Graphene's Band Structure: Tight Binding, Dirac Cones and Massless Electrons

Two triangular lattices, one honeycomb

Graphene is a single sheet of carbon atoms in a honeycomb arrangement, but a honeycomb is not itself a Bravais lattice — you cannot map every site onto every other site with a single set of lattice vectors, because alternating atoms have differently oriented bonds. The trick is to see it as two interpenetrating triangular sublattices, A and B, each a perfectly good Bravais lattice, offset from each other by one carbon-carbon bond length δ ≈ 0.142 nm. Every A atom is bonded to three B neighbours, and vice versa.

That two-site basis is the whole reason graphene's electronic structure is unusual. A simple triangular lattice with one atom per site would give an ordinary, gapped or metallic band; the two-atom basis lets the two sublattices' wavefunctions interfere, and at special points in momentum space that interference makes the energy gap vanish exactly.

Tight binding on the honeycomb

The standard model keeps only nearest-neighbour hopping between the 2pz orbitals that form graphene's π bonds — the orbitals that actually carry the conduction electrons — with hopping energy t ≈ 2.7-2.8 eV. Writing the Bloch Hamiltonian in the A/B sublattice basis gives a 2×2 matrix with zero diagonal (the two sublattices are chemically identical) and an off-diagonal term that sums the phase from the three nearest-neighbour vectors:

H(k) = | 0        f(k) |          f(k) = t · Σ_j exp(i k·δ_j)   (j = 1,2,3)
       | f*(k)    0    |

E±(k) = ± |f(k)| = ± t · sqrt( 3 + 2cos(k·a1) + 2cos(k·a2) + 2cos(k·(a1−a2)) )

Diagonalising the 2×2 matrix gives two bands, E+ (conduction) and E− (valence), symmetric about zero by construction — this electron-hole symmetry is a direct consequence of the bipartite A/B structure, not a coincidence. The dispersion is periodic over graphene's hexagonal Brillouin zone, itself the reciprocal-lattice image of the real-space honeycomb.

Where the gap closes: K and K'

|f(k)| is zero wherever the three phase terms cancel exactly, which happens at two inequivalent corners of the hexagonal Brillouin zone, conventionally called K and K′ (also written K and K-prime, or the two valleys). At those points the conduction and valence bands touch with zero gap — this is the Dirac point, and because undoped graphene has exactly one electron per pz orbital, the Fermi level sits precisely there. Graphene is therefore neither a metal (no partially filled band away from a special point) nor an insulator (no gap) — it is a semimetal.

Why the cone is linear, and why that means "massless"

Expand f(k) in a small momentum q measured from K: the constant and quadratic terms vanish by the geometry of the honeycomb, and only a term linear in q survives:

E±(q) ≈ ± ħ vF |q|              vF = 3 t a / (2ħ) ≈ 1×10⁶ m/s  (≈ c/300)

A linear E(q) is exactly the dispersion of a relativistic massless particle, E = c|p|, with the Fermi velocity vF playing the role of the speed of light. Expanding the same 2×2 Hamiltonian near K gives the two-dimensional massless Dirac equation, H = ħ vF (σx qx + σy qy), where σx, σy are Pauli matrices acting not on real electron spin but on the sublattice degree of freedom — which A/B site the amplitude sits on. That fictitious two-valued label is called pseudospin, and it is why graphene's low-energy carriers are described as massless Dirac fermions even though graphene is made of perfectly ordinary, non-relativistic carbon.

Pseudospin has a real, measurable consequence: it locks to the direction of momentum (chirality), which suppresses ordinary backscattering off smooth potentials (an electron reversing k would have to flip pseudospin, which a slowly varying potential cannot do) and produces Klein tunnelling — a Dirac electron can pass through an arbitrarily tall, sufficiently smooth potential barrier with transmission probability approaching 1 at normal incidence, the opposite of ordinary quantum tunnelling, where transmission falls exponentially with barrier height.

Doping, gaps and the density of states

Because the low-energy density of states is proportional to |E| (it is a cone, so the area of a constant-energy slice grows linearly with |E|), it vanishes exactly at the Dirac point — graphene has no carriers at zero doping and zero temperature, which is what makes it a semimetal rather than a metal. Applying a gate voltage shifts the Fermi level up into the conduction cone (electron doping) or down into the valence cone (hole doping), and the carrier density tunes continuously and symmetrically through zero — a knob no conventional semiconductor offers so cleanly. Breaking the A/B sublattice symmetry, for example by placing graphene on hexagonal boron nitride so the two sublattices sit in inequivalent registry with the substrate, opens a genuine gap at K by giving the diagonal of H(k) two different on-site energies; this is one of the few practical ways to make graphene switchable like a semiconductor.

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