶格模型,进而研究了单壁碳纳米管的次级能隙与半径和手性的关系。本文的主要工作包括:1、利用柱面晶格模型与平面晶格的对应关系,利用键长相等的条件,构造了对应的倒格子,其仍为柱面上的三角格子。另外,我们给出了柱面晶格的布里渊区,并讨论了第一布里渊区角点相对于平面石墨单层第一布里渊区角点移动的情况。对非手性碳纳米管,扶手型管的第一布里渊区角点在圆周方向上不移动而在轴向发生移动,同时锯齿型管在圆周方向上发生移动而在轴向上不发生移动;对手性碳纳米管,两者均发生移动。2、基于柱面晶格模型,考察了最近邻跳跃近似下的π电子紧束缚模型,模型Hamiltonian可以对角化,这里主要研究了次级能隙出现的原因,并分析了次级能隙与碳纳米管手征和半径之间的关系。具体内容包括:1)考虑了曲率效应对跃迁幅的修正,在柱面上用Fourier变换严格对角化模型Hamiltonian得到了单壁碳纳米管的能带。能带包括价带和导带,由轴向波矢和圆周方向的角量子数两个量子数描述。2)为分析次级能隙出现的原因,我们讨论了当三个跃迁幅不相等时,在动量空间的连续情况下零能隙存在的条件即三个跃迁幅形成三角形,给出了价带和导带简并点的位置,并得出次级能隙出现的原因是曲率效应导致带边移动的结果。3)分析了扶手型和锯齿型单壁碳纳米管的能带和能隙,得出扶手型碳纳米管的能隙严格为零,金属锯齿型碳纳米管的次级能隙随半径的变化规律同Kane与Mele和Ding等人讨论的理论结果一致,实验上Ouyang等人给出(9,0)、(12,0)和(15,0)的次级能隙分别为0.080±0.005, 0.042±0.004和0.029±0.004 eV,我们的计算结果分别为0.0792, 0.0446和0.0285 eV,与实验结果符合的很好。4)研究了手性第II类金属管次级能隙与半径和手征的关系。结果不能解析求解,通过数值结果可以看出,在当l = 1, 2, 3, 4(l为整数其满足m ? n = 3l),一组对应的R=6.4804, 6.8924, 7.3439, 7.8281 A,其相应的能隙分别为0.0068, 0.0110, 0.0131, 0.0137eV,因此半径增大,能隙不一定减小,还与l值有关。我们用数值拟合得到了次级能隙与碳纳米管半径和l关系的近似表达式,可以在一定程度上描述手性第II类金属管次级能隙的规律。更多还原

【Abstract】 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 .更多还原