Why are there two Dirac points K and K′?

The honeycomb Brillouin zone has two inequivalent corners related by inversion. Time-reversal symmetry pairs them; each hosts a linear band crossing when Δ = 0.

What opens the band gap?

A staggered sublattice potential (e.g. on h-BN substrate or in biased bilayer) sets ε_A = −Δ and ε_B = +Δ. At K and K′ where |φ(k)| = 0, the gap is 2Δ independent of hopping τ.

It is the sum of phase factors from the three nearest-neighbor hoppings between A and B sites. Its magnitude sets the band splitting away from the Dirac points.

Is this quantitative for real graphene?

Only qualitatively. Real graphene needs fitted τ and a ≈ 1.42 Å, includes trigonal warping and interactions; this page teaches the canonical two-band Dirac physics.

Graphene Tight-Binding Band Structure

Honeycomb π tight binding: E(k) along Γ–M–K–Γ, Dirac cones at K and K′ when Δ=0, gap 2Δ from sublattice stagger, and a Brillouin-zone energy map.

About this model

Graphene’s low-energy physics is captured by nearest-neighbor tight binding on a honeycomb lattice with two sublattices A and B. With nearest-neighbor vectors (0,a) and (±√3a/2,−a/2), the structure factor φ(k) = e^{iak_y} + 2e^{-iak_y/2} cos(√3 a k_x/2) vanishes at the Dirac points K and K′, producing linear π and π* bands E_±(k) = ±√(Δ² + τ²|φ(k)|²) when sublattice onsite energies ±Δ are included. Pristine graphene (Δ = 0) has touching cones with Fermi velocity v_F ≈ (3/2)τa; a finite stagger opens a gap 2Δ at K and K′ because |φ| = 0 there. This simulator plots E(k) along the standard path Γ → M → K → Γ, colors the π band in the first Brillouin zone, and marks K and K′. It is a two-band, single-π-orbital model — no next-nearest hopping, trigonal warping, or spin–orbit coupling.

Who it's for: Advanced undergraduate or graduate students in condensed-matter or solid-state physics after Bloch’s theorem and before the quantum Hall effect or bilayer graphene.

Key terms

Graphene
Honeycomb lattice
Tight-binding model
Dirac cone
Brillouin zone
Band gap
Fermi velocity

How it works

Graphene tight-binding band structure on a honeycomb lattice: π and π* bands along Γ–M–K–Γ, Dirac cones at K and K′ when Δ=0, and a tunable gap from sublattice staggered potential Δ.

Frequently asked questions

Why are there two Dirac points K and K′?: The honeycomb Brillouin zone has two inequivalent corners related by inversion. Time-reversal symmetry pairs them; each hosts a linear band crossing when Δ = 0.
What opens the band gap?: A staggered sublattice potential (e.g. on h-BN substrate or in biased bilayer) sets ε_A = −Δ and ε_B = +Δ. At K and K′ where |φ(k)| = 0, the gap is 2Δ independent of hopping τ.
What is φ(k)?: It is the sum of phase factors from the three nearest-neighbor hoppings between A and B sites. Its magnitude sets the band splitting away from the Dirac points.
Is this quantitative for real graphene?: Only qualitatively. Real graphene needs fitted τ and a ≈ 1.42 Å, includes trigonal warping and interactions; this page teaches the canonical two-band Dirac physics.

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Frequently asked questions

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Frequently asked questions