⬡ Graphene Band Structure & Dirac Cones

⬡ Graphene Band Structure & Dirac Cones

About this simulation

This viewer solves the real nearest-neighbour tight-binding Hamiltonian for graphene's honeycomb lattice — a triangular Bravais lattice with a two-atom (A/B) basis — and renders both the atomic geometry and the resulting E(k) band surface in true 3D. The two bands touch exactly at the K and K′ Brillouin-zone corners, producing the famous linear "Dirac cones" that make graphene's charge carriers behave as massless Dirac fermions with a constant Fermi velocity.

How it works

• Real-space atoms are generated from the actual honeycomb Bravais vectors a₁, a₂ and a two-atom basis, then bonded by nearest-neighbour distance search.
• The band surface is a genuine vertex-height mesh: for every (kx, ky) on a grid, the structure factor f(k) = Σ exp(i k·δᵢ) is evaluated and E±(k) = ±t|f(k)| sets the mesh height.
• Toggling the sublattice gap adds a mass term Δ so E±(k) = ±√(t²|f(k)|² + Δ²), modelling a substrate (e.g. hBN) that breaks the A/B symmetry.
• The k-point marker samples the live surface and a numerical derivative gives the local band velocity, which converges to the constant Fermi velocity only near K/K′.
• Three.js with OrbitControls renders both real 3D scenes; the geometry is fully recomputed whenever t or Δ change.

Key equations

f(k) = e^(ik·δ₁) + e^(ik·δ₂) + e^(ik·δ₃)
E±(k) = ±t|f(k)| (gapped: ±√(t²|f(k)|² + Δ²))
v_F = 3ta / (2ħ) ≈ 0.9–1.1 × 10⁶ m/s for t ≈ 2.7–2.9 eV, a = 1.42 Å.

Controls

• Hopping t — nearest-neighbour hopping integral; sets bandwidth and Fermi velocity.
• Gap Δ and the Sublattice gap toggle — opens a real bandgap at K/K′.
• Lattice size — how many unit cells the real-space honeycomb tiles.
• k_x / k_y sliders, snap buttons, or a click on the band panel's base plane — move the k-point marker.
• Reset — restore default parameters and camera.

Did you know?

Because f(K) = 0 exactly by symmetry — not by tuning — the Dirac point in pristine graphene is topologically protected: no smooth, symmetry-preserving perturbation of the nearest-neighbour hopping alone can open a gap. It takes something that actually distinguishes the A and B sublattices, like an hBN substrate or a staggered on-site potential, to do that.