Fullerene C60 and Carbon Nanotubes: Chirality and Band Gap

What is modelled

Carbon in three forms: the C60 cage (buckminsterfullerene), single-wall nanotubes defined by a chiral vector (n, m), and the graphene sheet they are rolled from. A nanotube is built atom by atom from the graphene lattice, and its electronic bands are computed from a nearest-neighbour tight-binding model of the pi electrons, so the metal-or-semiconductor rule and the band gap come out of numbers rather than being looked up.

Equations

• Graphene lattice constant a = √3 acc = 2.46 Å with a_cc = 1.42 Å; lattice vectors a1 = a(√3/2, 1/2) and a2 = a(√3/2, −1/2). The chiral vector C = n a1 + m a2 becomes the circumference.
• Diameter d = |C|/π = a √(n2 + nm + m2) / π; chiral angle θ = atan(√3 m / (2n + m)); period along the axis T = √3 |C| / dR with dR = gcd(2n + m, 2m + n); atoms per cell N = 4(n2 + nm + m2)/dR.
• Tight binding: E(k) = ±γ0 |1 + ei k·a1 + ei k·a2| with γ₀ = 2.7 eV. Rolling the sheet allows only the wave vectors on parallel cutting lines k = μ K1 + κ K2/|K2| (μ = 0 … N/2 − 1); the graph shows these one-dimensional sub-bands.
• A tube is metallic when a cutting line passes through a Dirac point, which happens exactly when (n − m) mod 3 = 0; otherwise the gap is about Eg ≈ 2 acc γ0 / d. An ideal metallic tube conducts 4e2/h = 155 µS ballistically.
• C60 is a truncated icosahedron: 60 atoms, 90 bonds, 12 pentagons and 20 hexagons, with Euler's relation V − E + F = 2; its radius for edge length a is (a/4) √(58 + 18√5).

Checks run in the page

• C60 topology and radius rebuilt independently of the Blender model (counts, Euler, bond types, radius formula).
• Atoms of a built tube found by enumerating lattice sites against the formula for N, with three neighbours for every interior atom and chord-versus-arc bond lengths within 1.5 %.
• The metallic rule against the numerical minimum of the bands for all tubes with n up to 12.
• The gap formula against the numerical gap.

Honest simplifications

• Zone folding of flat graphene ignores the curvature of the tube: small tubes (below about 1 nm) acquire small curvature gaps, and the nominally metallic (n, 0) and chiral tubes open gaps of tens of meV. Excitons dominate the optical gap of real semiconducting tubes. γ₀ is a fitted constant (values of 2.6 to 3.0 eV appear in the literature).
• Atom and bond positions are ideal: all bonds have the same length (the C60 drawn here uses 1.43 Å instead of the measured 1.40 and 1.45 Å), the tube is a rigid cylinder with open ends, and nothing vibrates. The view of the hexagon-pentagon and hexagon-hexagon bonds only shows which bonds have double-bond character.
• Tubes with more than about 4500 atoms per unit cell are not drawn but their numbers and bands are still computed.

What to try

• Compare (9,0), (10,0) and (11,0): only one in three is metallic, and the gap of the others shrinks as the diameter grows.
• Slide the roll-up control from the flat sheet to the tube and watch the red chiral vector close into the circumference.
• Pick (12,6) and (6,5): both are chiral but one is a metal and one a semiconductor.