Glossary

3.2 Wannier functions

4 min read•august 21, 2024

are localized basis functions in condensed matter physics, offering an alternative to extended Bloch states. They bridge real-space and reciprocal-space descriptions of electronic states in periodic systems, providing insights into chemical bonding and local properties.

These functions are obtained by Fourier transforming Bloch states over the Brillouin zone. Their mathematical formulation involves complex transformations, considering and periodicity. Wannier functions possess unique properties like spatial and , making them valuable in various applications.

Definition and concept

  • Wannier functions serve as localized basis functions in condensed matter physics, providing an alternative representation to extended Bloch states
  • These functions play a crucial role in understanding electronic structure and properties of crystalline solids
  • Wannier functions bridge the gap between real-space and reciprocal-space descriptions of electronic states in periodic systems

Localized basis functions

Top images from around the web for Localized basis functions

  • Lattice Structures in Crystalline Solids | Chemistry for Majors View original

    Is this image relevant?

  • Lattice Structures in Crystalline Solids (10.6) – Chemistry 110 View original

    Is this image relevant?

  • Lattice Structures in Crystalline Solids | Chemistry for Majors View original

    Is this image relevant?

1 of 3

Top images from around the web for Localized basis functions

  • Lattice Structures in Crystalline Solids | Chemistry for Majors View original

    Is this image relevant?

  • Lattice Structures in Crystalline Solids (10.6) – Chemistry 110 View original

    Is this image relevant?

  • Lattice Structures in Crystalline Solids | Chemistry for Majors View original

    Is this image relevant?

1 of 3
  • Wannier functions represent localized electronic states in crystalline solids
  • Provide a real-space description of electronic wavefunctions in periodic systems
  • Enable intuitive understanding of chemical bonding and local electronic properties
  • Resemble atomic orbitals but account for the periodic potential of the crystal lattice

Relation to Bloch functions

  • Wannier functions are linear combinations of Bloch functions
  • Transform between extended Bloch states and localized Wannier functions through a unitary transformation
  • Preserve the complete information contained in the Bloch functions
  • Allow for a more intuitive interpretation of electronic structure in real space

Fourier transform of Bloch states

  • Wannier functions obtained by Fourier transforming Bloch states over the Brillouin zone
  • Mathematical expression: wn(r−R)=V(2π)3∫BZe−ik⋅Rψnk(r)dkw_n(\mathbf{r} - \mathbf{R}) = \frac{V}{(2\pi)^3} \int_{\text{BZ}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r}) d\mathbf{k}
  • Inverse transformation reconstructs Bloch functions from Wannier functions
  • Provides a connection between reciprocal space (k-space) and real space representations

Mathematical formulation

  • Mathematical framework of Wannier functions involves complex transformations and considerations of symmetry and periodicity
  • Understanding the mathematical formulation is crucial for accurately constructing and utilizing Wannier functions in condensed matter physics
  • Mathematical properties of Wannier functions determine their usefulness in various applications and numerical methods

Wannier function construction

  • General form of Wannier functions: wn(r−R)=1N∑ke−ik⋅Rψnk(r)w_n(\mathbf{r} - \mathbf{R}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{n\mathbf{k}}(\mathbf{r})
  • Involves a sum over all k-points in the Brillouin zone
  • Requires careful choice of phase factors for Bloch functions
  • Results in a set of localized functions centered at lattice sites R

Gauge freedom

  • Wannier functions not uniquely defined due to in Bloch functions
  • Bloch functions can be multiplied by a k-dependent phase factor without changing physical observables
  • Gauge transformation: ψnk(r)→eiϕn(k)ψnk(r)\psi_{n\mathbf{k}}(\mathbf{r}) \rightarrow e^{i\phi_n(\mathbf{k})} \psi_{n\mathbf{k}}(\mathbf{r})
  • Different gauge choices lead to different sets of Wannier functions with varying degrees of localization

Periodic boundary conditions

  • Wannier functions must satisfy in the crystal lattice
  • Periodicity requirement: wn(r+R)=wn(r)w_n(\mathbf{r} + \mathbf{R}) = w_n(\mathbf{r}) for any lattice vector R
  • Ensures consistency with the underlying periodic structure of the crystal
  • Imposes constraints on the allowed forms of Wannier functions

Properties of Wannier functions

  • Wannier functions possess unique properties that make them valuable tools in condensed matter physics
  • These properties arise from their construction and relationship to Bloch functions
  • Understanding these properties is essential for effectively utilizing Wannier functions in various applications

Spatial localization

  • Wannier functions are spatially localized around specific lattice sites
  • Decay exponentially with distance from their center in many cases
  • Localization extent depends on the band structure and choice of gauge
  • Enables efficient description of local electronic properties and interactions

Orthogonality

  • Wannier functions form an orthonormal set of basis functions
  • Orthogonality condition: ⟨wn(r−R)∣wm(r−R′)⟩=δnmδRR′\langle w_n(\mathbf{r} - \mathbf{R}) | w_m(\mathbf{r} - \mathbf{R}') \rangle = \delta_{nm} \delta_{\mathbf{R}\mathbf{R}'}
  • Ensures linear independence and completeness of the Wannier basis
  • Simplifies calculations and matrix elements in the Wannier representation

Symmetry considerations

  • Wannier functions can be constructed to respect the symmetry of the crystal lattice
  • Symmetry-adapted Wannier functions transform according to irreducible representations of the crystal's space group
  • Symmetry properties aid in understanding and classifying electronic states
  • Can be used to study topological properties and symmetry-protected phases of matter

Applications in condensed matter

  • Wannier functions find extensive applications in various areas of condensed matter physics
  • Their localized nature and connection to Bloch states make them powerful tools for analyzing electronic structure and properties
  • Applications range from simple tight-binding models to complex topological materials

Tight-binding models

  • Wannier functions serve as a natural basis for constructing tight-binding Hamiltonians
  • Enable efficient description of electronic band structures in solids
  • Hopping integrals between Wannier functions determine band dispersions
  • Facilitate the study of electron correlation effects in strongly interacting systems

Chemical bonding analysis

  • Wannier functions provide insights into the nature of chemical bonding in solids
  • Center positions and spreads of Wannier functions reveal bonding characteristics
  • Allow for identification of covalent, ionic, and metallic bonding patterns
  • Aid in understanding the electronic origins of material properties

Topological insulators

  • Wannier functions play a crucial role in characterizing
  • Obstruction to constructing exponentially localized Wannier functions indicates non-trivial topology
  • Hybrid Wannier functions used to study edge states and bulk-boundary correspondence
  • Enable calculation of topological invariants and classification of topological phases

Key Terms to Review (21)

Band Theory: Band theory explains the electronic properties of solids, particularly how energy levels are structured in materials like metals, semiconductors, and insulators. It describes how the overlapping atomic orbitals create energy bands, with the conduction band and valence band defining the material's conductivity. Understanding this concept is crucial for grasping the behavior of various materials in different contexts, such as electrical conduction, optical properties, and magnetic behaviors.
Bloch's Theorem: Bloch's Theorem states that in a periodic potential, such as that found in crystalline solids, the wave functions of electrons can be expressed as a plane wave modulated by a periodic function. This means that the electron wave functions exhibit periodicity in accordance with the lattice structure, leading to the concept of energy bands and gaps which are crucial for understanding the electronic properties of materials.
D. J. Thouless: D. J. Thouless is a renowned physicist known for his contributions to the fields of condensed matter physics and quantum mechanics, particularly in understanding topological phases of matter and the mathematical framework for Wannier functions. His work has significantly influenced the study of Chern insulators and other exotic states of matter, showcasing the deep connection between topology and physical properties in solid-state systems.
Density Functional Theory: Density Functional Theory (DFT) is a computational quantum mechanical modeling method used to investigate the electronic structure of many-body systems, particularly in condensed matter physics. DFT simplifies the complex interactions of electrons in a system by expressing the energy of the system as a functional of the electron density rather than the many-body wave function, making it an essential tool for studying various physical phenomena, such as electronic properties, structural phase transitions, and correlations in materials.
Effective Mass: Effective mass is a concept used in solid-state physics to describe how the motion of charge carriers, such as electrons or holes, responds to external forces within a material. This term captures the effect of the periodic potential of the crystal lattice on the dynamics of these particles, allowing us to treat them as if they possess a different mass than their actual mass. The effective mass plays a crucial role in determining how particles behave under various conditions, linking it to phenomena like energy bands, wave functions, and excitonic effects.
Extended states: Extended states refer to the wave functions of electrons that can spread out over a larger region of space, typically in periodic potentials like crystals. These states are characterized by their delocalized nature, allowing electrons to move freely across a material, which is essential for electrical conductivity and other transport properties in solids.
Gauge freedom: Gauge freedom refers to the property in certain physical theories that allows for the choice of different mathematical descriptions or fields without altering the observable predictions of the theory. This flexibility often manifests in the ability to make transformations on potentials or fields that do not affect physical outcomes, reflecting an underlying symmetry in the system. In condensed matter physics, this concept is particularly relevant when dealing with electronic states and their descriptions, such as in the formulation of Wannier functions.
Gregory Wannier: Gregory Wannier was a prominent physicist known for developing the concept of Wannier functions, which are mathematical constructs used to describe the localization of electronic states in solids. His work has significantly impacted the understanding of condensed matter systems, providing insights into how electrons behave in periodic potentials and enabling deeper analysis of phenomena like electrical conductivity and magnetism.
Interference effects: Interference effects refer to the phenomenon where two or more overlapping waves combine to form a new wave pattern, which can result in regions of constructive or destructive interference. This concept is crucial in understanding how wave functions overlap and influence physical properties in systems such as electrons in solids, where the interference between wave functions can lead to unique behaviors like localization and delocalization.
K·p perturbation theory: k·p perturbation theory is a mathematical approach used in solid state physics to analyze the electronic properties of crystals by treating the wave vector 'k' and momentum operator as perturbations. This technique helps to understand how the energy bands in a crystal are influenced by external factors, such as electric fields or defects, and provides a way to describe how the wave functions of electrons evolve in the presence of periodic potentials. It's particularly useful for studying semiconductor materials and their behavior under various conditions.
Localization: Localization refers to the phenomenon where wave functions, such as those of electrons in a disordered medium, become confined to a limited region of space, preventing them from spreading out and moving freely. This concept is crucial in understanding various physical behaviors of materials, particularly in relation to insulating states and the nature of electronic states in solids, impacting how we think about electron transport and conductivity.
Localized states: Localized states refer to electronic states that are confined to a particular region in a material rather than being spread out over the entire system. These states arise due to various interactions such as disorder, impurities, or specific lattice structures, which prevent the electrons from delocalizing and moving freely. Understanding localized states is crucial for grasping concepts like band gaps and the behavior of materials in different phases.
Optical Properties: Optical properties refer to the behavior of materials when interacting with light, including phenomena such as absorption, reflection, refraction, and transmission. Understanding these properties is crucial in studying how different materials can manipulate light, which directly relates to concepts like effective mass, density of states, and the behavior of electrons in quantum systems.
Orthogonality: Orthogonality refers to the concept where two functions or vectors are perpendicular to each other in a given space, meaning their inner product is zero. This principle is important in quantum mechanics and solid-state physics because it allows for the separation of states and simplifies calculations involving wave functions, especially when discussing the properties of systems like lattices and localized states.
Periodic Boundary Conditions: Periodic boundary conditions (PBC) are a mathematical concept used in simulations and theoretical models that allows a system to be treated as if it repeats infinitely in all directions. This approach is especially useful in condensed matter physics, as it simplifies calculations by eliminating edge effects and mimicking an infinite lattice structure. By applying PBC, one can study the properties of materials without the complications that arise from boundaries, leading to more accurate results in the analysis of physical phenomena.
Symmetry: Symmetry refers to the balanced and proportionate similarity of structure on either side of a dividing line or around a center point. It plays a crucial role in understanding the properties and behaviors of physical systems, where it helps to simplify complex problems by identifying invariant characteristics under certain transformations. This concept is fundamental in classifying crystal structures and analyzing electronic states, revealing deep connections between geometry, physics, and materials science.
Tight-binding model: The tight-binding model is a theoretical framework used to describe the electronic structure of solids, particularly in the context of crystal lattices where electrons are assumed to be tightly bound to their respective atoms. This model helps explain how electrons can hop between neighboring sites in a lattice and leads to the formation of energy bands, which are critical for understanding various electronic properties of materials.
Topological Insulators: Topological insulators are materials that behave as insulators in their bulk while supporting conducting states on their surfaces or edges. This unique property arises from the topological order of the electronic band structure, which distinguishes them from ordinary insulators, allowing for robust surface states that are protected against scattering by impurities or defects.
Wannier Centers: Wannier centers refer to the position of the centers of Wannier functions, which are localized wave functions that represent quantum states in a periodic system. These centers are significant because they provide insight into the distribution of electronic charge within a material, linking the behavior of electrons to their spatial arrangement in real space. Understanding wannier centers is essential for analyzing electronic properties, topological phases, and localization phenomena in solid-state physics.
Wannier Functions: Wannier functions are a complete set of orthonormal wave functions that describe localized states in a crystal lattice, allowing us to represent electronic states in a way that emphasizes their spatial localization. They are particularly useful in condensed matter physics for analyzing properties of materials, as they provide insights into the electronic structure and interactions within a solid.
Wannier Transform: The Wannier Transform is a mathematical operation that transforms a wave function in momentum space into a localized wave function in real space, called a Wannier function. This transformation is crucial for studying the properties of electronic systems, particularly in understanding localization, band structure, and electronic interactions within solids. It provides an alternative perspective to the traditional Bloch functions, allowing for better insight into the spatial distribution of electrons in periodic potentials.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide