Honeycomb Lattice (formerly Graphene)

Overview

The nearest-neighbor tight-binding model on the honeycomb lattice is the standard model for the electronic band structure of graphene. It features two sublattices (A, B), bipartite structure, and the celebrated Dirac cones at the $K$ and $K'$ points of the Brillouin zone.

\[H = -t \sum_{\langle i,j \rangle \in A\text{-}B} \bigl(c^\dagger_i c_j + c^\dagger_j c_i\bigr)\]

Lattice properties: 2 sublattices per unit cell, bipartite (A vs B), coordination number 3, no flat band.

Key physics: The bipartite (chiral) symmetry guarantees that the spectrum is symmetric about $E = 0$. The two bands touch linearly at the $K$ and $K'$ points, forming massless Dirac cones with a linear dispersion $E \sim \pm v_F |\mathbf{k} - \mathbf{K}|$ near the Fermi level at half filling.

Bloch Spectrum

Statement

The $2 \times 2$ Bloch Hamiltonian in the sublattice basis is

\[H(\mathbf{k}) = \begin{pmatrix} 0 & f(\mathbf{k}) \\ f(\mathbf{k})^* & 0 \end{pmatrix}\]

with $f(\mathbf{k}) = -t\bigl[e^{i\mathbf{k}\cdot\boldsymbol{\delta}_1} + e^{i\mathbf{k}\cdot\boldsymbol{\delta}_2} + e^{i\mathbf{k}\cdot\boldsymbol{\delta}_3}\bigr]$, where $\boldsymbol{\delta}_{1,2,3}$ are the three A-to-B nearest-neighbor displacement vectors.

Using the unit-cell basis $(\mathbf{a}_1, \mathbf{a}_2)$ adopted by Lattice2D's Honeycomb topology and defining $\theta_1 = \mathbf{k} \cdot \mathbf{a}_1 = 2\pi m/L_x$, $\theta_2 = \mathbf{k} \cdot \mathbf{a}_2 = 2\pi n/L_y$, the two eigenvalues at each allowed momentum are

\[E_{mn,\pm} = \pm\, t\,\sqrt{3 + 2\cos\!\left(\frac{2\pi m}{L_x}\right) + 2\cos\!\left(\frac{2\pi n}{L_y}\right) + 2\cos\!\left(\frac{2\pi n}{L_y} - \frac{2\pi m}{L_x}\right)}\]

The full spectrum of $2 L_x L_y$ eigenvalues is obtained by ranging over $m \in \{0, \ldots, L_x - 1\}$ and $n \in \{0, \ldots, L_y - 1\}$.

Derivation

See Bloch Honeycomb Dispersion for the full derivation of $|f(\mathbf{k})|^2$ from the unit-cell geometry.

Dirac points

When $L_x$ and $L_y$ are both divisible by 3, the $K$ and $K'$ points $(m/L_x, n/L_y) = (1/3, 2/3)$ and $(2/3, 1/3)$ are commensurate with the finite Brillouin zone, and $|f(\mathbf{K})|^2 = 0$ exactly. Each Dirac point contributes two zero-energy modes (one per band), giving 4 zero modes total. For example, at $3 \times 3$ the spectrum contains exactly 4 zero eigenvalues.

References

• P. R. Wallace, "The Band Theory of Graphite", Phys. Rev. 71, 622 (1947) – original tight-binding calculation.
• A. H. Castro Neto et al., "The electronic properties of graphene", Rev. Mod. Phys. 81, 109 (2009) – comprehensive review.

QAtlas API

# Sorted single-particle spectrum, 3×3 honeycomb PBC
λ = QAtlas.fetch(Graphene(), TightBindingSpectrum(); Lx=3, Ly=3, t=1.0)
# → 18 eigenvalues, including 4 zero modes at Dirac points

Verification

Connections

• Lattice family: Part of the tight-binding model catalogue. The honeycomb lattice is bipartite like Lieb but unlike Kagome and Triangular.
• Flat-band contrast: Unlike the Kagome ($+2t$) and Lieb ($0$) lattices, the honeycomb has no flat band – both bands are fully dispersive.
• Methods: Computed via the Bloch Hamiltonian method; the generic builder provides an independent cross-check.