Boron Nitride and Bilayer Graphene

From a tight-binding problem set, computed with PythTB.

The honeycomb lattice has two sites per unit cell, call them A and B, and the simplest tight-binding model hops an electron only between neighboring A and B sites with amplitude $t$. Everything interesting about graphene, boron nitride, and their stacks follows from what you do to those two sublattices.

Sublattice asymmetry

In graphene the two sublattices are both carbon, so their onsite energies are equal and nothing distinguishes A from B. The two bands then touch at the corners of the Brillouin zone, the K points, and the spectrum is gapless. Hexagonal boron nitride has the same lattice but boron on one sublattice and nitrogen on the other, which the model captures with onsite energies $+\Delta$ and $-\Delta$ (the calculation uses $\Delta = 2$ with $t = -1$). The bands become

$$E_\pm(\mathbf{k}) = \pm\sqrt{\Delta^2 + |f(\mathbf{k})|^2}$$

where $f(\mathbf{k})$ is the sum of hopping phase factors over the three A-B bonds. At the K points $f$ vanishes, so where graphene has band touching, hBN has a gap of $2\Delta$. Sampling the bands on a $20 \times 20$ k-mesh and histogramming the eigenvalues gives the density of states, and the gap is unmistakable.

This is the cleanest statement of why graphene conducts and boron nitride insulates. Same lattice, same bonds, and a gap opened purely by making the two sublattices inequivalent.

Bernal-stacked bilayer

The second modification keeps both sublattices carbon but stacks a second graphene layer in the Bernal arrangement, where one sublattice of the top layer sits directly above a sublattice of the bottom layer. Those aligned atoms couple with an interlayer hopping $\gamma_1$, and the four bands work out to

$$E(\mathbf{k}) = \pm\frac{1}{2}\left(\sqrt{4|f(\mathbf{k})|^2 + \gamma_1^2} \mp \gamma_1\right)$$

with the same in-plane factor $f(\mathbf{k})$ as before (the calculation uses $\gamma_1 = 2$).

Two of the four bands are pushed away from zero energy by $\pm\gamma_1$, and the two that remain still touch at K, but the touching changes character. The monolayer’s linear Dirac cones become parabolic band touching in the bilayer, which is why monolayer and bilayer graphene have qualitatively different low-energy electrons even though both are gapless. A gap can then be opened in the bilayer by an electric field between the layers, which plays the same symmetry-breaking role that the boron-nitrogen difference plays in hBN.

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