Nanotube zone folding

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.