2.3 Tight-binding model

Fundamentals of the Tight-Binding Model

The tight-binding model is a quantum mechanical approach for describing electronic properties of solids. It starts from the opposite limit of the free electron model: instead of treating electrons as nearly free, it assumes electrons are tightly bound to individual atoms and only weakly interact with neighboring sites. This makes it a natural bridge between atomic physics and solid-state band theory.

The model expresses crystal wavefunctions as superpositions of atomic orbitals, allowing electrons to "hop" between adjacent atoms. By neglecting electron-electron interactions and focusing on valence electrons, it provides a tractable framework that still captures the essential physics of band formation.

Concept and Assumptions

The tight-binding model rests on several key assumptions:

• Electrons in the solid remain mostly localized on their parent atoms, with only small overlap between neighboring sites.
• Wavefunctions are built as linear combinations of atomic orbitals (LCAO), so the starting point is the known solution for isolated atoms.
• Electron-electron interactions are ignored; you work within a single-particle approximation.
• Only valence electrons matter. Core electrons are so tightly bound to nuclei that they don't contribute to bonding or conduction.

The physical picture is straightforward: each atom contributes its atomic orbitals, and the crystal's electronic structure emerges from the way those orbitals overlap and hybridize across the lattice.

Linear Combination of Atomic Orbitals

The LCAO method constructs crystal wavefunctions from atomic orbitals $\phi_n(\mathbf{r} - \mathbf{R})$ centered at each lattice site $\mathbf{R}$. To satisfy Bloch's theorem for a periodic crystal, you form the Bloch sum:

$\psi_k(\mathbf{r}) = \sum_{\mathbf{R}} e^{i\mathbf{k} \cdot \mathbf{R}} \phi_n(\mathbf{r} - \mathbf{R})$

Each term in the sum is the same atomic orbital, but shifted to a different lattice site and weighted by a phase factor $e^{i\mathbf{k} \cdot \mathbf{R}}$. That phase factor encodes the crystal momentum $\mathbf{k}$ and ensures the wavefunction has the correct translational symmetry. The result is a state that looks atomic near each site but behaves as a Bloch wave across the whole crystal.

Bloch's Theorem Application

Bloch's theorem states that for any Hamiltonian with the periodicity of the lattice, eigenstates take the form:

$\psi_k(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r})$

where $u_k(\mathbf{r})$ has the same periodicity as the lattice. This is powerful because it reduces the problem from an infinite crystal to solving for $u_k(\mathbf{r})$ within a single unit cell. Every electronic state is then labeled by a crystal momentum $\mathbf{k}$, which lives inside the first Brillouin zone. The tight-binding Bloch sums constructed above automatically satisfy this theorem.

Hamiltonian in the Tight-Binding Model

The tight-binding Hamiltonian captures two competing effects: the energy cost of sitting on an atom (on-site energy) and the energy gain from delocalizing across neighbors (hopping). The balance between these determines the band structure.

Nearest-Neighbor Approximation

The most common simplification limits electron hopping to adjacent atomic sites only. This works because the overlap between atomic orbitals drops off rapidly with distance, so second-neighbor and longer-range hoppings are typically much smaller. For many covalent solids, this approximation captures the essential band structure while keeping the math manageable.

Hopping Integrals

The hopping integral between sites $i$ and $j$ is defined as:

$t_{ij} = \langle \phi_i | H | \phi_j \rangle$

This matrix element quantifies the probability amplitude for an electron to tunnel from one atomic orbital to another. A few things to note:

• Hopping integrals decrease rapidly with interatomic distance, which is why the nearest-neighbor approximation works.
• The magnitude of $t$ directly controls the bandwidth: larger hopping means wider bands and more delocalized electrons.
• The sign and symmetry of $t_{ij}$ depend on the orbital types involved (s, p, d) and their relative orientation.

On-Site Energy Terms

The on-site energy is the diagonal matrix element:

$\epsilon_i = \langle \phi_i | H | \phi_i \rangle$

This represents the energy of an electron localized on atom $i$. It includes contributions from the isolated atomic energy level plus any shifts from the local crystal field. In a monatomic crystal with one orbital per site, all on-site energies are identical. When different atom types or orbitals are present, differences in $\epsilon_i$ set the relative positions of different bands and can open band gaps.

Band Structure Calculation

Band structure gives you $E(\mathbf{k})$, the relationship between electron energy and crystal momentum. This is the central output of the tight-binding model and determines whether a material is a metal, semiconductor, or insulator.

One-Dimensional Chain

The simplest case: identical atoms spaced by lattice constant $a$, each contributing one s-orbital.

1. Write the Bloch sum and apply the tight-binding Hamiltonian with on-site energy $\epsilon_0$ and nearest-neighbor hopping $t$.
2. The only neighbors of each atom are at $\pm a$, so the sum over neighbors gives two terms with phases $e^{\pm ika}$.
3. The resulting dispersion relation is:

$E(k) = \epsilon_0 + 2t \cos(ka)$

This cosine band has a total width of $4|t|$ (from $\epsilon_0 - 2|t|$ to $\epsilon_0 + 2|t|$) and is periodic in $k$ with period $2\pi/a$, consistent with the first Brillouin zone $-\pi/a \leq k \leq \pi/a$. Note that $t$ is typically negative for s-orbitals, so the band minimum sits at $k = 0$.

Two-Dimensional Lattices

Extending to 2D introduces richer physics:

• Square lattice: Each atom has four nearest neighbors. The dispersion becomes $E(\mathbf{k}) = \epsilon_0 + 2t[\cos(k_x a) + \cos(k_y a)]$, which you can plot over the 2D Brillouin zone.
• Hexagonal lattice (graphene): The unit cell contains two atoms, so you get two bands. These bands touch at the corners of the Brillouin zone (the K and K' points), forming the famous Dirac cones with linear dispersion $E \propto |\mathbf{k} - \mathbf{K}|$ near those points.

The lattice geometry directly controls the symmetry of the band structure and can produce features like saddle points and flat bands that have no 1D analog.

Three-Dimensional Crystals

In 3D, the full periodicity of the crystal lattice produces intricate band structures with multiple overlapping bands. Realistic calculations typically require:

• Multiple orbitals per atom (s, p, d)
• Longer-range hopping terms beyond nearest neighbors
• Careful treatment of symmetry to identify high-symmetry paths in the Brillouin zone

The tight-binding approach can predict whether a 3D material is a metal (bands crossing the Fermi level), a semiconductor (small gap), or an insulator (large gap).

Density of States

The density of states (DOS) counts how many electronic states are available at each energy. It's formally defined as:

$D(E) = \sum_n \int \frac{d^3k}{(2\pi)^3} \delta(E - E_n(\mathbf{k}))$

The DOS determines many measurable properties: electronic specific heat, electrical conductivity, optical absorption, and more. A high DOS at the Fermi energy means lots of states are available for conduction; a gap in the DOS means the material is insulating.

Calculation Methods

• Analytical: For simple models (1D chain, 2D square lattice), you can evaluate the integral directly. The 1D chain gives a DOS with inverse-square-root divergences at the band edges.
• Numerical integration: For 3D systems, integrate over constant-energy surfaces in k-space. The tetrahedron method divides the Brillouin zone into tetrahedra and interpolates linearly, giving smooth and accurate results.
• Green's function techniques: Useful for disordered systems or when you need the local DOS at a specific site.

DOS vs. Energy Plots

Plotting $D(E)$ reveals the character of a material at a glance:

• Van Hove singularities appear where $\nabla_k E(\mathbf{k}) = 0$ (band edges, saddle points). These show up as kinks, jumps, or divergences in the DOS depending on dimensionality.
• Metals have a finite DOS at the Fermi energy; semiconductors and insulators have a gap.
• The shape of the DOS near the Fermi level strongly influences transport and thermodynamic properties.

Applications in Materials Science

Metals and Semiconductors

Tight-binding models predict whether a material conducts or not based on its band filling and band gap. For example, copper's overlapping s and d bands produce metallic behavior, while silicon's sp3-hybridized bands open a gap of about 1.1 eV. The model also handles doping naturally: adding impurity on-site energies shifts states into the gap, explaining how small concentrations of dopants dramatically change conductivity.

Graphene and Carbon Nanotubes

Graphene is the textbook tight-binding success story. A nearest-neighbor model on the honeycomb lattice with a single $p_z$ orbital per carbon atom and hopping parameter $t \approx -2.7$ eV reproduces the Dirac cone dispersion remarkably well.

Carbon nanotubes are conceptually rolled-up graphene sheets. The tight-binding model predicts that a nanotube is metallic or semiconducting depending on its chirality (the rolling direction), specifically on whether the allowed k-lines in the rolled geometry pass through the Dirac points. This prediction has been confirmed experimentally.

Topological Insulators

Tight-binding models on lattices with spin-orbit coupling can capture band inversion and topological phase transitions. Materials like $\text{Bi}_2\text{Se}_3$ and HgTe quantum wells have been modeled this way, predicting topologically protected surface states with spin-momentum locking. These surface states are robust against non-magnetic disorder, making them interesting for spintronics.

Limitations and Extensions

Validity of Approximations

The basic tight-binding model works best for materials with well-localized orbitals (covalent and ionic solids). It struggles in several situations:

• Strongly correlated systems where electron-electron interactions dominate (e.g., transition metal oxides). The single-particle picture breaks down.
• Metallic systems where electrons are highly delocalized and long-range interactions matter.
• Materials with significant core-valence overlap, where ignoring core electrons introduces errors.

Spin-Orbit Coupling Inclusion

For heavy elements (large atomic number), relativistic effects become significant. Spin-orbit coupling modifies the Hamiltonian by adding spin-dependent hopping terms. This leads to:

• Band splitting at high-symmetry points
• Topological phase transitions (e.g., from normal insulator to topological insulator)
• Phenomena like the Rashba effect (spin splitting at surfaces/interfaces) and the quantum spin Hall effect

Many-Body Effects

To go beyond the single-particle picture, the most common extension adds a Hubbard-U term:

$H_U = U \sum_i n_{i\uparrow} n_{i\downarrow}$

This penalizes double occupancy of a site, capturing on-site Coulomb repulsion. The resulting Hubbard model is central to understanding Mott insulators (materials that band theory predicts to be metallic but are actually insulating due to strong correlations) and is widely studied in the context of high-temperature superconductivity.

Comparison with Other Models

Tight-Binding vs. Free Electron Model

The two models represent opposite limits. Real materials often fall somewhere in between, and the better choice depends on how localized or delocalized the relevant electrons are.

Tight-Binding vs. Density Functional Theory

Tight-binding is semi-empirical: you fit the hopping parameters to experiment or to DFT calculations. DFT is a first-principles method that computes electronic structure from the atomic numbers alone, with no fitting parameters (in principle).

• DFT is more accurate and predictive but computationally expensive, scaling as $O(N^3)$ or worse with system size.
• Tight-binding is orders of magnitude faster, making it practical for very large systems (thousands to millions of atoms), nanostructures, and transport calculations.
• A common workflow is to run DFT on a small system, extract tight-binding parameters, then use those parameters for large-scale tight-binding simulations.

Computational Implementations

Matrix Formulation

The tight-binding problem reduces to a matrix eigenvalue equation. For each $\mathbf{k}$-point:

1. Construct the Hamiltonian matrix $H(\mathbf{k})$ in the basis of atomic orbitals, with on-site energies on the diagonal and hopping integrals (multiplied by appropriate phase factors) on the off-diagonal.
2. If orbitals on different sites are not orthogonal, also construct the overlap matrix $S(\mathbf{k})$.
3. Solve the generalized eigenvalue problem $H(\mathbf{k})\mathbf{c} = E(\mathbf{k})S(\mathbf{k})\mathbf{c}$.
4. The eigenvalues give the band energies $E_n(\mathbf{k})$; the eigenvectors give the orbital composition of each band.

For large systems, the Hamiltonian matrix is sparse (most elements are zero because only nearby atoms are coupled), so sparse matrix storage and algorithms are essential.

Numerical Methods

• Direct diagonalization (e.g., LAPACK routines) works for small to medium matrices but scales as $O(N^3)$.
• Iterative methods like Lanczos or Davidson algorithms efficiently find a subset of eigenvalues for large systems without diagonalizing the full matrix.
• k-space integration for the DOS uses techniques like the tetrahedron method or Gaussian broadening.
• Fast Fourier transforms convert between real-space and k-space representations efficiently.

Software Packages

Several widely used packages implement tight-binding calculations:

• PythTB (Python): Simple and well-documented, good for learning and prototyping models.
• Kwant (Python): Designed for quantum transport in nanostructures, handles scattering and conductance calculations.
• TBmodels (Python): Focuses on constructing and manipulating tight-binding models, integrates well with Wannier90 output.
• Wannier90: Constructs maximally localized Wannier functions from DFT output, providing tight-binding parameters for realistic materials.

Experimental Validation

Angle-Resolved Photoemission Spectroscopy (ARPES)

ARPES directly maps the occupied band structure $E(\mathbf{k})$ by measuring the energy and momentum of photoelectrons ejected from a sample. Comparing ARPES data with tight-binding dispersion curves is one of the most direct ways to validate model parameters. ARPES has confirmed tight-binding predictions for graphene's Dirac cones, topological surface states, and Fermi surface topologies in many materials.

Scanning Tunneling Spectroscopy (STS)

STS measures the local density of states at a surface by recording the differential conductance $dI/dV$ as a function of bias voltage. This can be compared directly with the DOS calculated from tight-binding models. STS is particularly valuable for validating predictions of edge states in nanoribbons and surface states in topological insulators, where the local electronic structure varies spatially.

Optical Spectroscopy

Optical absorption and reflectivity measurements probe interband transitions. The absorption spectrum depends on the joint density of states (the convolution of occupied and unoccupied DOS weighted by transition matrix elements). Tight-binding calculations predict the energies and strengths of these transitions, and comparison with measured spectra validates the band structure, especially the size of band gaps in semiconductors.