Nanotubes electronic structure

Compared to the vibrational spectroscopy of carbon nanotubes, their absorption and luminescence spectroscopy kind of lives in the shadows. This is, however, not due to a lack of information these methods could provide to the understanding of the nanotubes electronic structure. There are rather experimental complications that arise from the inhomogeneity of the available materials. 

Sumpter, B. G., D.-e. Jiang, and V. Meunier. New insight into carbon-nanotube electronic-structure selectivity. Small 4, 2008 2035-2042. 

Structurally, carbon nanotubes of small diameter are examples of a onedimensional periodic structure along the nanotube axis. In single wall carbon nanotubes, confinement of the stnreture in the radial direction is provided by the monolayer thickness of the nanotube in the radial direction. Circumferentially, the periodic boundary condition applies to the enlarged unit cell that is formed in real space. The application of this periodic boundary condition to the graphene electronic states leads to the prediction of a remarkable electronic structure for carbon nanotubes of small diameter. We first present... 

As the nanotube diameter increases, more wave vectors become allowed for the circumferential direction, the nanotubes become more two-dimensional and the semiconducting band gap disappears, as is illustrated in Fig. 19 which shows the semiconducting band gap to be proportional to the reciprocal diameter l/dt. At a nanotube diameter of dt 3 nm (Fig. 19), the bandgap becomes comparable to thermal energies at room temperature, showing that small diameter nanotubes are needed to observe these quantum effects. Calculation of the electronic structure for two concentric nanotubes shows that pairs of concentric metal-semiconductor or semiconductor-metal nanotubes are stable [178]. 

Experimental measurements to test the remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because... 

Abstract—The fundamental relations governing the geometry of carbon nanotubes are reviewed, and explicit examples are pre.sented. A framework is given for the symmetry properties of carbon nanotubes for both symmorphic and non-symmorphic tubules which have screw-axis symmetry. The implications of symmetry on the vibrational and electronic structure of ID carbon nanotube systems are considered. The corresponding properties of double-wall nanotubes and arrays of nanotubes are also discussed. 

Of particular importance to carbon nanotube physics are the many possible symmetries or geometries that can be realized on a cylindrical surface in carbon nanotubes without the introduction of strain. For ID systems on a cylindrical surface, translational symmetry with a screw axis could affect the electronic structure and related properties. The exotic electronic properties of ID carbon nanotubes are seen to arise predominately from intralayer interactions, rather than from interlayer interactions between multilayers within a single carbon nanotube or between two different nanotubes. Since the symmetry of a single nanotube is essential for understanding the basic physics of carbon nanotubes, most of this article focuses on the symmetry properties of single layer nanotubes, with a brief discussion also provided for two-layer nanotubes and an ordered array of similar nanotubes. 

Regarding the electronic structure, the number of energy bands for ( ,0) zigzag carbon nanotubes is In, the number of carbon atoms per unit cell, with symmetries... 

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the optimal spacing between them. Figure 5 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc P6/mmni Dh,), and P6/mcc... 

Key Words —Carbon nanotube, electronic properties, structural properties, strain energy, band gap, band structure, electronic structure. 

Before we can analyze the electronic structure of a nanotube in terms of its helical symmetry, we need to find an appropriate helical operator S>(h,ip), representing a screw operation with a translation h units along the cylinder axis in conjunction with a rotation if radians about this axis. We also wish to find the operator S that requires the minimum unit cell size (i.e., the smallest set of carbon atoms needed to generate the entire nanotube using S) to minimize the computational complexity of calculating the electronic structure. We can find this helical operator by first... 

We will now discuss the electronic structure of single-shell carbon nanotubes in a progression of more sophisticated models. We shall begin with perhaps the simplest model for the electronic structure of the nanotubes a Hiickel model for a single graphite sheet with periodic boundary conditions analogous to those im-... 

The previous analysis of the electronic structure of the carbon nanotubes assumed that we could neglect curvature effects, treating the nanotube as a single... 

Now, let us return to our discussion of carrying out an electronic structure calculation for a nanotube using helical symmetry. The one-electron wavefunc-tions can be constructed from a linear combination of Bloch functions linear combination of nuclear-centered functions Xj(r),... 

As an example of a nanotube representative of the diameters experimentally found in abundance, we have calculated the electronic structure of the [9,2] nanotube, which has a diameter of 0.8 nm. Figure 8 depicts the valance band structure for the [9,2] nanotube. This band structure was calculated using an unoptimized nanotube structure generated from a conformal mapping of the graphite sheet with a 0.144 nm bond distance. We used 72 evenly-spaced points in the one-... 

Experimental measurements to test these remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because of the strong dependence of the predicted properties on tubule diameter and chirality. Ideally, electronic or optical measurements should be made on individual single-wall nanotubes that have been characterized with regard to diameter and chiral angle. Further ex-... 

Electronic Structures of Single-Walled Carbon Nanotubes... 

Studies on the electronic structure of carbon nanotube (CNT) is of much importance toward its efficient utilisation in electronic devices. It is well known that the early prediction of its peculiar electronic structure [1-3] right after the lijima s observation of multi-walled CNT (MWCNT) [4] seems to have actually triggered the subsequent and explosive series of experimental researches of CNT. In that prediction, alternative appearance of metallic and semiconductive nature in CNT depending on the combination of diameter and pitch or, more specifically, chiral vector of CNT expressed by two kinds of non-negative integers (a, b) as described later (see Fig. 1). 

Yang, X. and Dong, J. (2005) Geometrical and electronic structures of the (5, 3) single-walled gold nanotube from first-principles calculations. Physical Review B -Condensed Matter, 71,233403-1-233403-4. 

D.L. Carroll, X. Blase, J.C. Charlier, P. Redlich, P.M. Ajayan, S. Roth, and M. Ruhle, Effects of nanodomain formation on the electronic structure of doped carbon nanotubes. Phys. Rev. Lett. 81, 2332-2335 (1998). 

M.S. Strano, C.A. Dyke, M.L. Usrey, P.W. Barone, MJ. Allen, H. Shan, C. Kittrell, R.H. Hauge, J.M. Tour, R.E. Smalley, Electronic structure control of single-walled carbon nanotube functionalization. Science 301, 1519-1522 (2003). 

Arnold, M. S. Green, A. A. Hulvat, J. F. Stupp, S. I. Hersam, M. C. 2006. Sorting carbon nanotubes by electronic structure using density differentiation. Nature Nanotechnol. 1 60-65. 

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