2. Fundamentals of Carbon Nanotubes

CARBON MATERIALS are found in a variety of forms such as fullerenes, graphite, carbon fibres, carbon nanotubes, and diamond. The reason why carbon assumes many structural forms is that a carbon atom can form several distinct types of orbital hybridization. The $sp^n$ hybridization is essential for determining the dimensionality of not only carbon based molecules but also carbon based solids. Carbon is the only element in the periodic table that has isomers from zero-dimensions to three-dimensions, see Table 2. In $sp^n$ hybridization, $(n+1)$ $\sigma$ bonds per carbon atom are formed, which form a skeleton for the local structure of the $n$-dimensional structure. In $sp$ hybridization, two $\sigma$ bonds form a one-dimensional chain structure, which is known as a carbyne. Interestingly, $sp^2$ hybridization, which forms a planer structure in two-dimensional graphite, also forms a planar local structure in the closed polyhedra of the (zero-dimensional) fullerene family and the one-dimensional cylinders called carbon nanotubes (CNTs). Carbon fibers which are macroscopic one-dimensional materials are closely related to CNTs, because of their characteristic high length to diameter ratio. A carbon fiber, however, consists of many graphite planes and microscopically exhibits electronic properties that are predominantly two-dimensional. Amorphous graphite, showing mainly $sp^2$ hybridization, consisting of randomly stacked graphite layer segments. Because of the weak inter-planer interaction between two graphite planes, they can move easily relative to each other, thereby forming a solid lubricant. In this sense, amorphous graphite can behave like a two-dimensional material. Four $\sigma$ bonds defining a regular tetrahedron are sufficient to form a three-dimensional structure known as the diamond structure. Amorphous carbon is a disordered, three-dimensional material in which both $sp^2$ and $sp^3$ hybridization is present.

CNTs are unique nano-structures that can be considered conceptually a prototype one-dimensional quantum wire. The fundamental building block of CNTs is the very long, all-carbon cylindrical single-wall CNT (SW-CNT), one atom in wall thickness and tens of atoms around the circumference (typical diameter $\sim\mathrm{1.4~nm}$). Initially, CNTs gained great interest in research community because of their exotic electronic properties, and this interest continued as other remarkable properties were discovered and promise of practical applications developed. In this chapter, a brief historical review of CNT research is presented and some basic definitions relevant to the structural properties of CNTs are provided. Finally, application of CNTs in electronics, especially CNT based transistors, are discussed.

Table 2.1: Important carbon isomers [12].

Dimension	0-D	1-D	2-D	3-D
Isomer	Fullerene	Nanotubes	Graphite	Diamond
Hybridization	$sp^2$	$sp^2$	$sp^2$	$sp^3$
Density $\mathrm{[g/cm^3]}$	$1.72$	$1.2-2.0$	$2.26$	$3.52$
Bond Length (Å)	$1.40$ (C=C)	$1.42$ (C=C)	$1.42$ (C=C)	$1.54$ (C-C)
Electronic Properties	Semiconductor $E_\mathrm{g}=\mathrm{1.9~eV}$	Semiconductor or Metal	Semi-metal	Insulator $E_\mathrm{g}=\mathrm{5.47~eV}$

Subsections

• 2.1 Historical Overview
• 2.2 Theoretical Background
• 2.3 Nanotube Growth Methods
  • 2.3.1 Arc-Discharge and Laser Ablation
  • 2.3.2 Chemical Vapor Deposition
  
  
• 2.4 Electronic Structure
  • 2.4.1 Electronic Band Structure of Graphene
  • 2.4.2 Electronic Band Structure of SW-CNTs
  
  
• 2.5 Phonon Properties
  • 2.5.1 Phonon Dispersion Relations of Graphene
  • 2.5.2 Phonon Dispersion Relations of SW-CNTs
  
  
• 2.6 Electron-Phonon Interaction
  • 2.6.1 Electron-Phonon Matrix Elements
  
  
• 2.7 Transport Properties
  • 2.7.1 Ballistic Transport
  • 2.7.2 Diffusive Transport
  
  
• 2.8 CNTs in Electronics
  • 2.8.1 Fabrication and Performance of CNT-FETs
  • 2.8.2 SCHOTTKY Barrier Model of CNT-FET Operation
  • 2.8.3 Environmental Influences on the Performance of CNT-FETs
  • 2.8.4 Prototype CNT-FET Circuits

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors