Carbon Nanotube Chirality & Electronic Structure Simulator

Chiral Vector (n, m)

n 10

m 10

Tube length 4.0 nm

Structure

Chirality

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Chiral angle θ

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Diameter

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Circumference

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Electronic character — (n−m) mod 3 = –

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Atoms shown

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C–C bonds

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How it works

A single-wall carbon nanotube is a sheet of graphene — a hexagonal lattice of carbon atoms — rolled up so that one lattice point maps exactly onto another. The roll is defined by the chiral vector C_h = n·a₁ + m·a₂, where a₁, a₂ are the graphene lattice vectors (length a ≈ 0.246 nm) and (n, m) are the integers you control above. Rolling the sheet so C_h becomes the tube's circumference fixes every geometric and electronic property that follows.

d = a·√(n² + nm + m²) / π
θ = atan( √3·m / (2n + m) )
metallic  if (n − m) mod 3 = 0
semiconducting  otherwise

n = m ("armchair") — chiral angle θ = 30°, the hexagon edges run parallel to the tube axis in a zig-zag ring. Always metallic, since n − m = 0.
m = 0 ("zigzag") — θ = 0°, hexagon edges run along the axis. Metallic only if n is a multiple of 3.
0 < m < n ("chiral") — the lattice spirals around the tube at an angle between 0° and 30°.
The metallic/semiconducting split comes from where the allowed circumferential electron wavevectors land relative to graphene's Dirac points — when (n − m) is a multiple of 3, an allowed state sits exactly on the Dirac point and the tube conducts like a 1‑D metal; otherwise a bandgap opens and it behaves as a semiconductor. This is the real, textbook zone-folding result for single-wall carbon nanotubes.
This engine does not model an idealized cylinder — it builds the actual graphene lattice for your (n, m), rolls it with the true chiral vector geometry, and derives every bond from lattice adjacency, including the seam where the sheet closes on itself.

Sublattice A Sublattice B Metallic Semiconducting

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