Introduction to the Properties of Carbon Nanotubes

Carbon atoms can bond with their neighboring atoms to create sheets (graphene), cylinders (nanotubes), and balls (fullerenes) with unique properties. This article provides the minimum information required for an electrical engineer to parse a scholarly article on single-wall carbon nanotubes.

A note about the information presented in this article: In an effort to keep the article as reader-friendly to an electrical engineering audience as possible, I have taken certain liberties that an advanced reader might object to. Please feel free to bridge any gaps you might feel I have left for the reader in the comments below.

Purpose and Introduction

Since their rediscovery in 1991, scientists have been exploring the chemical, physical, and material properties of carbon nanotubes. 1998 saw the first field-effect transistors and, by 2003, NEC had used them to create transistors with an order of magnitude greater conductance than silicon MOSFETs. In 2013, scientists succeeded in created a functioning 20-instruction carbon nanotube computer.
This article attempts to provide the background information needed to better understand what nanotubes are and the terminology used to describe them so that you can approach scholarly articles that interest you. This article assumes the reader has not taken physical chemistry or condensed matter physics.

This website also has some high-quality visualizations: http://toutestquantique.fr/en/

Building Blocks

Chemistry of Carbon Atoms

Neutral carbon atoms have six electrons that surround the nucleus and are held in the atom by the electric force. Electrons that are far from the nucleus require an order-of-magnitude less energy to remove than electrons that lie close to the nucleus.
Since it requires less energy to remove the outermost electrons, it should come as no surprise that the outermost electrons (valence electrons) are the ones that interact with other atoms.

Electrons that are closer to the nucleus must acquire progressively greater amounts of energy to be able to leave their parent atom, creating an ion. Data from Wolfram Alpha

 

Quantum Theory

Electrons move in unique repetitious patterns in the region of space outside the nucleus of an atom in orbitals that can be described and predicted with mathematical wave functions that are defined by the unique properties of individual electrons. Each orbit can be occupied by a single electron and each orbit has a certain amount of energy associated with it—two electrons never occupy the same orbital, but they may have identical energies. Electrons must absorb energy to be able to move to a higher orbital, and if they absorb enough energy, they can leave the atom entirely. Electrons tend to spontaneously move to lower energy orbitals if space is available for them, and as they do, they will expel the extra energy as a photon, i.e., a particle of electromagnetic energy (which is sometimes visible light).

Electrons are found in the region of space that defines their specific orbitals, never in between, a fact confirmed in part by examining the emission spectrum of carbon. These spectral lines correspond to photons that have an energy equal to the energy difference of two orbitals.

Visible carbon emission spectrum. Photons of a particular wavelength are emitted as electrons move to lower energy orbitals in carbon atoms. Image courtesy of Wolfram Alpha.

In the lowest energy configuration of carbon, two electrons fill the 1s orbital closest to the nucleus, two electrons fill the 2s orbital, and the remaining two electrons fill the 2p orbital. The outermost four electrons are available to bond with other carbon atoms. By absorbing a small amount of energy, the wave functions change and new hybrid orbitals are formed.

Electron energy diagrams for ground state, sp², and sp³ hybrid orbitals. Electrons fill the lowest energy orbitals first. But with the addition of small amounts of energy (~1-2 eV), electrons can move to create hybridized orbitals when the 2s and 2p orbitals merge.

3D hybrid orbital models. Note the sp² trigonal planar and sp³ tetrahedron shapes. Image by Mark Hughes.

 

Carbon-Carbon Bonds

Carbon atoms can bond with three or four neighboring atoms to create carbon allotropes with unique material properties.
A carbon atom can bond:

• With the nearest four carbon atoms in a tetrahedral pattern (sp³ hybrid orbital) to create the 3D crystalline structure diamond
• With the three nearest carbon atoms in a trigonal planar pattern (sp² hybrid orbital) to create the 2D crystal lattice structure graphene
• With three nearest carbon atoms in a new shape (sp² and sp³ mixed state) to create q-carbon, nanotubes, and fullerenes

Two neutral carbon atoms separated by several atomic widths will have identical energy levels and subsequently identical spectral lines. As they are brought close together, the electrons begin to influence each other as their orbital wave functions begin to overlap.

As carbon atoms are linked together in a crystal, more electrons move in between atoms, obeying the rule that the electron may only take a path that does not cause destructive interference and only one electron can exist in a particular orbital at a time. S orbitals split into twice as many orbitals as there are atoms (one for each electron in the S-orbitals), some with higher energy and some with lower energy than before. P orbitals split into six times as many orbitals (Px, Py, Pz) as there are atoms, again, some with higher energy, and some with lower energy.

Additional orbits, that obey the rules that no two electrons can be in the same orbital and destructive interference cannot occur, are able to come into existence. The spread in energy of these new orbitals is on the order of just a few electron volts and inversely related to the distance between atoms. Given that there are 10²³ atoms in a mole, there are 10²³ energy levels in the 1s shell alone that all have unique energies in the range of 10⁰ eV; that leaves an unimaginably high number of unique energies tightly packed into a small range.

Artistic interpretation of increased spectral lines as additional carbon atoms are brought near each other. Electrons only exist in allowed bands of energy. There are no electrons in the forbidden bands between allowed bands. Image by Mark Hughes

What begins as discrete energy levels of individual atoms quickly broadens into energy bands as multiple atoms are linked together. The space between bands where no electrons are allowed is called the bandgap. It is impossible and unnecessary to identify individual electron orbitals when they are so tightly spaced.

Valence Band and Conduction Band

A material can only gain energy if it is the amount necessary to move from one energy band to a nearby band with empty energy levels. If the band containing the valence electrons is full (or the amount of energy introduced is less than needed to move between bands), the material will act as an insulator. If the band containing the valence electrons has so many available levels that it cannot possibly be filled, the material acts as a conductor.
Electrons can gain energy to move between bands through the application of an external electric field, through the absorption of a photon, or through thermal energy at temperatures above absolute zero. If the bandgap is small (~10⁰ eV), thermal energy is sufficient to allow electrons to spontaneously jump to a higher energy band, giving the material the properties of a semiconductor. The nearly filled band the electrons leave is called the valence band and the nearly empty band they jump to is called the conduction band.
If a large gap exists between the valence and conduction band, electrons are not free to move between energy bands and the material is an insulator. If the energy levels broaden and the bandgap is nonexistent (valence and conduction bands overlap), the material acts as a metal. If the gap is small but still present, it functions as a semiconductor.

Image showing the valence and conduction bands of generic materials. Image created by Nanite

Single-wall carbon nanotube zone folding band structure graphs: (a) a metal (overlapping orbitals), (b) a semi-metal (small overlap), and (c) a semiconductor (small gap). Image courtesy of InTechOpen

Graphene to Nanotube

Graphene sheets are two-dimensional, honeycomb-shaped arrays of carbon atoms with each carbon atom bonded with three neighboring atoms in a repeating hexagonal array.
Start at any given carbon atom in a hexagonal arrangement (e.g., top center) and you can move (n, m) units away to find another carbon atom with an identical geometry in the hexagonal arrangement (e.g., top center). Move to the right in a zig-zag pattern and you will find an identical arrangement of carbon atoms which increases only with the index n. Move 30° down and to the right in an arm-chair pattern, and you will again find an identical arrangement of carbon atoms where n=m.
The distance and direction to the first repetitions are called lattice vectors and the multipliers (n, m) are the indices.
Carbon atoms in this arrangement have electrons in sp² hybrid orbitals (planar trigonal) with the single electron in the 2p¹ orbital disassociated from its parent atom and free to move around the crystal lattice.

Image of graphene sheet chiral index by Mark Hughes

Nanotubes are described by their chiral index (n,m). Imagine rolling a sheet of graphene into a cylinder around a given axis so that a single carbon atom is mapped on top of another carbon atom at position (n,m). This creates a repetitious pattern of carbon atoms whose electron orbitals interact. It's no longer an sp² trigonal planar shape, but it is not quite an sp³ tetrahedron shape. The orbitals are in a mixed sp²-sp³ shape but still with an electron disassociated from the parent atom that moves freely between atoms.

You can visualize a nanotube as a rolled-up sheet of graphene. As an example: Imagine rolling the image above to form a cylinder so that the carbon atom at the (0,0) position overlaps and replaces the carbon atom with position (6,6); this would be described as an armchair [6,6] single-wall carbon nanotube.
It is worth noting that nanotubes are not manufactured by rolling graphene sheets. This is merely a visualization tool for describing groups of nanotubes and predicting their properties.

Nanotube Properties

An electron that moves around the circumference of a nanotube has a different wave function than an electron that moves along the length of the tube. The nanotube then has varying characteristics around the circumference, parallel to the axis, and laterally through the axis.

Thermal Conductivity	Along Length	Through Axis
Single Wall Nanotube	3500Wm⋅K

1.5Wm⋅K

Copper	400Wm⋅K

Aluminum	235Wm⋅K

Gold	320Wm⋅K

Single-wall carbon nanotubes are ~10 times more thermally conductive along their length than copper, aluminum, or gold. In the lateral direction across their axis, they are ~150-250 times less thermally conductive than copper, aluminum, or gold.

 

Current Density	Along Length	Through Axis
Single Wall Nanotube	107−1013Acm2

	negligible
Copper	5×102−5×103Acm2

The current density of carbon nanotubes is as much as ~10 orders of magnitude higher than that of copper.

 

More about the properties of carbon nanotubes can be found in the book Physical Properties of Carbon Nanotubes.

Applications and Conclusion

Carbon nanotubes are likely already in your home. Researchers have already created transistors, computers, televisions, and cancer-detecting chips with carbon nanotubes.
If you are at the end of your career, it is unlikely that you will purchase a nanotube semiconductor for incorporation into a new product design. But if you are an entry-level or mid-career engineer, you will be able to purchase nanotube-based NRAM from Fujitsu as early as 2018 and the market should continue to grow from there.

Image courtesy of Nantero.

This article avoided the use of technical terms wherever possible to appeal to a wider audience. Some of the technical terms that are used in relation to carbon nanotubes can be found by clicking the links scattered throughout the article.