5.2 Tight-binding model

The tight-binding model is a quantum mechanical approach used to describe the electronic structure of solids. It assumes electrons are tightly bound to atomic orbitals and uses linear combinations of these orbitals to approximate electronic wave functions in crystals.

This model helps us understand energy band formation and electronic properties of materials. It bridges the gap between atomic and solid-state perspectives, providing insights into how atomic orbitals interact and form delocalized electronic states in crystalline structures.

Tight-binding model overview

• The tight-binding model is a quantum mechanical approach used to describe the electronic structure of solids, particularly in systems where electrons are tightly bound to atomic orbitals
• It assumes that the electronic wave functions in a solid can be approximated by a linear combination of atomic orbitals (LCAO) centered at each atomic site
• The model provides a framework for understanding the formation of energy bands and the electronic properties of materials, bridging the gap between the atomic and solid-state perspectives

Atomic orbitals in crystals

• In a crystal, the atomic orbitals of individual atoms overlap and interact with each other, leading to the formation of delocalized electronic states
• The tight-binding model considers the atomic orbitals as the basis functions for constructing the electronic wave functions in the solid
• The atomic orbitals are assumed to be localized around each atomic site, with their spatial extent determined by the effective potential of the atom

Bloch functions from atomic orbitals

• Bloch functions are the eigenstates of the Hamiltonian in a periodic potential, such as a crystal lattice
• In the tight-binding model, Bloch functions are constructed as linear combinations of atomic orbitals, taking into account the periodicity of the lattice
• The Bloch functions are characterized by a wave vector $k$ and a band index $n$, representing the crystal momentum and the energy band, respectively

Wannier functions vs Bloch functions

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• Wannier functions are localized functions that are obtained by Fourier transforming the Bloch functions over the Brillouin zone
• While Bloch functions are extended and have a well-defined crystal momentum, Wannier functions are localized around each atomic site and do not have a well-defined momentum
• Wannier functions provide a real-space representation of the electronic states and are useful for studying localized properties and constructing effective models

Tight-binding Hamiltonian

• The tight-binding Hamiltonian describes the energy of the system in terms of the atomic orbitals and their interactions
• It consists of two main contributions: the on-site energies of the atomic orbitals and the hopping integrals between neighboring orbitals
• The Hamiltonian matrix elements are determined by the overlap integrals between the atomic orbitals and the effective potential of the crystal

Diagonal elements of Hamiltonian matrix

• The diagonal elements of the tight-binding Hamiltonian matrix represent the on-site energies of the atomic orbitals
• They correspond to the energy of an electron occupying a particular atomic orbital in the absence of interactions with other orbitals
• The on-site energies are determined by the effective potential of the atom and can vary depending on the atomic species and the orbital type (s, p, d, etc.)

Off-diagonal elements of Hamiltonian matrix

• The off-diagonal elements of the tight-binding Hamiltonian matrix represent the hopping integrals between neighboring atomic orbitals
• They quantify the strength of the electronic coupling between orbitals on different atomic sites
• The hopping integrals depend on the overlap between the atomic orbitals and the distance between the atomic sites, typically decaying exponentially with increasing distance

Energy bands from tight-binding model

• The tight-binding model allows for the calculation of the energy bands of a solid by solving the eigenvalue problem of the Hamiltonian matrix
• The energy bands are obtained as a function of the wave vector $k$, representing the allowed energy states for electrons in the crystal
• The shape and dispersion of the energy bands depend on the hopping integrals and the geometry of the crystal lattice

Energy dispersion relations

• The energy dispersion relations describe how the energy of the electronic states varies with the wave vector $k$
• In the tight-binding model, the dispersion relations are determined by the eigenvalues of the Hamiltonian matrix as a function of $k$
• The dispersion relations provide information about the effective mass, group velocity, and density of states of the electrons in the solid

Bandwidth of energy bands

• The bandwidth of an energy band represents the range of energies spanned by the band
• It is determined by the strength of the hopping integrals between the atomic orbitals
• Larger hopping integrals lead to broader energy bands, indicating stronger electronic coupling and more delocalized states
• Narrower bandwidths correspond to more localized electronic states and can lead to interesting phenomena such as Mott insulators

Overlap integral in tight-binding model

• The overlap integral quantifies the spatial overlap between two atomic orbitals
• It is a measure of the strength of the electronic coupling between the orbitals and determines the magnitude of the hopping integrals in the tight-binding Hamiltonian
• The overlap integral depends on the spatial extent and symmetry of the atomic orbitals involved
• Larger overlap integrals indicate stronger electronic coupling and more significant contributions to the energy bands

Orthogonality of Wannier functions

• Wannier functions, being localized functions centered at each atomic site, form an orthonormal basis set
• The orthogonality of Wannier functions means that the overlap integral between two different Wannier functions is zero
• This property allows for the construction of localized orbital bases and the formulation of effective models in terms of Wannier functions
• The orthogonality of Wannier functions simplifies the calculation of various physical quantities, such as the density matrix and the polarization

Accuracy of tight-binding approximation

• The accuracy of the tight-binding approximation depends on the validity of the assumptions made in the model
• The model assumes that the electronic wave functions can be well-approximated by a linear combination of atomic orbitals, which is more accurate for systems with strongly localized electrons
• The accuracy of the model can be improved by including higher-order hopping terms, considering the overlap between more distant neighbors, or incorporating additional corrections

Comparison to nearly-free electron model

• The nearly-free electron model is another approach to describe the electronic structure of solids, assuming that the electrons are weakly perturbed by the periodic potential of the lattice
• In contrast, the tight-binding model assumes that the electrons are strongly localized around the atomic sites and that the electronic wave functions can be constructed from atomic orbitals
• The tight-binding model is more suitable for systems with strongly correlated electrons or narrow energy bands, while the nearly-free electron model is more appropriate for systems with weakly bound electrons and broad energy bands

Applications of tight-binding model

• The tight-binding model has been successfully applied to a wide range of materials and phenomena in condensed matter physics
• It provides a framework for understanding the electronic properties of materials, such as the band structure, density of states, and transport properties
• The model has been particularly useful in studying low-dimensional systems, such as graphene and carbon nanotubes, where the electronic properties are strongly influenced by the atomic structure

Graphene band structure from tight-binding

• Graphene, a two-dimensional allotrope of carbon, has a unique electronic structure that can be well-described by the tight-binding model
• The tight-binding model for graphene considers the $\pi$ orbitals of the carbon atoms, which give rise to the valence and conduction bands
• The model predicts the existence of Dirac points in the band structure of graphene, where the valence and conduction bands touch at specific points in the Brillouin zone
• The linear dispersion relation near the Dirac points leads to the remarkable electronic properties of graphene, such as high electron mobility and the presence of massless Dirac fermions

Transition metal oxides

• Transition metal oxides are a class of materials known for their rich electronic and magnetic properties, often driven by strong electron correlations
• The tight-binding model has been applied to study the electronic structure of transition metal oxides, particularly in the context of metal-insulator transitions and magnetism
• By considering the d orbitals of the transition metal ions and their hybridization with the oxygen p orbitals, the tight-binding model can capture the essential features of the electronic structure
• The model has been used to investigate phenomena such as the Mott insulating state, orbital ordering, and the interplay between electronic and lattice degrees of freedom in transition metal oxides

Limitations of tight-binding model

• Despite its success in describing various electronic properties, the tight-binding model has certain limitations
• The model relies on the assumption of strongly localized atomic orbitals, which may not always be valid, especially in systems with significant overlap between orbitals or strong electron delocalization
• The model typically considers only a limited number of orbitals and hopping terms, neglecting higher-order contributions that may be important in some cases
• The tight-binding model does not explicitly account for electron-electron interactions, which can play a crucial role in strongly correlated systems
• The model may not capture all the details of the electronic structure, particularly in systems with complex band structures or multiple orbital contributions