💎Mathematical Crystallography Unit 9 – Reciprocal Space & Brillouin Zones

Key Concepts

• Reciprocal space represents the Fourier transform of the real space lattice
• Brillouin zones are primitive cells in reciprocal space, containing all unique k-points
• Reciprocal lattice vectors ($\vec{b_1}, \vec{b_2}, \vec{b_3}$) are defined as perpendicular to real space lattice planes
  • Magnitude of reciprocal lattice vectors is inversely proportional to interplanar spacing ($|\vec{b}| = \frac{2\pi}{d}$)
• Wigner-Seitz cell construction method used to determine the first Brillouin zone
• Symmetry operations in real space have corresponding operations in reciprocal space
  • Translation, rotation, and reflection symmetries are preserved
• Bloch's theorem describes the behavior of electron wavefunctions in periodic potentials
• Fermi surfaces, which represent the energy states of electrons, are often constructed in reciprocal space

Reciprocal Lattice Basics

• Reciprocal lattice is a Fourier transform of the real space lattice
• Each point in reciprocal space corresponds to a set of lattice planes in real space
• Reciprocal lattice vectors are defined as:
  • $\vec{b_1} = 2\pi \frac{\vec{a_2} \times \vec{a_3}}{\vec{a_1} \cdot (\vec{a_2} \times \vec{a_3})}$
  • $\vec{b_2} = 2\pi \frac{\vec{a_3} \times \vec{a_1}}{\vec{a_1} \cdot (\vec{a_2} \times \vec{a_3})}$
  • $\vec{b_3} = 2\pi \frac{\vec{a_1} \times \vec{a_2}}{\vec{a_1} \cdot (\vec{a_2} \times \vec{a_3})}$
• Reciprocal lattice is crucial for understanding diffraction patterns (X-ray, electron, neutron)
• The reciprocal lattice of a face-centered cubic (FCC) lattice is a body-centered cubic (BCC) lattice, and vice versa
• Reciprocal space is used to describe electronic band structures and phonon dispersions

Brillouin Zone Definition

• Brillouin zones are primitive cells in reciprocal space, containing all unique k-points
• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
  • Constructed by drawing perpendicular bisector planes between the origin and neighboring reciprocal lattice points
• Brillouin zones are named in order of increasing distance from the origin (first, second, third, etc.)
• The first Brillouin zone is of particular importance as it contains all the unique electronic states
• High symmetry points within the Brillouin zone (Γ, X, L, etc.) are often used to describe electronic band structures
• The volume of the Brillouin zone is inversely proportional to the volume of the real space unit cell
• Brillouin zone boundaries represent Bragg planes, where diffraction occurs

Construction Methods

• Wigner-Seitz cell construction is the most common method for determining the first Brillouin zone
  • Draw perpendicular bisector planes between the origin and neighboring reciprocal lattice points
  • The smallest enclosed volume around the origin is the first Brillouin zone
• Ewald sphere construction is used to visualize the conditions for diffraction
  • A sphere with radius $\frac{1}{\lambda}$ is drawn in reciprocal space
  • Diffraction occurs when the Ewald sphere intersects with reciprocal lattice points
• Brillouin zone folding is used to map higher Brillouin zones back into the first Brillouin zone
  • Useful for understanding complex band structures and phonon dispersions
• Brillouin zone interpolation methods (Wannier interpolation, Shirley interpolation) are used to efficiently compute electronic properties
• Brillouin zone integration techniques (tetrahedron method, special k-points) are employed to calculate properties like density of states and total energy

Symmetry in Reciprocal Space

• Symmetry operations in real space have corresponding operations in reciprocal space
  • Translation symmetry in real space leads to discrete points in reciprocal space
  • Rotational symmetry in real space is preserved in reciprocal space
  • Reflection symmetry in real space is also maintained in reciprocal space
• Space group symmetries are used to reduce the computational burden by considering only the irreducible Brillouin zone
• Time-reversal symmetry relates k-points with opposite wavevectors ($\vec{k}$ and $-\vec{k}$)
  • Leads to Kramers degeneracy in systems with time-reversal symmetry
• Inversion symmetry in real space leads to parity selection rules for electronic transitions
• Nonsymmorphic space groups have symmetry elements with fractional translations, leading to additional degeneracies at Brillouin zone boundaries
• Symmetry-adapted plane waves can be used to efficiently represent electronic wavefunctions

Applications in Solid State Physics

• Electronic band structure calculations are performed in reciprocal space
  • Dispersion relations $E(\vec{k})$ describe the energy of electronic states as a function of wavevector
• Fermi surfaces, which represent the energy states of electrons at the Fermi level, are constructed in reciprocal space
  • Shape of the Fermi surface determines electronic properties like electrical conductivity and heat capacity
• Phonon dispersion relations $\omega(\vec{q})$ describe the energy of lattice vibrations as a function of wavevector
  • Used to understand thermal properties like heat capacity and thermal conductivity
• Optical properties (absorption, reflection) are determined by transitions between electronic states in reciprocal space
• Magnetic properties (spin waves, magnon dispersions) are also described in reciprocal space
• Superconductivity is often understood in terms of electron pairing in reciprocal space (Cooper pairs)
• Topological properties (Berry curvature, Chern numbers) are defined in reciprocal space and are used to characterize materials like topological insulators

Computational Techniques

• Density functional theory (DFT) is widely used to compute electronic structure in reciprocal space
  • Kohn-Sham equations are solved self-consistently to determine the electron density and energy
• Plane-wave basis sets are commonly employed in DFT calculations due to their completeness and computational efficiency
  • Pseudopotentials are used to represent the core electrons and reduce the computational cost
• Wannier functions, which are localized in real space, can be constructed from Bloch wavefunctions in reciprocal space
  • Used to compute tight-binding Hamiltonians and interpolate electronic properties
• Brillouin zone integration methods (tetrahedron method, special k-points) are used to efficiently compute properties like density of states and total energy
• Maximally localized Wannier functions (MLWFs) are used to construct minimal tight-binding models and compute topological invariants
• Wannier interpolation techniques enable the efficient computation of electronic properties on dense k-point grids
• Machine learning techniques (neural networks, Gaussian process regression) are being increasingly used to predict electronic properties in reciprocal space

Advanced Topics and Extensions

• Topological invariants (Chern numbers, Z2 invariants) are defined in reciprocal space and are used to characterize topological phases of matter
  • Computed using Berry curvature and Wannier charge centers
• Weyl points and Dirac points are special degeneracies in the electronic band structure that give rise to exotic properties
  • Weyl semimetals and Dirac semimetals are examples of materials hosting these degeneracies
• Floquet engineering involves driving a system periodically in time to create new effective Hamiltonians in reciprocal space
  • Used to create topological phases and manipulate electronic properties
• Moire superlattices, formed by stacking 2D materials with a twist angle, have a reciprocal space description that enables the understanding of their electronic properties
• Electron-phonon coupling, which is responsible for superconductivity and charge density waves, is described by scattering processes in reciprocal space
• Excitonic effects, arising from electron-hole interactions, are described by transitions in reciprocal space
• Spin-orbit coupling, which leads to the splitting of electronic states, is incorporated into reciprocal space calculations
• Reciprocal space descriptions are being extended to quasicrystals and aperiodic systems, where the concept of Brillouin zones is generalized