(n,m) chirality → metallic or semiconducting band structure via zone folding of graphene's Brillouin zone.
Chirality (n,m)
Metallic
Quick select:
Metallic if (n-m) mod 3 = 0
Diameter: 0.81 nm
Chiral angle: 30.0°
Band gap: 0.00 eV
Trans. modes: 12
Type: Armchair (6,6)
Zone folding: Roll graphene sheet along chiral vector C_h = n·a₁ + m·a₂. Periodic boundary condition
C_h·k = 2πj quantizes k⊥ into N = 2(n²+nm+m²)^{1/2} / a allowed lines cutting the graphene Brillouin zone.
Metallic condition: (n−m) mod 3 = 0 → a quantized line passes through the K-point (Dirac cone) → zero gap.
Otherwise semiconducting with E_g ≈ 2γ₀a/(√3 d) where γ₀ ≈ 2.7 eV (hopping).
Types: Armchair (n,n): always metallic; Zigzag (n,0): metallic iff n mod 3=0; Chiral: general case.