nanotube chirality

Roll a sheet of striped wrapping paper into a tube and the stripes can end up running straight along the tube, straight around it, or spiralling at an angle — all from the same sheet, just rolled differently. A carbon nanotube is graphene rolled into a tube, and in the same way the honeycomb pattern can meet itself straight-on or at a slant. Nanotube chirality is the label for exactly which way the sheet was rolled — how the hexagons wind around the tube — and it fixes both the tube's diameter and its electronic character.

Crystallographers pin it down with two whole numbers, the chiral indices (n, m). Mark a starting hexagon on the flat graphene sheet; the chiral vector C equals n times the first lattice vector plus m times the second, and you roll the sheet so that the starting hexagon lands exactly on the hexagon that the vector points to. The diameter follows directly: d equals a times the square root of (n squared plus n times m plus m squared), all divided by pi, with a about 0.246 nm. Three cases have names: (n, 0) tubes are zigzag, (n, n) tubes are armchair, and anything in between is chiral, with the honeycomb spiralling around the tube.

Here is why it is not just bookkeeping: the rolling direction decides whether the nanotube conducts like a metal or a semiconductor. The rule is clean — a tube is metallic when n minus m is a multiple of 3, and semiconducting otherwise. So of all possible tubes, about one in three comes out metallic and two in three semiconducting, purely from geometry, with no change in chemistry at all. This is one of the sharpest examples anywhere of structure dictating property: two nanotubes made of identical carbon atoms can be a wire or a switch depending only on how the sheet was rolled.