FCC and Graphene Bands

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

Tight binding is the cheapest honest model of a band structure. Put one orbital on each lattice site, let electrons hop only to nearest neighbors with amplitude $t$, and the Hamiltonian in k-space is already diagonal in the band index. For a Bravais lattice with one site per cell the single band is written as

$$E(\mathbf{k}) = \varepsilon_0 + t \sum_{\boldsymbol{\delta}} e^{i\mathbf{k}\cdot\boldsymbol{\delta}}$$

where the sum runs over the nearest-neighbor vectors $\boldsymbol{\delta}$ and $\varepsilon_0$ is the onsite energy. Everything that follows comes from evaluating this sum on different lattices.

The FCC band

The face-centered cubic lattice has primitive vectors $(\tfrac{1}{2},\tfrac{1}{2},0)$, $(0,\tfrac{1}{2},\tfrac{1}{2})$, $(\tfrac{1}{2},0,\tfrac{1}{2})$ and twelve nearest neighbors. With $\varepsilon_0 = 0$ and $t = -1$, sweeping $\mathbf{k}$ along the standard path through the high-symmetry points gives the single band below.

The bandwidth is set by the coordination. More neighbors means more terms in the sum, so close-packed FCC produces a wide band, and the shape between the labeled points is a fingerprint of the lattice geometry that shows up in real FCC metals well before any serious theory is applied.

Graphene and its density of states

Graphene needs the two-site honeycomb cell, and its two bands touch at the K points of the Brillouin zone. Histogramming the eigenvalues over a uniform k-mesh gives the density of states.

Two features carry the physics. The density of states drops to zero at zero energy without a gap, which is the signature of the Dirac points, where the bands touch only at isolated k-points. And the two sharp peaks at $E = \pm t$ are van Hove singularities, coming from saddle points of the dispersion at the M points. Adding a next-nearest-neighbor hopping (a tenth of $t$ in the problem set) breaks the symmetry between the two bands, tilting the spectrum so the electron and hole sides are no longer mirror images.

Zigzag nanotubes

Rolling graphene into a cylinder quantizes the momentum around the circumference, so a nanotube’s bands are one-dimensional slices of the graphene dispersion. Whether a slice passes through a Dirac point decides everything. For a $(n, 0)$ zigzag tube the allowed transverse momenta hit the Dirac point only when $n$ is divisible by 3, so those tubes are metallic and the rest are semiconductors. Plotting the lowest subband for $n = 1, 2, 4, 10, 16$, none of which is divisible by 3, the dispersion stays pinched off from zero energy with a small gap, and zooming in near the minimum confirms the gap never closes. Two thirds of all zigzag tubes are semiconductors by this counting argument alone, which is a remarkable amount of device physics to fall out of one hopping parameter.

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