Optical excitations of finite carbon nanotubes
Introduction
Carbon nanotubes have drawn much attention since they were discovered by Iijima in 1991 [1]. They could be used as components in the submicrometer-scaled devices and in nanocomposites. Each carbon nanotube is a rolled-up graphite sheet in the cylindrical form. The radius is in the range 3–100 Å, and the length has several micrometers. A long carbon nanotube is regarded as a one-dimensional quantum wire, owing to very small energy level spacings. The transition from a one-dimensional quantum wire to a zero-dimensional quantum dot could be achieved by cutting a long nanotube to a shorter nanotube. The scanning tunneling microscopy (STM) has been successfully utilized to produce finite carbon nanotubes with lengths ∼100 Å [2]. The reduction in length or dimensionality could induce many interesting physical phenomena, e.g., quantized standing waves [3], [4], [5], novel magnetic properties [6], and quantum transport properties [7]. The electronic structures have been calculated from the first-principles local-density approximation [4], [5] and the tight-binding model [6], [7], [8], [9], [10]. The zero-dimensional discrete states due to the finite size are expected to dominate the main features of optical excitations.
There are a lot of experimental and theoretical studies on optical properties of infinite carbon nanotubes [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. The experimental measurements [11], [12] show that optical excitations from the cross polarized light are very important in understanding absorption spectra, or very useful in identifying electronic structures. In this work, the π-electronic optical excitations of finite carbon nanotubes in the presence of the cross polarized light are studied within the gradient approximation [17], [18], [19], [20], [21], [22]. The dependence of the optical spectral function (A(ω)) on the length (h), the radius (r), the chiral angle (θ), and the magnetic flux (φ) is investigated in detail. The predicted results could be verified by the experimental measurements, as done for infinite carbon nanotubes [11], [12], [13], [14], [15], [16].
Section snippets
Theory
A finite carbon nanotube, as shown in Fig. 1(a), is formed by rolling a graphite sheet from the origin to the vector . and are the primitive lattice vectors of a graphite sheet. The nanotube axis is parallel to . The radius and chiral angle are, respectively, and . b=1.42 Å is the C–C bond length. The length is determined by the total number of carbon atoms (NA). For example, the length of the (m,m) armchair nanotube (the (m,0) zigzag
Results and discussion
The absorption spectrum of the (9, 9; 162) armchair carbon nanotube is shown in Fig. 2(a) by the solid curve. It is mainly determined by the number of excitation channels and the dipole matrix element. The former is associated with the joint density of states (JDOS) DJ(ω)=∑L[Γ/[Ec(L±1)−Ev(L)−ω]2+Γ2]. ωex=Ec(L±1)−Ev(L) is the inter-π-state excitation energy. As a result of the discrete feature of electronic states, JDOS exhibits many delta-function-like divergent peaks at ωex’s (Fig. 1(c)). Such
Conclusion
In conclusion, we have calculated absorption spectra for finite carbon nanotubes with various geometric structures. They exhibit rich peak structures due to the discrete inter-π-state excitations. The absorption peaks strongly depend on the length, the radius, the chiral angle, and the magnetic flux. They gradually group together in the increasing of length. They change from the delta-function-like peaks into the square-root divergent peaks at h→∞. The threshold excitation energy decreases in
Acknowledgements
This work was supported in part by the National Science Council of Taiwan, the Republic of China under Grant Nos. NSC 92-2112-M-006-014.
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