How (n,m) chirality indices determine metallic vs. semiconducting character
A SWCNT is formed by rolling graphene along chiral vector C = na₁ + ma₂. The tube is:
• Metallic if (n−m) mod 3 = 0
• Semiconductor otherwise
Periodic boundary condition around circumference quantizes k_⊥: C·k = 2πj (j integer). These "allowed lines" slice the graphene Brillouin zone.
If a line passes through a K or K' point (Dirac cone), the tube is metallic with zero gap. Otherwise, a bandgap opens: E_g ≈ 2γa_CC / d.
Graphene's conical dispersion E = ±ℏv_F|k| near K makes armchair (n,n) tubes always metallic — the zone-folding rule.