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<graphene> is a two-dimensional honeycomb lattice with two sublattices. Nearest-neighbour hopping gives two bands $E_\pm(\mathbf k)=\pm t|\sum_{j=1}^3e^{i\mathbf k\cdot\boldsymbol\delta_j}|$. They touch at two inequivalent valleys. Expanding near valley $\tau=\pm1$ gives
$$
\boxed{H_\tau=\hbar v_F(\tau q_x\sigma_x+q_y\sigma_y),\qquad
E_\pm=\pm\hbar v_F|\mathbf q|.}
$$
The two-component spinor describes sublattice amplitudes, not real <spin>. This is a massless two-dimensional <Dirac Hamiltonian> with material <velocity> $v_F$ in place of $c$. The emergent band <quasiparticles> are relativistic in this mathematical sense; the <electron>'s vacuum rest mass is not set to zero. Real <spin> and valley each contribute a degeneracy of two in the ideal low-energy model.
A <single-walled carbon nanotube> is a rolled <graphene> sheet with <carbon nanotube chiral vector> $\mathbf C_h=n\mathbf a_1+m\mathbf a_2$. Periodicity requires $\mathbf k\cdot\mathbf C_h=2\pi\ell$, selecting lines through <graphene> momentum space and producing one-dimensional <transverse electronic modes>. Near a valley,
$$
\boxed{E_\pm(q_\parallel)=\pm\hbar v_F\sqrt{q_\parallel^2+q_\perp^2}.}
$$
A quantization line through one of the <Dirac valleys in graphene> gives a metallic branch; ideal <nanotube zone folding> gives the criterion $n-m\equiv0\pmod3$. Otherwise the nearest line misses the valley and opens $E_g=2\hbar v_F|q_\perp|$. The transverse momentum gives a <carbon-nanotube Dirac mass>, with $m_{\rm eff}=\hbar|q_\perp|/v_F$. For a large-radius ideal semiconducting tube, $|q_\perp|\simeq2/(3d)$ and $E_g\simeq4\hbar v_F/(3d)$, showing the inverse-diameter gap. Curvature, strain, <spin-orbit coupling> and interactions can change the simple classification; nominally metallic non-armchair tubes can acquire small gaps. One-dimensional <transverse electronic mode> thresholds give <Van Hove singularities>, unlike <graphene>'s low-energy <density of states> proportional to $|E|$.
A gate changes occupation and the <Fermi level> in both systems. Ideal monolayer <graphene> can be tuned through <charge neutrality> from <holes> to <electrons>, but an ordinary scalar gate does not open a bulk <band gap>. Including <spin> and valley,
$$
n=\operatorname{sgn}(E_F)\frac{E_F^2}{\pi(\hbar v_F)^2},\qquad
D(E)=\frac{2|E|}{\pi(\hbar v_F)^2}.
$$
The gate relation includes geometric and <quantum capacitances>, so one should not equate gate <voltage> to <Fermi energy> without the <capacitance> model. A <semiconducting carbon nanotube> can instead be depleted when its <Fermi level> lies in a gap and turned on in <electron> or <hole> bands. Contact <Schottky barriers> can strongly influence the observed nanotube field effect. A truly gapless ideal metallic tube cannot be switched off simply by moving the <Fermi level> through its band crossing. These gate responses reflect band structure and contacts, not a change of wrapping indices.
\Image[/past-exam-of-the-physics-of-nanoelectronic-systems-course-of-the-university-of-cambridge/2013/paper-1-graphene-nanotube-bands.png]
{title=Original low-energy graphene cut and metallic or gapped nanotube subbands}
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