One Dimensional Band Structure of Nanotubes

The characterization of semiconducting and metallic SWNTs with subtle changes in structure confirms the remarkable electronic behavior of the nanotubes, and represents a significant step forward in understanding these 1D materials. In addition, the ability to probe simultaneously atomic structure and electronic properties provides a unique opportunity to investigate further several interesting properties of these 1D materials. These properties include, for example, the detailed DOS of the nanotubes, the role of symmetry breaking distortions on the electronic character of nanotubes, and the electronic properties of defect and end electronic states. Several of these interesting issues are addressed in the following sections and are compared quantitatively with theory.

By making tunneling spectroscopy measurements over an extended energy range, the one-dimensional nature of the energy bands are observed as sharp peaks in the DOS [10,24,25]. When spectroscopic measurements are made on atomically-resolved nanotubes, it is also possible to compare the experimental DOS quantitatively with that resulting from a simple ^-only tight-binding calculation. Kim et al. [25] reported the first detailed experimental comparison with theory on a metallic tube with indices (13,7). This nanotube is the upper isolated tube that rests on the Au surface shown in Fig. 5a. Current vs. voltage measurements exhibited a linear response at V = 0 as expected for a metal and showed steps at larger voltages that correspond to a series of sharp peaks in the dl/dV. These peaks correspond to the VHS resulting from the extremal points in the 1D energy bands.

A direct comparison of these experimental data to the theoretical electronic band structure calculated by a n-only tight-binding model was made [25]. Significantly, the spectroscopy data show good agreement with the calculated DOS for the (13,7) tube (Fig. 5b). The agreement between the VHS positions determined from the calculations and dl/dV data are especially good below EF, where the first seven peaks correspond well. The peak splitting due to the anisotropy around K is also reproduced in the dl/dV. Notably, the experimental gap between the first VHS in this metallic tube, Eg2 ~ 1.6 eV, is in agreement with predictions for metallic tubes [26]; that is, EgP1 = 6Yoac-c/d = 1.6eV, where Yo = 2.5 eV, the value determined from the semiconducting energy gap data [11]. Above the Fermi energy some deviation between the experimental data and calculations exist, but the observed differences may be due to band repulsion, which arises from curvature-induced hybridization [27].

Kim et al. [25] also compared their results to a published a + n calculation for a (13,7) SWNT [28] and a n-only calculation for a closely related

Fig. 5. STM imaging and spectroscopy on a metallic SWNT. (a) Tunneling spectra were recorded on the isolated upper tube. The inset shows an atomic resolution image of this tube. A portion of a hexagonal lattice is overlaid to guide the eye. (b) Comparison of the DOS obtained from experiment (upper curve) and a n-only tight-binding calculation for the (13,7) SWNT (second curve from, top). The broken vertical lines indicate the positions of VHS in the tunneling spectra after consideration of thermal broadening convolution. The calculated DOS for a (12,6) tube is included for comparison [25]

Fig. 5. STM imaging and spectroscopy on a metallic SWNT. (a) Tunneling spectra were recorded on the isolated upper tube. The inset shows an atomic resolution image of this tube. A portion of a hexagonal lattice is overlaid to guide the eye. (b) Comparison of the DOS obtained from experiment (upper curve) and a n-only tight-binding calculation for the (13,7) SWNT (second curve from, top). The broken vertical lines indicate the positions of VHS in the tunneling spectra after consideration of thermal broadening convolution. The calculated DOS for a (12,6) tube is included for comparison [25]

set of indices. Although detailed comparison is difficult due to the large DOS broadening, all peaks within ± 2eV match well with the n-only calculation. This comparison suggests that curvature-induced hybridization is only a small perturbation within the experimental energy scale (\ V \< 2 V) for the (13,7) tube. The sensitivity of the VHS to variations in the (n, m) indices was investigated by calculating the DOS of the next closest metallic SWNT to the experimental diameter and angle; that is, a (12,6) tube. Significantly, the calculated VHS for this (12,6) tube deviate much more from the experimental DOS peaks than in the case of the (13,7) tube. It is worth noting that the poor agreement in this case demonstrates that subtle variations in diameter and helicity do produce experimentally distinguishable changes in the DOS.

Van Hove singularities in the electronic DOS of semiconducting nano-tubes have also been observed [10,29]. Odom et al. [29] have characterized spectroscopically a small-diameter (10,0) semiconducting nanotube (Fig. 6a), and directly compared the DOS with a tight-binding calculation.

The normalized conductance exhibits relatively good agreement with the calculated (10,0) DOS below EF but poorer agreement above (Fig. 6b). However, the n-only DOS calculation does not include n/a and n*/a* mixing due to curvature. This hybridization of n/a orbitals is believed to produce more pronounced effects on the conduction band [27], and this might explain the observed deviations. Additional work is needed to resolve this point. These results show clearly that the VHS peaks in the electronic band structure, which are characteristic of 1D systems, can be measured experimentally and agree well with the DOS calculated using n-only tight-binding models.

Fig. 6. STM image and spectroscopy of a semiconducting nanotube. (a) Image of a SWNT on the surface of a rope. (b) Comparison of the DOS obtained from experiment (upper curve) and calculation for the (10,0) SWNT (lower curve) [29]

Fig. 6. STM image and spectroscopy of a semiconducting nanotube. (a) Image of a SWNT on the surface of a rope. (b) Comparison of the DOS obtained from experiment (upper curve) and calculation for the (10,0) SWNT (lower curve) [29]

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