Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 2 / Section A(i) / 13

The carbon nanotube chiral vector identifies equivalent sites of graphene that become one point on rolling:The primitive vectors meet at $60^{\circ }$, $b$ is the C–C bond length, and one can choose $0\le m\le n$. The carbon nanotube chiral angle satisfies $\tan \theta =\sqrt{3}m/(2n+m)$. Armchair tubes have $(n,n)$ and $\theta =30^{\circ }$; zigzag tubes have $(n,0)$ and $\theta =0$. Other pairs are chiral, with mirror-related left- and right-handed forms. Armchair and zigzag tubes themselves have mirror symmetry.

The circumferential boundary condition requires $\mathbf{k}\cdot {\mathbf{C}}_h=2\pi \ell$. In nanotube zone folding, an allowed wavevector line intersects one of the Dirac valleys in graphene precisely when $n-m$ is divisible by three. Thus all armchair tubes are metallic in this approximation. Zigzag tubes are metallic for $n$ divisible by three and otherwise semiconducting; general chiral tubes follow the same modular rule. Curvature can open small gaps in nominally metallic non-armchair tubes. Strain, interactions and broken symmetries can modify the ideal result; metallic band structure does not imply zero device resistance.

For an armchair single-walled carbon nanotube, $|{\mathbf{C}}_h|=3bn$. HenceThe integer bounds give $(n,m)=(8,8),(9,9),(10,10),(11,11),(12,12)$.

The neighboring values $d_7\simeq 0.9492\mathrm{nm}$ and $d_{13}\simeq 1.7628\mathrm{nm}$ lie outside the interval. “Exact” here means exact within the stipulated cylindrical model and input bond length; the decimals are rounded. No van der Waals wall thickness is added because the diameter is measured between atomic centers.

Four advantages are strong electrostatic gate control, rapid charge transport, high current-carrying capability and mechanical flexibility.

A semiconducting single-walled carbon nanotube has a nanometre-scale channel cross-section, allowing efficient gate control in a field-effect transistor. A clean short tube can have a long mean free path and correspondingly little channel scattering. Strong covalent bonding supports large current densities for interconnect applications, while high axial thermal conductivity can help distribute heat. Small mass, large aspect ratio and mechanical compliance also permit conducting or semiconducting networks on flexible substrates. Metallic interconnects and low-leakage transistors require different electronic tube populations.

Manufacturing must control diameter, carbon nanotube chiral vector, purity, alignment and placement. Metallic contaminants short-circuit transistor networks; low network density reduces current, whereas excessive density makes isolation difficult. Tube–metal contact resistance, schottky barriers, tube–tube junctions, catalyst residues and substrate charge traps can dominate the intrinsic channel response. Thermal boundary resistance can limit cooling despite high axial thermal conductivity. Compare sorted and unsorted networks at matched density and contact geometry, measure contact and channel contributions separately, and assess uniformity across many devices. High intrinsic mobility alone is not evidence of a reproducible integrated circuit.