Single-walled carbon nanotube

A carbon nanotube comprising one rolled graphene wall. The wrapping indices determine its allowed circumferential electronic wave vectors and therefore its metallic or semiconducting band structure.

An enriched population selected by wrapping indices or electronic type. Sorting can narrow optical transition distributions or remove metallic leakage paths from semiconductor networks. Purity, yield, residual sorting agents and the processed-device environment remain separate performance variables.

An optical excitation between nanotube electronic subbands, often labelled $E_{ii}$ for prominent axial-polarization transitions. One-dimensional Van Hove singularities enhance the joint density of states, but selection rules, oscillator strengths and strong excitons interactions determine actual absorption peaks. Dielectric environment, bundling, doping and defects can change those peaks.

An approximation that samples the graphene bands on wave vectors satisfying $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, where ${\mathbf{C}}_h$ is the carbon nanotube chiral vector. It produces one-dimensional subbands and the ideal metallicity criterion $n-m\equiv 0(\mathrm{mod}3)$; curvature and interactions can modify its predictions.

Circumferential momentum quantization of a graphene Dirac cone produces nanotube dispersion $E_{\pm }(q_{\parallel })=\pm \sqrt{(\hslash v_F{q}_{\parallel }{)}^2+(\hslash v_F{q}_{\perp }{)}^2}$. Comparing with a massive one-dimensional Dirac Hamiltonian identifies the displayed effective mass and gap $2m_{\mathrm{eff}}{v}_F^2$. This is an emergent band mass, not a changed vacuum electron rest mass. A quantization line through the valley gives zero mass in the ideal zone-folding model; curvature, strain and interactions can change that idealization.

The graphene translation ${\mathbf{C}}_h=n{\mathbf{a}}_1+m{\mathbf{a}}_2$ that becomes a tube circumference when its endpoints are identified. It determines wrapping indices and diameter $|{\mathbf{C}}_h|/\pi$; it is perpendicular to the tube axis.

The angle of the wrapping vector from the zigzag direction of graphene. For chiral indices $0\le m\le n$, $\theta ={\tan }^{-1}[\sqrt{3}m/(2n+m)]$, ranging from zero for zigzag tubes to 30 degrees for armchair tubes. Together with diameter, it determines the discrete wrapping geometry, but does not alone determine the observed colour of a nanotube suspension.

In ideal nanotube zone folding, a single-walled carbon nanotube with indices $(n,m)$ satisfying $n-m\not\equiv 0(\mathrm{mod}3)$. Circumferential quantization misses the Dirac valleys in graphene, opening a band gap.