Materials chemistry

Carbon nanotubes, graphene, 2D materials

Relate carbon bonding and dimensionality to the electronic, mechanical, and optical properties of nanotubes, graphene, and other atomically thin materials.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionOne carbon sheet, several dimensions

Imagine a hexagonal carbon net. Leave it flat and it is graphene; roll it into a cylinder and it is a nanotube. Curving or stacking changes which directions electrons and heat can travel.

The chiral vector (n,m) specifies how the hexagonal sheet is joined around the tube; ideal single-wall tubes are metallic or semiconducting depending on the indices.

SchoolStructure and bonding

Definition: Graphene

A single atomic layer of sp²-bonded carbon in a honeycomb lattice. The σ bonds form a strong in-plane framework; p orbitals form delocalized π bands.

Each carbon has three nearest neighbours. In ideal graphene, the π and π* bands meet at Dirac points, giving a zero-gap semimetal rather than an ordinary semiconductor.

Carbon architectures
Structure	Dimensionality / signature
Graphene	2D; Dirac bands
Nanotube	1D transport; chirality-dependent gap
Few-layer stack	Interlayer coupling matters

Example: Classifying a nanotube

Solution

In the nearest-neighbour zone-folding picture, n−m divisible by 3 gives a band crossing and metallic behaviour. Curvature can open a small gap except in armchair tubes; real defects and interactions also modify this rule.

UndergraduateChirality and electronic structure

\mathbf C_h=n\mathbf a_1+m\mathbf a_2,\\qquad d=\frac{a}{\pi}\sqrt{n^2+nm+m^2}

The chiral vector closes the graphene lattice around the circumference. With graphene lattice constant a≈0.246 nm, the diameter follows from its magnitude; quantized circumferential wavevectors intersect the graphene dispersion and determine subbands.

AdvancedBeyond a single ideal layer

Substrate screening, strain, edges, disorder, adsorbates, and interlayer stacking all reshape spectra. Moiré superlattices in twisted bilayers can produce narrow bands and correlated phases, but twist-angle inhomogeneity and competing orders complicate interpretation.

ResearchResearch frontier

References

Helical Microtubules of Graphitic Carbon · S. Iijima, 1991
Electric Field Effect in Atomically Thin Carbon Films · K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, 2004
The Electronic Properties of Graphene · A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, 2009
Moiré Bands in Twisted Double-Layer Graphene · R. Bistritzer, A. H. MacDonald, 2011