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The vectors from the B atom to its three nearest N neighbours can be chosen asIndeed ${\mathbf{n}}_1={\mathbf{d}}_N-{\mathbf{d}}_B$, ${\mathbf{n}}_2={\mathbf{n}}_1-{\mathbf{a}}_1$ and ${\mathbf{n}}_3={\mathbf{n}}_1-{\mathbf{a}}_2$. All three have length $a/\sqrt{3}$. Repeating the two-site basis over $\mathbf{R}=m{\mathbf{a}}_1+n{\mathbf{a}}_2$ produces the honeycomb pattern in the first panel of the original figure in solution.

Solving ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ givesThe first Brillouin zone is the Wigner–Seitz hexagon of this reciprocal lattice. Its vertices are $(0,\pm 4\pi /(3a))$ and $(\pm 2\pi /(\sqrt{3}a),\pm 2\pi /(3a))$, with independent signs in the latter set. The centre is $\Gamma =(0,0)$; one inequivalent corner is $K=(0,4\pi /(3a))$, with $K^{\prime }=-K$. Corners related by a reciprocal lattice translation are equivalent, leaving two inequivalent valleys.

At a valley $K$, $\gamma (K)=1+2\cos (2\pi /3)=0$. For h-BN the inequivalent sublattice energies therefore leave the direct two-band gapGraphene's equivalent carbon sites instead have ${ϵ}_B={ϵ}_N$ in this model, so the two bands touch at the valleys. Near a valley, $|\gamma (K+\mathbf{q})|\simeq (\sqrt{3}a/2)|\mathbf{q}|$: graphene has approximately linear, gapless Dirac cones while h-BN has gapped band edges. The model panel in the original figure in solution displays this distinction.

First, undoped h-BN is an electrical insulator with thermally suppressed carriers, whereas neutral graphene is a zero-gap semimetal whose transport is readily changed by gating. Disorder, temperature and contacts still affect its measured conductivity. Second, h-BN suppresses interband absorption below its large gap and is transparent over much of the visible range, whereas graphene permits low-energy interband absorption unless occupations block it. A quantitative optical threshold in real h-BN also involves additional bands and strong excitonic effects; a two-orbital model should not be presented as a numerical prediction of its complete optical spectrum.