5. Honeycomb bands and the Dirac approximation
Position in the course: Lesson 5 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A honeycomb lattice is not a Bravais lattice: it has two sites per primitive cell. A minimal graphene model uses one pz orbital per site, orthogonal orbitals, real nearest-neighbor hopping, and no spin–orbit or interaction terms. Spin then supplies two copies rather than changing the matrix shown. This model isolates the pi bands near charge neutrality.
Each A orbital couples to three B neighbors separated by bond vectors δj. Their Bloch phases add into f(k). The characteristic equation has the same two-by-two algebra as the SSH chain, but now momentum has two components. Energies are plus or minus |tf|. At a Brillouin-zone corner K, the three phases can be 1, exp(2πi/3), and exp(−2πi/3); their sum vanishes. The bands therefore meet without an arbitrary parameter cancellation.
Expand exp[i(K+q)·δ] to first order in q. The constant term sums to zero, leaving a linear complex combination of qx and qy. A basis phase choice brings the result to a Pauli-matrix Dirac form. The Pauli matrices here act on sublattice amplitudes, not directly on physical spin. Opposite valleys have related but convention-dependent matrix signs. The velocity is set by hopping and bond length, not the speed of light.
A staggered onsite term Δσz gives energies ±sqrt[(ħvFq)²+Δ²] and opens a gap 2|Δ|. Equal onsite shifts merely move both bands. Longer-range hopping can break particle-hole symmetry, while strain changes individual bond hoppings and can alter the low-energy structure. The Dirac approximation is valid only near the band touching and at wavelengths long compared with the lattice spacing. It does not reproduce the whole Brillouin zone or predict scattering lifetimes by itself.
2. Derivation step by step
Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.
2.1. Two valleys and two sublattices are distinct labels
A valley labels a neighborhood of a distinct band touching in momentum space. A sublattice labels components of the local two-orbital basis. Physical spin is yet another degree of freedom. Mixing those labels leads to incorrect state counts and selection rules. A boundary can couple valleys when it breaks translation symmetry on an atomic scale, so a one-valley continuum Hamiltonian may be insufficient at an edge. The phase of f depends on the cell convention while its magnitude and bands remain invariant. For quantitative use, compare the linear approximation with the full lattice dispersion over the actual energy window.
3. Worked example
The identity 1+e^(2πi/3)+e^(−2πi/3)=1+2cos(2π/3)=0 verifies the touching condition. Adding Δ=0.1 eV yields zone-corner levels ±0.1 eV, hence a 0.2 eV gap in the model; this is not a graphene experimental prediction.
4. Exercises with explained solutions
Exercise. Why does an equal onsite energy εI not open a gap?
Explained solution. The identity commutes with all matrices and shifts each eigenvalue by ε. At the touching point both levels remain equal. A gap requires a term distinguishing or coupling the degenerate components in a way that lifts their equality.
Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.
5. Misconceptions and limitations
Sublattice pseudospin is not physical spin. Low-energy linear dispersion alone does not establish a realistic relativistic many-body theory.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: density-functional foundations · Interacting Monte Carlo methods
- Slater–Koster, original orbital-geometry construction (1954)
- Su–Schrieffer–Heeger, solitons in polyacetylene (1979)
- Marzari et al., maximally localized Wannier functions review (2012)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
7. Related theory and practice
Quantum mechanics · Density functional theory · ABACUS · DFTB+