Optoelectronics with Carbon Nanotubes

An SWNT can be defined as a cylinder of a rolled up graphene sheet, which is a
monolayer of carbon atoms in a honeycomb structure (Figure I-1). Note that CNTs are not
actually manufactured from graphene and that this is a conceptual convenience; see the methods
section for details of carbon nanotube synthesis. In graphene, each carbon atom has four valence
electrons (i.e., 2s, 2px, 2py and 2pz), three of which (2s, 2px, and 2py) are sp 2 hybridized and form
covalent bonds with the nearest neighbor atoms, making graphene a very stable structure.
Graphite is many sheets of graphene that are stacked, while another allotrope, diamond, forms
covalent bonds with all four valence electrons with four neighboring atoms. Carbon nanotubes’
exceptional tensile strength is also due to the covalent bonds. The remaining fourth 2pz electron
in graphene/carbon nanotube is free and contributes to conduction.
The way in which an SWNT is “rolled up” from graphene defines its structure and
determines the electronic properties. Figure I-1 shows the graphene honeycomb structure in real
space with unit vectors a1 and a2, and the chiral vector C for a particular SWNT (i.e., a (4, 5)
SWNT). The structure can be visualized as a strip of graphene cut along the dotted lines
perpendicular to C and rolled up in the direction of C into a long cylinder so that the length |C|
becomes the circumference. The chiral vector uniquely defines a SWNT and can be expressed as
C nama (Eq. I.1)
1 2
where n and m are integers. The indices (n, m) are sufficient to define any SWNT; the figure
shows, for example, the (4, 5) nanotube. There are two specific types of nanotubes called
“armchair” where n = m, and “zigzag” where m = 0, named after the pattern of lines connecting
the atoms seen around the circumference. Armchair nanotubes are always metallic, but zigzag
tubes can be either semiconducting or metallic (see below). These are useful structures that
appear frequently in theoretical work because of their high degree of symmetry that simplifies
calculations. In practice, SWNTs are a family of many different structures with varied electronic
and optical properties, the fact that poses interesting and often challenging problems. For
example, there is little control over what types of SWNTs are produced in any fabrication
method. Although there is usually a limited range of diameters and the number of sidewalls (i.e.,
SWNT vs. MWNT) depending on the growth conditions and catalyst particles, the product is
always a mixture of metallic and semiconducting CNTs of different diameters, chiralities and
lengths, which makes it difficult to control their properties and identify their (n, m) designations.
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