2.3 Tight-binding model

The tight-binding model is a quantum mechanical approach used to describe electronic properties of solids. It bridges atomic physics and solid-state band theory by considering electrons tightly bound to atoms, with limited interactions between neighbors.

This model expresses crystal wavefunctions as superpositions of atomic orbitals, allowing for electron hopping between adjacent atoms. It neglects electron-electron interactions and focuses on valence electrons, providing a simplified framework for understanding electronic behavior in materials.

Fundamentals of tight-binding model

• Provides a quantum mechanical approach to describe electronic properties of solids in condensed matter physics
• Bridges the gap between atomic physics and solid-state band theory by considering electrons tightly bound to atoms

Concept and assumptions

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• Assumes electrons in a solid are tightly bound to their respective atoms, with limited interactions between neighboring atoms
• Treats electron wavefunctions as linear combinations of atomic orbitals, allowing for electron hopping between adjacent atoms
• Neglects electron-electron interactions, focusing on single-particle approximation
• Considers only valence electrons, ignoring core electrons tightly bound to nuclei

Linear combination of atomic orbitals

• Expresses crystal wavefunctions as superpositions of atomic orbitals from different lattice sites
• Utilizes basis functions $\phi_n(\mathbf{r} - \mathbf{R})$ centered at atomic positions $\mathbf{R}$
• Constructs Bloch functions $\psi_k(\mathbf{r}) = \sum_{\mathbf{R}} e^{i\mathbf{k} \cdot \mathbf{R}} \phi_n(\mathbf{r} - \mathbf{R})$ for periodic systems
• Allows for the description of electronic states in terms of atomic-like wavefunctions

Bloch's theorem application

• States that eigenfunctions of the Hamiltonian for a periodic potential have the form $\psi_k(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r})$
• Incorporates translational symmetry of the crystal lattice into electronic wavefunctions
• Reduces the problem to solving for $u_k(\mathbf{r})$ within a single unit cell
• Enables the description of electronic states in terms of crystal momentum $\mathbf{k}$

Hamiltonian in tight-binding model

• Describes the energy of electrons in a solid using a simplified quantum mechanical framework
• Focuses on the interplay between localized atomic states and electron hopping between neighboring atoms

Nearest-neighbor approximation

• Limits electron hopping to adjacent atomic sites, simplifying the model
• Assumes overlap integrals between non-neighboring atoms are negligible
• Reduces computational complexity while maintaining essential physics
• Provides a good approximation for many materials (covalent solids)

Hopping integrals

• Quantify the probability of electrons tunneling between neighboring atomic sites
• Represented by matrix elements $t_{ij} = \langle \phi_i | H | \phi_j \rangle$ between atomic orbitals
• Decrease rapidly with increasing distance between atoms
• Determine the width and shape of energy bands in the solid

On-site energy terms

• Represent the energy of electrons localized on individual atoms
• Appear as diagonal elements $\epsilon_i = \langle \phi_i | H | \phi_i \rangle$ in the Hamiltonian matrix
• Include contributions from atomic energy levels and local crystal field effects
• Influence the position of energy bands relative to each other

Band structure calculation

• Determines the relationship between electron energy and crystal momentum in a solid
• Reveals the electronic properties and behavior of materials in condensed matter physics

One-dimensional chain

• Models simplest case of atoms arranged in a linear chain with lattice constant $a$
• Yields dispersion relation $E(k) = \epsilon_0 + 2t \cos(ka)$ for s-orbital chain
• Demonstrates formation of energy bands with width $4t$
• Illustrates concepts of Brillouin zone and band gaps in periodic systems

Two-dimensional lattices

• Extends model to planar structures (square, hexagonal lattices)
• Produces more complex band structures with multiple bands and symmetry points
• Reveals anisotropic electronic properties dependent on lattice geometry
• Applies to materials like graphene, showcasing Dirac points and linear dispersion

Three-dimensional crystals

• Encompasses full 3D periodicity of crystalline solids
• Results in intricate band structures with multiple overlapping bands
• Requires consideration of multiple orbitals and longer-range interactions
• Predicts electronic properties of bulk materials (metals, semiconductors, insulators)

Density of states

• Describes the number of available electronic states per unit energy in a solid
• Plays crucial role in determining various physical properties of materials in condensed matter physics

Definition and significance

• Represents the number of electronic states per unit energy per unit volume
• Formally defined as $D(E) = \sum_n \int \frac{d^3k}{(2\pi)^3} \delta(E - E_n(\mathbf{k}))$
• Determines electronic, thermal, and optical properties of materials
• Influences phenomena like electrical conductivity and heat capacity

Calculation methods

• Employs numerical integration over constant energy surfaces in k-space
• Utilizes tetrahedron method for improved accuracy in 3D systems
• Applies analytical approaches for simple models (1D chain, 2D square lattice)
• Incorporates Green's function techniques for more complex systems

DOS vs energy plots

• Visualizes distribution of electronic states across energy spectrum
• Reveals characteristic features (van Hove singularities, band edges)
• Distinguishes between metals, semiconductors, and insulators
• Identifies energy ranges with high or low density of states, impacting material properties

Applications in materials science

• Demonstrates practical utility of tight-binding model in understanding and predicting material properties
• Bridges theoretical concepts with real-world applications in condensed matter physics

Metals and semiconductors

• Predicts metallic or semiconducting behavior based on band structure
• Explains conductivity differences between materials (Cu, Si, Ge)
• Models doping effects on electronic properties of semiconductors
• Aids in designing new materials for electronic and optoelectronic applications

Graphene and carbon nanotubes

• Describes unique electronic properties of 2D graphene sheets
• Predicts band structure of carbon nanotubes based on graphene folding
• Explains metallic or semiconducting behavior of nanotubes depending on chirality
• Models edge states and quantum confinement effects in nanoribbons

Topological insulators

• Captures band inversion and topological phase transitions
• Predicts existence of protected surface states in materials (Bi2Se3, HgTe)
• Models spin-momentum locking in topological surface states
• Aids in designing new topological materials for spintronics applications

Limitations and extensions

• Acknowledges constraints of the basic tight-binding model in condensed matter physics
• Explores methods to enhance the model's accuracy and applicability to complex systems

Validity of approximations

• Assesses accuracy of nearest-neighbor approximation for different materials
• Examines limitations of single-particle picture in strongly correlated systems
• Considers effects of neglecting core electrons in certain compounds
• Evaluates importance of long-range interactions in ionic or metallic systems

Spin-orbit coupling inclusion

• Incorporates relativistic effects on electron spin in heavy elements
• Modifies Hamiltonian to include spin-dependent hopping terms
• Predicts band splitting and topological phase transitions
• Explains phenomena like Rashba effect and quantum spin Hall effect

Many-body effects

• Extends model to include electron-electron interactions
• Incorporates Hubbard-U term for on-site Coulomb repulsion
• Models correlation effects in transition metal compounds
• Addresses phenomena like Mott insulation and high-temperature superconductivity

Comparison with other models

• Contextualizes tight-binding approach within broader landscape of electronic structure methods
• Highlights strengths and weaknesses relative to other theoretical frameworks in condensed matter physics

Tight-binding vs free electron model

• Compares localized orbital picture with delocalized plane wave basis
• Contrasts ability to describe band gaps and complex band structures
• Examines accuracy in modeling different classes of materials (metals vs insulators)
• Discusses computational efficiency and ease of implementation

Tight-binding vs density functional theory

• Compares semi-empirical approach with first-principles calculations
• Contrasts accuracy in predicting electronic and structural properties
• Examines ability to handle large systems and complex geometries
• Discusses trade-offs between computational cost and predictive power

Computational implementations

• Explores practical aspects of applying tight-binding model in computational condensed matter physics
• Bridges theoretical concepts with numerical techniques for solving electronic structure problems

Matrix formulation

• Expresses Hamiltonian as a matrix in basis of atomic orbitals
• Utilizes sparse matrix techniques for efficient storage and operations
• Implements periodic boundary conditions through Bloch's theorem
• Solves generalized eigenvalue problem to obtain band structure

Numerical methods

• Employs diagonalization algorithms for small to medium-sized systems
• Utilizes iterative methods (Lanczos, Davidson) for large-scale problems
• Implements k-space integration techniques for density of states calculations
• Applies fast Fourier transform for efficient real-space to k-space transformations

Software packages for tight-binding

• Reviews popular codes (PythTB, Kwant, TBmodels)
• Discusses features like band structure plotting and transport calculations
• Examines integration with other electronic structure methods (DFT)
• Explores user-friendly interfaces and scripting capabilities

Experimental validation

• Connects theoretical predictions of tight-binding model with experimental measurements
• Demonstrates importance of model in interpreting and guiding experiments in condensed matter physics

Angle-resolved photoemission spectroscopy

• Directly maps electronic band structure in momentum space
• Compares measured dispersion relations with tight-binding predictions
• Reveals Fermi surface topology and many-body effects
• Validates model parameters through fitting of experimental data

Scanning tunneling spectroscopy

• Probes local density of states on material surfaces
• Compares measured dI/dV spectra with calculated DOS from tight-binding
• Reveals spatial variations in electronic structure (defects, edges)
• Validates predictions of surface and edge states in topological materials

Optical spectroscopy techniques

• Measures interband transitions and optical conductivity
• Compares observed absorption spectra with tight-binding calculations
• Reveals information about joint density of states and selection rules
• Validates predictions of band gaps and optical properties in semiconductors