## Ka

Fig. 8. Splitting of the DOS in zigzag nanotubes. Two minimum energy positions are found in the conduction band for zigzag nanotubes, (n, 0) measured from the energy at the K point. Open circles denote metallic carbon nanotubes for k = |K1| vectors away from the K point along the K ^ M and K ^ r lines, which are the directions of the energy minima (see Fig. 6). (The inset shows an expanded view of the figure at small E/yo and small ka for semiconducting nanotubes. The closed circles denote semiconducting carbon nanotubes for k = |K1|/3 vectors. ) Note that the maximum of the horizontal axis corresponds to the M point, ka = 2n/3, which is measured from the K point. A nanotube diameter of 1 nm corresponds to a (13,0) carbon nanotube to the lowest 1D energy level are plotted for metallic (open circles) and semiconducting (closed circles) zigzag nanotubes (n, 0) by putting ka = |K1 |a and ka = |K1|a/3, respectively. The corresponding energy separation is plotted in Fig. 5 as solid squares. In the case of (3n + 1,0) and (3n — 1,0) semiconducting zigzag nanotubes, E^ comes respectively, from the K-r and K-M lines, while ES2 comes from K-M and K-r and so on. In the case of (3n, 0) metallic zigzag nanotubes, the DOS peaks come from both K-M and K-r. This systematic rule will be helpful for investigating the STS spectra in detail. Using Eqs. (7) and (15), the widths of EM and Ef1, denoted by A E^ and A E^, respectively, are determined by the zigzag nanotubes, and are analytically given by

Although this trigonal warping effect is proportional to (a/dt)2, the terms in (18) are not negligible, since this correction is the leading term in the expressions for the width A Epp (dt), and the factor 8 before 70 makes this correction significant in magnitude for dt=1.4nm. For example, E11(dt) is split by about 0.18eV for the metallic (18, 0) zigzag nanotube, and this splitting is large enough to be observable by STS experiments. Although the trigonal warping effect is larger for metallic nanotubes than for semiconducting nanotubes of comparable diameters, the energy difference of the third peaks E3s3(dt) = 870 sin2(2a/3dt) between the (17, 0) and (19,0) zigzag nanotubes is about 0.63 eV, using an average dt value of 1.43 nm, which becomes easily observable in the experiments. These calculations show that the trigonal warping effect is important for metallic single wall zigzag nanotubes with diameters dt < 2nm. More direct measurements  of the chirality by the STM technique and of the splitting of the DOS by STS measurements on the same nanotube would provide very important confirmation of this prediction.

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