Brillouin zones are key to understanding electron and phonon behavior in crystals. They represent the primitive cell in reciprocal space, helping us analyze band structures and material properties.

The first Brillouin zone contains unique wave vectors closest to the origin. Its boundaries, determined by perpendicular bisector planes, represent the maximum wavelength of waves that can propagate without scattering in the crystal.

Brillouin zones in reciprocal space

• Brillouin zones are a fundamental concept in solid state physics used to describe the behavior of electrons and phonons in periodic structures like crystals
• They represent the primitive cell in the reciprocal lattice, which is the Fourier transform of the real space Bravais lattice
• Understanding Brillouin zones is crucial for analyzing electronic band structures, phonon dispersion relations, and various physical properties of crystalline materials

First Brillouin zone boundaries

• The first Brillouin zone is defined as the Wigner-Seitz primitive cell in the reciprocal lattice space
• It contains all the unique wave vectors (k-points) that are closest to the origin of the reciprocal lattice
• The boundaries of the first Brillouin zone are determined by the perpendicular bisector planes of the reciprocal lattice vectors connecting the origin to its nearest reciprocal lattice points
• These boundaries represent the maximum wavelength of a wave that can propagate in the crystal without being scattered by the periodic potential

Brillouin zone construction

Wigner-Seitz cell in reciprocal space

• The Wigner-Seitz cell is a geometrical construction used to determine the primitive cell in both real and reciprocal space
• In reciprocal space, the Wigner-Seitz cell is constructed by drawing perpendicular bisector planes to the reciprocal lattice vectors connecting the origin to its nearest neighbors
• The smallest enclosed volume around the origin forms the first Brillouin zone, which contains all the unique k-points
• The Wigner-Seitz cell construction ensures that the Brillouin zone has the same symmetry as the reciprocal lattice

Brillouin zones vs lattice planes

Relationship of Brillouin zones to Bragg planes

• Brillouin zone boundaries are closely related to the Bragg planes in a crystal
• Bragg planes are defined by the condition for constructive interference of waves scattered by the periodic potential: $2d\sin\theta = n\lambda$
• The reciprocal lattice vectors perpendicular to the Bragg planes have a magnitude of $2\pi/d$, where $d$ is the interplanar spacing
• Brillouin zone boundaries coincide with the Bragg planes in reciprocal space, as they represent the maximum wave vector that can propagate without being scattered

Brillouin zones of common lattices

Cubic lattice Brillouin zones

• For a simple cubic lattice with lattice constant $a$, the first Brillouin zone is a cube with side length $2\pi/a$ centered at the origin of the reciprocal lattice
• The body-centered cubic (BCC) lattice has a Brillouin zone in the shape of a rhombic dodecahedron
• The face-centered cubic (FCC) lattice has a Brillouin zone in the shape of a truncated octahedron

Hexagonal lattice Brillouin zones

• Hexagonal lattices, such as the hexagonal close-packed (HCP) structure, have a Brillouin zone in the shape of a hexagonal prism
• The first Brillouin zone of a hexagonal lattice has six hexagonal faces perpendicular to the basal plane and two hexagonal faces parallel to the basal plane
• The high symmetry points in the hexagonal Brillouin zone are labeled as $\Gamma$, $M$, $K$, $A$, $L$, and $H$

Wave propagation in periodic structures

Dispersion relation in periodic lattices

• The dispersion relation describes the relationship between the wave vector $k$ and the frequency $\omega$ (or energy $E$) of a wave propagating in a periodic structure
• In a periodic potential, the dispersion relation is periodic in reciprocal space with the periodicity of the reciprocal lattice
• The dispersion relation is often represented within the first Brillouin zone, as it contains all the unique information
• The slope of the dispersion relation gives the group velocity of the wave, which determines the speed and direction of energy propagation

Bloch wave functions

• Bloch's theorem states that the eigenstates of a periodic Hamiltonian can be written as the product of a plane wave and a periodic function with the same periodicity as the lattice
• The Bloch wave function is given by $\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})$, where $u_{n\mathbf{k}}(\mathbf{r})$ is a periodic function and $n$ is the band index
• Bloch wave functions are used to describe the electronic states in a periodic potential, such as in crystalline solids
• The periodicity of the Bloch wave functions in reciprocal space leads to the formation of energy bands and bandgaps

Fermi surfaces within Brillouin zones

Electron energy bands and Fermi surfaces

• In crystalline solids, the electronic states form energy bands due to the periodic potential of the lattice
• The energy bands are a result of the splitting and shifting of atomic orbitals as they interact in the periodic potential
• The Fermi surface is the surface in reciprocal space that separates the occupied electronic states from the unoccupied states at absolute zero temperature
• The shape of the Fermi surface depends on the electronic band structure and the electron filling, and it can be complex and multi-dimensional within the Brillouin zone

Brillouin zone folding

Extended vs reduced zone schemes

• The dispersion relation and electronic band structure can be represented in two ways: the extended zone scheme and the reduced zone scheme
• In the extended zone scheme, the energy bands are plotted continuously in reciprocal space, extending beyond the first Brillouin zone
• In the reduced zone scheme, the energy bands are folded back into the first Brillouin zone using the periodicity of the reciprocal lattice
• The reduced zone scheme is often preferred as it contains all the unique information and simplifies the visualization of the band structure

Brillouin zone boundaries and degeneracies

High symmetry points and lines

• High symmetry points and lines in the Brillouin zone are points and lines of high symmetry in the reciprocal lattice
• These points and lines are often labeled with specific symbols, such as $\Gamma$ for the center of the Brillouin zone, $X$ for the center of a square face, and $L$ for the center of a hexagonal face
• At high symmetry points and lines, the energy bands can exhibit degeneracies, meaning that multiple electronic states have the same energy
• Degeneracies at high symmetry points and lines are important for understanding the electronic properties of materials, such as the presence of Dirac points or Van Hove singularities

Van Hove singularities

• Van Hove singularities are points in the electronic band structure where the density of states diverges or has a discontinuity
• They occur at critical points in the Brillouin zone where the gradient of the energy band vanishes, such as at saddle points or local extrema
• Van Hove singularities can have significant effects on the electronic and optical properties of materials, such as enhanced optical absorption or electronic instabilities
• The presence and type of Van Hove singularities depend on the symmetry and topology of the electronic band structure in the Brillouin zone

Phonon dispersion in Brillouin zones

Acoustic vs optical phonon branches

• Phonons are quantized lattice vibrations in a crystal, and their dispersion relation can be represented in the Brillouin zone
• There are two main types of phonon branches: acoustic and optical
• Acoustic phonon branches have frequencies that approach zero at the center of the Brillouin zone ($\Gamma$ point) and correspond to sound waves in the crystal
• Optical phonon branches have finite frequencies at the $\Gamma$ point and involve out-of-phase motion of atoms in the unit cell
• The number and type of phonon branches depend on the number of atoms in the unit cell and the symmetry of the crystal structure

Experimental probing of Brillouin zones

X-ray diffraction and Brillouin zones

• X-ray diffraction is a powerful technique for studying the structure of crystalline materials and probing the reciprocal lattice
• The diffraction pattern obtained from X-ray scattering is related to the Fourier transform of the electron density in the crystal
• The positions and intensities of the diffraction peaks provide information about the lattice parameters, symmetry, and atomic positions in the unit cell
• The Brillouin zone boundaries can be determined from the X-ray diffraction data by analyzing the positions of the diffraction peaks in reciprocal space

Neutron scattering and phonon dispersion

• Neutron scattering is a technique used to study the dynamics of atoms and molecules in condensed matter systems
• Inelastic neutron scattering can probe the phonon dispersion relation by measuring the energy and momentum transfer between the neutrons and the lattice vibrations
• The measured phonon dispersion curves provide information about the interatomic forces, lattice dynamics, and thermodynamic properties of the material
• Neutron scattering experiments can map out the phonon dispersion relation throughout the Brillouin zone, revealing the acoustic and optical phonon branches and any anomalies or soft modes