🧪 Graphene Dirac Cone Band Structure

This simulator computes graphene's real π-electron tight-binding dispersion relation, E±(kx,ky) = ±t·√(3 + 2cos(√3 ky a) + 4cos(3kx a/2)cos(√3 ky a/2)), across a grid spanning several Brillouin zones and renders it as two energy surfaces in 3D. Unlike a rolled-up carbon nanotube — where confining the momentum around the tube's circumference quantizes kx or ky into discrete allowed lines and can open a bandgap — this simulation keeps graphene flat and two-dimensional, so both kx and ky vary continuously and the full hexagonal pattern of the band structure is visible. The result is graphene's defining physical fact: the conduction and valence bands genuinely touch, with zero energy gap, at six symmetric Dirac points, and near those points the dispersion is linear (a cone) rather than the parabola of an ordinary semiconductor. Tune the hopping parameter t to see the bandwidth and cone steepness (Fermi velocity) change while the Dirac points themselves stay pinned in place, and drag to orbit the surface and inspect the cone-touching points directly.