Brillouin Zone
The first Brillouin zone is the Wigner–Seitz primitive cell of a crystal's reciprocal lattice: the region of wavevector (k) space containing all points closer to the origin than to any other reciprocal lattice point. Because wavevectors differing by a reciprocal lattice vector describe the same Bloch state, every distinct electron and phonon state can be labelled by a k inside this zone, and its boundaries are where Bragg reflection opens band gaps.
Curious
An electron in a crystal behaves as a wave with a wavelength and direction, described by its wavevector k. Because the crystal repeats itself with period a, waves whose wavevectors differ by 2π/a look identical at every atom. That means we only need a limited range of k values, from −π/a to +π/a in one dimension, to list every distinct state. That range is the first Brillouin zone, named after Léon Brillouin.
At the edges of the zone, the electron's wavelength fits the lattice exactly so that reflections from successive rows of atoms add up, the same condition as in Bragg diffraction. Standing waves form there, and that is where gaps open between energy bands.
Exploring
| Real-space lattice | Reciprocal lattice | First Brillouin zone shape |
|---|---|---|
| Simple cubic | Simple cubic | Cube |
| Body-centred cubic | Face-centred cubic | Rhombic dodecahedron |
| Face-centred cubic | Body-centred cubic | Truncated octahedron |
Deep Dive
Formal definition. Given primitive lattice vectors ai, reciprocal vectors bi satisfy ai·bj = 2πδij. Bloch's theorem (see Bloch's Theorem) shows that ψk+G and ψk describe the same state for any reciprocal lattice vector G, so k can be reduced into the first zone (the reduced-zone scheme). The zone boundaries are the perpendicular bisector planes of reciprocal lattice vectors, exactly the planes satisfying the Laue diffraction condition 2k·G = G².
Counting. A crystal of N primitive cells with periodic boundary conditions has exactly N allowed k values in the first zone. Each band therefore holds 2N electrons including spin, which is the counting behind the distinction between metals and insulators. Higher Brillouin zones, reached by crossing successive bisector planes, each have the same volume and are used in the extended-zone picture and in constructing Fermi surfaces.
Phonons and beyond. Lattice vibrations are also labelled by k in the Brillouin zone, and processes in which crystal momentum changes by a reciprocal lattice vector (Umklapp processes) limit thermal conductivity. In topological band theory, quantities integrated over the whole zone, such as the Chern number, classify materials like topological insulators.
Concept Relationship Map
Requires: Bravais lattices, Bloch's theorem
Enables: energy bands, density of states, Fermi surfaces
Commonly confused with: the Wigner–Seitz cell (its real-space counterpart)
Sources
- Brillouin, L. (1930). Journal de Physique et le Radium, 1, 377–400.
- Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley.
- Ashcroft, N.W., Mermin, N.D. (1976). Solid State Physics. Holt, Rinehart and Winston.