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Let ${\mathbf{a}}_1,{\mathbf{a}}_2$ be equal-length primitive translation vectors of the graphene lattice, separated by $60^{\circ }$. The carbon nanotube chiral vectoridentifies the two points joined when the sheet is rolled: $|{\mathbf{C}}_h|$ is the circumference, not the tube axis. The right-hand panel of the preceding original diagram shows ${\mathbf{C}}_h=6{\mathbf{a}}_1+6{\mathbf{a}}_2$ and ${\mathbf{C}}_h=7{\mathbf{a}}_1$.

The periodic boundary condition is $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$, with $\ell$ an integer. Applying it at either Dirac valley gives the nanotube zone folding criterionfor a metallic tube. Thus $(6,6)$ is an armchair metallic example, whereas $(7,0)$ is a zigzag semiconducting example. Other choices with $n-m\not\equiv 0(\mathrm{mod}3)$ also give a semiconducting carbon nanotube.

This criterion is the ideal flat-sheet approximation. Curvature can open a small gap in nominally metallic non-armchair tubes; ideal armchair tubes remain gapless in that approximation. The property is determined by the wrapping vector, not simply by calling a tube “chiral” or by its diameter alone.

A single-walled carbon nanotube confines the electron around its circumference while allowing motion along its axis. Nanotube zone folding therefore turns the graphene bands into one-dimensional subbands. A semiconducting carbon nanotube has no allowed circumferential wave vector through a Dirac valley; a metallic tube has a band crossing at the Fermi energy in the ideal model. Each massive subband edge gives a one-dimensional van Hove singularity in the density of states, and corresponding peaks appear in the joint density of states.

For light polarized along the tube, the main electric-dipole selection rule conserves circumferential angular momentum: transitions connect matching valence and conduction subbands. These finite-energy transitions are conventionally $S_{11},S_{22},\dots$ in a semiconducting carbon nanotube and $M_{11},M_{22},\dots$ in a metallic tube. A metallic band crossing does not make its first strong interband peak have zero energy. Perpendicular polarization admits different angular-momentum channels, often much weaker because of depolarization. Excitons, dielectric screening and trigonal warping modify the simple band-edge transition energies.

Experimentally, measure absorption versus photon energy; Raman spectroscopy becomes resonantly enhanced when the excitation matches an optical transition. A photoluminescence excitation spectrum of dispersed semiconducting tubes correlates excitation near $S_{22}$ with emission near $S_{11}$ after relaxation. Metallic tubes generally lack comparable bright emission because nonradiative relaxation is efficient. Combining optical energies with radial-breathing-mode Raman spectroscopy or electron diffraction assigns tube diameter and roll-up indices; a Kataura plot organizes those assignments.^[ref]