Graphene Dirac Cone Band Structure Simulator

Graphene π-Band Structure

Hopping parameter t 2.70 eV

k-space zoom 3 BZ

Bandwidth (max E)

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Gap at Dirac pt.

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Fermi velocity v_F

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Dirac points found

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How it works

Graphene's carbon atoms sit on a honeycomb lattice — two interpenetrating triangular sublattices (A, B). Solving the tight-binding Hamiltonian for one π-electron hopping between nearest neighbours with hopping energy t gives two energy bands as a function of crystal momentum (kx, ky):

E±(kx,ky) = ±t·√( 3 + 2cos(√3·ky·a) + 4cos(3kx·a/2)·cos(√3·ky·a/2) )

where a ≈ 0.142 nm is the carbon–carbon bond length. This is the exact formula being evaluated on every grid point of the surface you see — nothing here is illustrative.

At six special points on the edge of the hexagonal Brillouin zone (the Dirac points, K and K′), the expression under the square root hits exactly zero: the + and − bands meet with no energy gap. Expanding E near a Dirac point gives E ≈ ±ħv_F|q|, a straight-line (conical) dispersion instead of the parabola E ∝ q² of an ordinary semiconductor. That linearity is why graphene's charge carriers behave like massless relativistic (Dirac) fermions with a fixed effective velocity v_F = 3ta/2ħ ≈ c/300, and why undoped graphene is a zero-bandgap semimetal rather than an insulator or normal metal.

Hopping parameter t — sets the overall energy scale (bandwidth) of the π-bands; the accepted graphene value is ≈2.7 eV. The Dirac points' location in k-space never moves when you change t (it's fixed by the lattice geometry) — only the cone's steepness (v_F) and the band's overall height change.
k-space zoom — widens the plotted (kx,ky) window so you can see how the pattern of Dirac points repeats across the periodic reciprocal lattice, not just inside one Brillouin zone.
Dirac points found — computed live by scanning the grid for where E ≈ 0, exactly the same formula as the surface. Red markers sit precisely where the two colored sheets touch.
The inset plot is a straight cross-section of E(kx,ky) through the nearest Dirac point — the V-shape you see there is the linear Dirac cone, sliced flat.

Cross-section through nearest Dirac point (linear, not parabolic)

Conduction band (E+) Valence band (E−) Dirac point (zero gap)

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