## Trigonal Warping Effects in the DOS Windows

Within the linear k approximation for the energy dispersion relations of graphite, Epp of (17) depends only on the nanotube diameter, dt. However, the width of the Epp band in Fig. 5 becomes large with increasing Epp [11].

When the value of |Ki| = 2/dt is large, which corresponds to smaller values of dt, the linear dispersion approximation is no longer correct. When we then plot equi-energy lines near the K point (see Fig. 3), we get circular contours for small k values near the K and K' points in the Brillouin zone, but for large k values, the equi-energy contour becomes a triangle which connects the three M points nearest to the K-point (Fig. 6). The distortion in Fig. 3 of the equi-energy lines away from the circular contour in materials with a 3-fold symmetry axis is known as the trigonal warping effect.

In metallic nanotubes, the trigonal warping effects generally split the DOS peaks for metallic nanotubes, which come from the two neighboring lines near the K point (Fig. 6). For armchair nanotubes as shown in Fig. 6a, the two lines are equivalent to each other and the DOS peak energies are equal, while for zigzag nanotubes, as shown in Fig. 6b, the two lines are not equivalent, which gives rise to a splitting of the DOS peak. In a chiral nanotube the two lines are not equivalent in the reciprocal lattice space, and thus the splitting values of the DOS peaks are a function of the chiral angle.

^ (b) | |

M/ | |

i |
K |

M |
M |

K |

Fig. 6. The trigonal warping effect of the van Hove singularities. The three bold lines near the K point are possible k vectors in the hexagonal Brillouin zone of graphite for metallic (a) armchair and (b) zigzag carbon nanotubes. The minimum energy along the neighboring two lines gives the energy positions of the van Hove singularities

On the other hand, for semiconducting nanotubes, since the value of the k vectors on the two lines near the K point contribute to different spectra, namely to that of El^1(dt) and ES2(dt), there is no splitting of the DOS peaks for semiconducting nanotubes. However, the two lines are not equivalent Fig. 4b, and the ES2(dt) value is not twice that of ES1(dt). It is pointed out here that there are two equivalent K points in the hexagonal Brillouin zone denoted by K and K' as shown in Fig. 4, and the values of ES (dt) are the same for the K and K' points. This is because the K and K' points are related to one another by time reversal symmetry (they are at opposite corners from each other in the hexagonal Brillouin zone), and because the chirality of a nanotube is invariant under the time-reversal operation. Thus, the DOS for semiconducting nanotubes will be split if very strong magnetic fields are applied in the direction of the nanotube axis.

The peaks in the 1D electronic density of states of the conduction band measured from the Fermi energy are shown in Fig. 7 for several metallic (n, m) nanotubes, all having about the same diameter dt (from 1.31 nm to 1.43 nm), but having different chiral angles: 0 = 0°, 8.9°, 14.7°, 20.2°, 24.8°, and 30.0° for nanotubes (18,0), (15,3), (14,5), (13,7), (11.8), and (10,10), respectively. When we look at the peaks in the 1D DOS as the chiral angle is varied from the armchair nanotube (10,10) (0 = 30°) to the zigzag nanotube (18,0) (0 = 0°) of Fig. 7, the first DOS peaks around E = 0.9 eV are split into two peaks whose separation in energy (width) increases with decreasing chiral angle.

This theoretical result [11] is important in the sense that STS (scanning tunneling spectroscopy) [22] and resonant Raman spectroscopy experiments [25,27,28,29] depend on the chirality of an individual SWNT, and therefore trigonal warping effects should provide experimental information about the chiral angle of carbon nanotubes. Kim et al. have shown that the DOS of a (13, 7) metallic nanotube has a splitting of the lowest energy peak in their STS spectra [22], and this result provides direct evidence for the trigonal warping effect. Further experimental data will be desirable for a systematic study of this effect. Although the chiral angle is directly observed by scanning tunneling microscopy (STM) [32], corrections to the experimental observations are necessary to account for the effect of the tip size and shape and for the deformation of the nanotube by the tip and by the substrate [33]. We expect that the chirality-dependent DOS spectra are insensitive to such effects.

In Fig. 8 the energy dispersion relations of (6) along the K-r and K-M directions are plotted. The energies of the van Hove singularities corresponding

1 1 1 1 1 1 1 1 1 : (13,7) |
^Jii | |

^ , (U- | ||

: ' '(10,10) ' ' ' ' : lYr-i-l , . |
i. u^ |

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Energy [eV]

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Energy [eV]

Fig. 7. The 1D electronic density of states vs energy for several metallic nanotubes of approximately the same diameter, showing the effect of chirality on the van Hove singularities: (10,10) (armchair), (11,8), (13,7), (14,5), (15,3) and (18,0) (zigzag). We only show the density of states for the conduction n* bands

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