Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 2 / Section A(i) / 13 / ii / Solution

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.