12.3 Carbon nanotubes and graphene

Structure of carbon nanotubes

Carbon nanotubes (CNTs) are a cylindrical allotrope of carbon that can be thought of as graphene sheets rolled into seamless tubes. This structural relationship to graphene is more than a metaphor: the electronic and mechanical properties of CNTs derive directly from the properties of the parent graphene lattice, modified by the geometry of rolling.

Cylindrical shape

CNTs have a hollow cylindrical shape with an extremely high aspect ratio. Diameters range from about 1 nm to tens of nanometers, while lengths can extend to several centimeters. This combination of nanoscale width and macroscale length gives CNTs many of their useful properties.

Single-walled vs multi-walled nanotubes

• Single-walled nanotubes (SWNTs) consist of one graphene layer rolled into a cylinder, with typical diameters of 0.8–2 nm.
• Multi-walled nanotubes (MWNTs) are made of multiple concentric graphene cylinders nested inside each other, with an interlayer spacing of approximately 0.34 nm (similar to the layer spacing in graphite). MWNT diameters range from about 2–100 nm.

SWNTs are more interesting for fundamental physics because their properties depend sensitively on chirality, while MWNTs tend to be easier to produce in bulk and are mechanically robust.

Chirality and diameter

The chirality of a nanotube describes how the graphene sheet is oriented relative to the tube axis. It's specified by the chiral vector $(n, m)$, which tells you the direction along which the graphene sheet is "rolled up."

The chiral vector determines the nanotube diameter through:

$d_t = \frac{a}{\pi}\sqrt{n^2 + nm + m^2}$

where $a \approx 0.246$ nm is the graphene lattice constant.

Chirality is not just a geometric detail. It directly controls whether the nanotube is metallic or semiconducting, which is one of the most striking structure-property relationships in nanoscience.

Electronic properties of carbon nanotubes

The electronic structure of CNTs comes from graphene's band structure, but with an added constraint: electrons are confined around the circumference of the tube. This quantum confinement in one dimension creates discrete allowed momentum values around the tube while leaving the axial direction continuous, producing a quasi-one-dimensional electronic system.

Metallic vs semiconducting nanotubes

Whether a nanotube is metallic or semiconducting depends entirely on its chirality $(n, m)$:

• Metallic if $n - m = 3k$, where $k$ is an integer (including zero)
• Semiconducting otherwise

This means roughly one-third of all possible chiralities produce metallic tubes and two-thirds produce semiconducting tubes. Metallic nanotubes have a finite density of states at the Fermi level, giving them high conductivity. Semiconducting nanotubes have a band gap, making them candidates for transistor channels.

Band structure and density of states

The band structure is derived by "slicing" graphene's 2D band structure with the allowed quantized momentum values around the circumference. Each allowed slice produces a 1D sub-band.

A key consequence of this 1D confinement is the appearance of van Hove singularities: sharp peaks in the density of states at the onset of each sub-band. These singularities are directly observable using scanning tunneling spectroscopy (STS) and are responsible for strong optical absorption at specific wavelengths.

The energy spacing between symmetric van Hove singularities scales inversely with diameter:

$E_{ii} = \frac{2a_0 \gamma_0}{d_t}$

where $a_0 \approx 0.142$ nm is the C–C bond length, $\gamma_0 \approx 2.7$ eV is the nearest-neighbor hopping energy, and $d_t$ is the nanotube diameter.

Ballistic transport and conductivity

In conventional conductors, electrons scatter frequently off defects and phonons. In high-quality CNTs, the mean free path can reach several micrometers, meaning electrons travel the entire length of a short nanotube without scattering. This is ballistic transport.

• Metallic CNTs can achieve conductivities on the order of $10^9$ S/m, comparable to copper.
• The combination of ballistic transport, high current-carrying capacity, and nanoscale diameter makes CNTs strong candidates for interconnects in future nanoelectronics.

Synthesis methods for carbon nanotubes

Three main methods are used to synthesize CNTs. Each involves vaporizing or decomposing a carbon source and allowing nanotubes to nucleate and grow, but they differ in energy source, scalability, and product quality.

Arc discharge

1. Two graphite electrodes are placed in an inert atmosphere (typically helium).
2. A high-current arc is struck between them, reaching temperatures above 3000°C.
3. Carbon vaporizes from the anode and deposits on the cathode as a mixture of SWNTs and MWNTs.

Arc discharge produces high yields but the product contains significant impurities (amorphous carbon, catalyst residues) that require post-synthesis purification.

Laser ablation

1. A graphite target doped with metal catalyst (Ni, Co) is placed in an inert gas flow inside a furnace.
2. A high-power pulsed laser vaporizes the target.
3. The vaporized carbon condenses in the presence of catalyst particles, forming primarily SWNTs.

Laser ablation produces high-quality SWNTs with a narrow diameter distribution, but at lower yields than arc discharge or CVD.

Chemical vapor deposition

1. Metal catalyst nanoparticles (Fe, Co, or Ni) are deposited on a substrate.
2. A carbon-containing gas (methane, ethylene, or acetylene) flows over the catalyst at 600–1200°C.
3. The gas decomposes on the catalyst surface, and carbon atoms assemble into nanotubes that grow from the catalyst particles.

CVD is the most widely used method because it offers control over nanotube diameter, length, and alignment by tuning temperature, pressure, gas composition, and catalyst size. It's also the most scalable approach for industrial production.

Applications of carbon nanotubes

Field emission displays

CNTs are excellent field emitters because their sharp tips and high aspect ratio concentrate electric fields, allowing electrons to tunnel out at relatively low applied voltages. CNT-based field emission displays (FEDs) were explored as alternatives to LCDs, offering low power consumption and fast response times. The main challenge has been achieving uniform emission across large arrays of nanotubes.

Nanosensors and nanoelectronics

• Sensors: The high surface-to-volume ratio of CNTs makes them extremely sensitive to adsorbed molecules. A single molecule binding to a CNT can measurably change its conductance, enabling gas sensors, chemical sensors, and biosensors.
• Transistors: Semiconducting SWNTs can serve as the channel in field-effect transistors (FETs). Their high carrier mobility and nanoscale diameter make them attractive for next-generation logic devices.
• Interconnects: Metallic CNTs are being investigated as replacements for copper wiring at the nanoscale, where copper suffers from increased resistivity due to surface scattering.

Composite materials and nanofibers

Adding even small amounts of CNTs to polymer, ceramic, or metal matrices can dramatically improve mechanical properties like tensile strength, elastic modulus, and fracture toughness. CNT-reinforced composites are being developed for lightweight structural materials, energy storage electrodes, and filtration membranes. CNT-based nanofibers produced by electrospinning combine high specific surface area with excellent mechanical performance.

Structure of graphene

Graphene is a single atomic layer of carbon atoms arranged in a two-dimensional hexagonal lattice. It serves as the conceptual building block for other carbon allotropes: stack graphene layers and you get graphite, roll one into a cylinder and you get a nanotube, wrap one into a sphere and you get a fullerene.

Honeycomb lattice

Each carbon atom in graphene is sp²-hybridized, forming three strong $\sigma$ bonds with its neighbors in the plane at 120° angles. The remaining $p_z$ orbital on each atom points perpendicular to the sheet and overlaps with neighboring $p_z$ orbitals to form a delocalized $\pi$ electron network across the entire sheet. This delocalized network is what gives graphene its remarkable electrical conductivity.

Monolayer vs bilayer graphene

• Monolayer graphene has a linear (conical) dispersion relation near the Dirac points, producing massless Dirac fermions with no band gap.
• Bilayer graphene (two layers, typically in AB/Bernal stacking) has a quadratic dispersion relation. Crucially, an external perpendicular electric field can open a tunable band gap in bilayer graphene, which monolayer graphene lacks.

The number of layers and their stacking order significantly modify the electronic properties, so distinguishing monolayer from few-layer graphene matters for device applications.

Lattice constants and bond lengths

• Lattice constant: $a \approx 2.46$ Å (the distance between equivalent atoms in neighboring unit cells)
• C–C bond length: approximately 1.42 Å, intermediate between a single bond (1.54 Å in diamond) and a double bond (1.34 Å in ethylene)

The short, strong bonds contribute to graphene's extraordinary mechanical strength: its intrinsic tensile strength is about 130 GPa, making it the strongest material ever measured.

Electronic properties of graphene

Graphene's electronic properties are among the most studied in condensed matter physics. They arise from the symmetry of the honeycomb lattice and the resulting band structure.

Linear dispersion relation

Near the six corners of the Brillouin zone (the K and K' points, also called Dirac points), the energy-momentum relationship is linear:

$E = \hbar v_F k$

where $v_F \approx 10^6$ m/s is the Fermi velocity and $k$ is the wave vector measured from the Dirac point.

This is the same form as the relativistic energy-momentum relation for massless particles, except with $v_F$ replacing the speed of light. Charge carriers in graphene therefore behave as massless Dirac fermions. This leads to several unusual phenomena:

• The half-integer quantum Hall effect, observed even at room temperature
• Suppression of backscattering, which contributes to high mobility
• Klein tunneling, where carriers can pass through potential barriers with near-unity probability

High carrier mobility

Graphene's carrier mobility can exceed 200,000 cm²/(V·s) at room temperature in suspended samples, far surpassing silicon (~1,400 cm²/(V·s)) and even III-V semiconductors like GaAs (~8,500 cm²/(V·s)). This extraordinary mobility results from weak electron-phonon coupling and the suppression of backscattering by the honeycomb lattice symmetry.

High mobility translates directly to fast device operation, making graphene attractive for high-frequency transistors and photodetectors.

Ambipolar electric field effect

Graphene is ambipolar: by applying a gate voltage, you can continuously shift the Fermi level from below the Dirac point (hole conduction) through the Dirac point (minimum conductivity) to above it (electron conduction). There's no need to chemically dope the material to switch carrier type.

This tunability enables novel device architectures such as electrostatically defined p-n junctions and frequency multipliers, and it's the basis for graphene's use in sensors where small changes in the chemical environment shift the Dirac point.

Synthesis methods for graphene

Mechanical exfoliation

The original method for isolating graphene, famously demonstrated by Geim and Novoselov in 2004 using adhesive tape:

1. Press adhesive tape onto highly oriented pyrolytic graphite (HOPG).
2. Peel repeatedly to thin the graphite flakes.
3. Press the tape onto a $\text{SiO}_2$/Si substrate and peel away.
4. Identify monolayer flakes using optical microscopy (the ~300 nm oxide thickness creates visible contrast for single layers).

This produces the highest-quality graphene with minimal defects, but flakes are small (typically a few to tens of micrometers) and the process is not scalable.

Epitaxial growth on SiC

1. A silicon carbide (SiC) substrate is heated above 1300°C in ultra-high vacuum or an argon atmosphere.
2. Silicon atoms sublimate preferentially from the surface.
3. The remaining carbon atoms rearrange into graphene layers on the SiC surface.

This method produces wafer-scale graphene directly on a semi-insulating substrate, which is convenient for electronics. The drawbacks are the high cost of SiC wafers and the strong coupling between the first graphene layer and the substrate, which can modify its electronic properties.

Chemical vapor deposition on metals

CVD on metal foils (especially copper) is the dominant method for producing large-area graphene:

1. A copper foil is heated to ~1000°C in a low-pressure furnace.
2. A carbon precursor gas (typically methane mixed with hydrogen) flows over the surface.
3. Methane decomposes on the copper surface, and carbon atoms form a graphene film.
4. Growth on copper is self-limiting: once a monolayer forms, further decomposition is suppressed, giving good thickness control.
5. The graphene is transferred to a target substrate using a polymer support layer, after which the copper is etched away.

CVD can produce continuous graphene films over areas of square meters with good electronic quality, making it the most practical route for industrial applications.

Applications of graphene

High-frequency transistors

Graphene's high carrier mobility and saturation velocity make it well-suited for radio-frequency (RF) transistors. Graphene RF transistors have demonstrated cutoff frequencies of several hundred GHz. However, the absence of a band gap in monolayer graphene means these transistors cannot be fully switched off, limiting their use in digital logic. Current research focuses on band gap engineering (through nanoribbons, bilayer gating, or chemical functionalization) and reducing contact resistance.

Transparent conductive electrodes

Monolayer graphene absorbs only about 2.3% of visible light while maintaining high sheet conductivity, making it a candidate to replace indium tin oxide (ITO) in:

• Solar cells
• LEDs and OLEDs
• Touch screens
• Flexible and wearable displays

Graphene's mechanical flexibility gives it a significant advantage over brittle ITO for bendable and stretchable devices. The main challenges are achieving sufficiently low sheet resistance (ITO typically offers ~10–100 Ω/sq) and developing reliable large-area transfer processes.