Physics of Carbon Nanotube Devices

Physics of Carbon Nanotube Devices

by Francois Leonard

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ISBN-13: 9780815519683
Publisher: Elsevier Science
Publication date: 11/18/2008
Series: Micro and Nano Technologies
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 300
File size: 12 MB
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THE PHYSICS OF CARBON NANOTUBE DEVICES


By François Léonard

Elsevier

Copyright © 2009 William Andrew Inc.
All rights reserved.
ISBN: 978-0-8155-1968-3



CHAPTER 1

Introduction


Carbon nanotubes are high aspect ratio hollow cylinders with diameters ranging from one to tens of nanometers, and with lengths up to centimeters. As the name implies, carbon nanotubes are composed entirely of carbon, and represent one of many structures that carbon adopts in the solid state. Other forms of solid carbon include for example diamond, graphite, and buckyballs. These many different forms arise because of the ability of carbon to form hybridized orbitals and achieve relatively stable structures with different bonding configurations. Carbon nanotubes exist because of sp2 hybridization, the same orbital structure that leads to graphite. In this chapter, we discuss the atomic and electronic structure of carbon nanotubes, and establish the basic nanotube properties that will be utilized in the following chapters on nanotube devices.


1.1 Structure of Carbon Nanotubes

To understand the atomic structure of carbon nanotubes, one can imagine taking graphite, as shown in Fig. 1.1, and removing one of the two-dimensional planes, which is called a graphene sheet; a single graphene sheet is shown in Fig. 1.2 (a). A carbon nanotube can be viewed as a strip of graphene (strip in Fig. 1.2) that is rolled-up to form a closed cylinder. The basis vectors [??]1 = a(√3, 0) and [??]2 = a(√3/2, 3/2) generate the graphene lattice, where a = 0.142 nm is the carbon–carbon bond length. The two atoms marked A and B in the figure are the two atoms in the unit cell of graphene. In cutting the rectangular strip, one defines a "circumferential" vector [??] = n]??]1 + m]??]2 corresponding to the edge of the graphene strip that will become the nanotube circumference. The nanotube radius is obtained from [??] as

R = C/2π = (√3/2π) an2 + m2 + nm. (1.1)


There are two special cases shown in Fig. 1.2 that deserve special mention. First, when the circumferential vector lies purely along one of the two basis vectors, the carbon nanotube is said to be of "zigzag" type. The example in Fig. 1.2 (a) shows the generation of a (10,0) zigzag nanotube. Second, when the circumferential vector is along the direction exactly between the two basis vectors, n = m, and the carbon nanotube is said to be of "armchair" type. The example in Fig. 1.2 (b) shows the generation of a (5,5) armchair nanotube, whereas in Fig. 1.2 (c) a chiral nanotube is shown where the strip is generated by mn.

In a planar graphene sheet, the bonds to the three nearest neighbors of a carbon atom, [??]1 = a(0, 1), [??]2 = a(√3/2, -1/2) and [??]3 = a(√3/2, -1/2) (Fig. 1.2 (a)) are equivalent. Rolling up a graphene sheet however causes differences between the three bonds. In the case of zigzag nanotubes the bonds oriented at a nonzero angle to the axis of the cylinder are identical, but different from the axially oriented bonds which remain unaffected upon rolling up the graphene strip. For armchair nanotubes the bonds oriented at a nonzero angle to the circumference of the cylinder are identical, but different from the circumferentially oriented bonds. All three bonds are slightly different for other chiral nanotubes.

We discussed the single wall nanotube, which consists of a single layer of rolled-up graphene strip. Nanotubes, however are found in other closely related forms and shapes as shown in Fig. 1.3. Fig. 1.3 (b) shows a bundle of single wall nanotubes with the carbon nanotubes arranged in a triangular lattice. The individual tubes in the bundle are attracted to their nearest neighbors via van der Waals interactions, with typical distances between nanotubes being comparable to the interplanar distance of graphite which is 3.1 [Angstrom]. The cross-section of an individual nanotube in a bundle is circular if the diameter is smaller than 15 [Angstrom] and deforms to a hexagon as the diameter of the individual tubes increases. A close allotrope of the single wall carbon nanotube is the multi wall carbon nanotube (MWNT), which consists of nested single wall nanotubes, in a Russian doll fashion as shown in Fig. 1.3 (c). Again, the distance between walls of neighboring tubes is comparable to the interplanar distance of graphite. Carbon nanotubes also occur in more complex shapes such as junctions between nanotubes of two different chiralities (Fig. 1.3 (d)) and Y-junctions (Fig. 1.3 (e)) These carbon nanotube junctions are atomically precise in that each carbon atom preserves its sp2 hybridization and thus makes bonds with its three nearest neighbors without introducing dangling bonds. The curvature needed to create these interesting shapes arises from pentagon–heptagon defects in the hexagonal network.


1.2 Electronic Properties of Carbon Nanotubes

Elemental carbon has six electrons with orbital occupancy 1s2 2s2 2p2, and thus has four valence electrons. The 2s and 2p orbitals can hybridize to form sp, sp2 and sp3 orbitals, and this leads to the different structures that carbon materials adopt. For example, the sp hybridization leads to linear carbon molecules, while the sp3 hybridization gives the diamond structure. The sp2 hybridization is responsible for the graphene and carbon nanotube structures. In graphene and nanotubes, each carbon atom has three 2sp2 electrons and one 2p electron. The three 2sp2 electrons form the three bonds in the plane of the graphene sheet (Fig. 1.4), leaving an unsaturated pz orbital (Fig. 1.2 (d)). This pz orbital, perpendicular to the graphene sheet and thus the nanotube surface, forms a delocalized π network across the nanotube, which is responsible for its electronic properties. These electronic properties can be well described starting from a tight-binding model for graphene, as we now discuss.


1.2.1 Graphene Electronic Structure

A carbon atom at position [??]s has an unsaturated pz orbital described by the wave function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the orthogonal tight-binding representation, the interaction between orbitals on different atoms vanishes unless the atoms are nearest neighbors. With H the Hamiltonian, this can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a vector connecting nearest-neighbors between the A and B sublattices (Fig. 1.2 (a)), and where we have set the on-site interaction energy to zero, without loss of generality. The tight-binding parameter γ represents the strength of the nearest-neighbor interactions. The A and B sublattices correspond to the set of all A and B atoms in Fig. 1.2 (a).

To calculate the electronic structure, we construct the Bloch wavefunction for each of the sublattices as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)


where s = A or B refers to each sublattice, and [??]s refers to the set of points belonging to sublattice s. The total wavefunction is then a linear combination of these two functions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)


where λ[??] is the mixing parameter to be determined below. The Hamiltonian matrix elements are calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

From Eq. (1.2) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

and the Hamiltonian matrix elements are therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.7)

leading to the Schrödinger equation H]??] = E]??] in matrix form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)

Diagonalization of this matrix leads to the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)


This band structure for graphene is plotted in Fig. 1.5 as a function of kx and ky. The valence and conduction bands meet at six points( [±4π/3√3a, 0); (±2π/3√3a, ±2π/3a)] at the corner of the first Brillouin zone. Graphene is thus a peculiar material: bands cross at the Fermi level, but the Fermi surface consists only of points in k-space, and the density of states at these so-called Fermi points vanishes. Graphene can be described as a gapless semiconductor, or a as a semi-metal with zero overlap. Because of the symmetry of the graphene lattice, the Brillouin zone has 2π/3 rotational symmetry, and there are only two nonequivalent Fermi points.

The values of λ[??] that correspond to the two branches in Eq. (1.9) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.10)


The lower branch, corresponding to lower energies, has opposite sign of the wavefunction on the two atoms of the unit cell while the wavefunction for the upper branch has the same sign on the two atoms of the unit cell. Thus the low energy branch has bonding character while the higher energy branch has antibonding character.

Near the Fermi points, the band structure can be approximated as

E ≈ 3γa/2 |[??] - [??]F| (1.11)


and is thus isotropic and linear. This relation will be useful to obtain the bandgap of semiconducting nanotubes as described in the next section.


1.2.2 Carbon Nanotube Electronic Structure

To obtain the electronic structure of carbon nanotubes, we start from the band structure of graphene and quantize the wavevector in the circumferential direction:

[??] · [??] = kxCx + kyCy = 2πp (1.12)


where [??], the circumferential vector, is shown in Fig. 1.2 and p is an integer. Eq. (1.12) provides a relation between kx and ky defining lines in the (kx, ky) plane. Each line gives a one-dimensional energy band by slicing the two-dimensional band structure of graphene shown in Fig. 1.5. The particular values of Cx, Cy and p determine where the lines intersect the graphene band structure, and thus, each nanotube will have a different electronic structure. Perhaps the most important aspect of this construction is that nanotubes can be metallic or semiconducting, depending on whether or not the lines pass through the graphene Fermi points. This concept is illustrated in Fig. 1.6 where the first Brillouin zone of graphene is shown as a shaded hexagon with the Fermi points at the six corners. In the left panel, the lines of quantized circumferential wavevectors do not intersect the graphene Fermi points, and the nanotube is semiconducting, with a bandgap determined by the two lines that come closer to the Fermi points. The right panel illustrates a situation where the lines pass through the Fermi points, leading to crossing bands at the nanotube Fermi level, and thus metallic character.
(Continues...)


Excerpted from THE PHYSICS OF CARBON NANOTUBE DEVICES by François Léonard. Copyright © 2009 William Andrew Inc.. Excerpted by permission of Elsevier.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

Introduction; Metallic Carbon Nanotubes for Current Transport; Physics of Nanotube/Metal Contacts; Electronic Devices; Electromechanical Devices; Field Emission; Optoelectronic Devices; Chemical and Biological Sensors; References; Index

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