5.2 Tight-binding model

Tight-binding model overview

The tight-binding model calculates electronic band structure by starting from the atomic limit: electrons bound to individual atoms. Instead of treating electrons as nearly free (like the free electron model does), it builds up the solid's wave functions from atomic orbitals. This makes it especially powerful for materials where electrons remain fairly localized, such as d-electron systems and low-dimensional structures like graphene.

The core idea is the linear combination of atomic orbitals (LCAO) method. You take the known orbitals of isolated atoms, place them on a periodic lattice, and allow electrons to "hop" between neighboring sites. The result is a set of energy bands whose width and shape depend on how strongly those orbitals overlap.

Atomic orbitals in crystals

When atoms are brought together to form a crystal, their atomic orbitals begin to overlap. An electron that was confined to a single atom now has some probability of tunneling to a neighboring site. This overlap is what transforms discrete atomic energy levels into continuous energy bands.

In the tight-binding framework, you use these atomic orbitals as your basis functions. Each orbital is assumed to be well-localized around its atomic site, with a spatial extent set by the atom's effective potential. The key approximation is that this localization is strong enough that each orbital overlaps appreciably only with orbitals on nearby atoms, not distant ones.

Bloch functions from atomic orbitals

Because the crystal has translational periodicity, the electronic eigenstates must satisfy Bloch's theorem. Each eigenstate can be labeled by a wave vector $\mathbf{k}$ (the crystal momentum) and a band index $n$.

In the tight-binding model, you construct Bloch functions as weighted sums of atomic orbitals across all lattice sites. For a single orbital $\phi$ on a lattice with sites at positions $\mathbf{R}$, the Bloch function is:

$\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k} \cdot \mathbf{R}} \, \phi(\mathbf{r} - \mathbf{R})$

The phase factor $e^{i\mathbf{k} \cdot \mathbf{R}}$ encodes the periodicity. Summing over all $N$ lattice sites with this phase automatically produces a state that satisfies Bloch's theorem.

Wannier functions vs Bloch functions

Bloch functions and Wannier functions are two complementary representations of the same electronic states.

• Bloch functions are extended throughout the crystal and have a well-defined crystal momentum $\mathbf{k}$. They're the natural choice for describing transport and band dispersion.
• Wannier functions are obtained by Fourier transforming Bloch functions over the Brillouin zone. They are localized around individual lattice sites and do not have a well-defined momentum.

Wannier functions are useful when you need a real-space picture, for instance when constructing effective tight-binding Hamiltonians from first-principles calculations, or when studying localized phenomena like defect states and polarization.

Tight-binding Hamiltonian

The tight-binding Hamiltonian captures two physical contributions:

• On-site energies (diagonal elements): the energy of an electron sitting in a particular atomic orbital, as if the neighboring atoms weren't there.
• Hopping integrals (off-diagonal elements): the matrix elements that describe an electron tunneling from an orbital on one site to an orbital on a neighboring site.

In matrix form, for a single-orbital model, the Hamiltonian in the Bloch basis becomes:

$H(\mathbf{k}) = \varepsilon_0 + \sum_{\boldsymbol{\delta}} t_{\boldsymbol{\delta}} \, e^{i\mathbf{k} \cdot \boldsymbol{\delta}}$

where $\varepsilon_0$ is the on-site energy, $t_{\boldsymbol{\delta}}$ is the hopping integral to a neighbor at displacement $\boldsymbol{\delta}$, and the sum runs over neighboring sites.

Diagonal elements of the Hamiltonian matrix

The diagonal element $\varepsilon_0$ represents the on-site energy:

$\varepsilon_0 = \langle \phi(\mathbf{r} - \mathbf{R}) | \hat{H} | \phi(\mathbf{r} - \mathbf{R}) \rangle$

This is the energy an electron would have if it occupied that orbital with no hopping to other sites. It depends on the atomic species and the orbital type (s, p, d, etc.). In a multi-atom basis, different sublattice sites can have different on-site energies, which directly affects whether band gaps open.

Off-diagonal elements of the Hamiltonian matrix

The off-diagonal hopping integral between sites $\mathbf{R}$ and $\mathbf{R}'$ is:

$t = \langle \phi(\mathbf{r} - \mathbf{R}) | \hat{H} | \phi(\mathbf{r} - \mathbf{R}') \rangle$

This quantifies how easily an electron tunnels between the two sites. Hopping integrals typically decay exponentially with the distance $|\mathbf{R} - \mathbf{R}'|$, which is why the tight-binding model often truncates the sum to nearest neighbors (or at most next-nearest neighbors). The sign and magnitude of $t$ control the shape of the resulting bands.

Energy bands from the tight-binding model

Solving the eigenvalue problem $H(\mathbf{k})\,\psi = E(\mathbf{k})\,\psi$ at each $\mathbf{k}$ point gives you the energy bands $E_n(\mathbf{k})$.

For the simplest case of a one-dimensional chain with lattice constant $a$ and nearest-neighbor hopping $t$, the dispersion relation is:

$E(k) = \varepsilon_0 + 2t\cos(ka)$

This single cosine band illustrates the essential physics: atomic levels broaden into bands, and the band shape is dictated by the lattice geometry and hopping parameters.

Energy dispersion relations

The dispersion relation $E(\mathbf{k})$ tells you how energy varies with crystal momentum. From it you can extract:

• Effective mass: $m^* = \hbar^2 \left( \frac{d^2 E}{dk^2} \right)^{-1}$, which governs how electrons respond to external fields. Near band edges, the curvature of $E(k)$ determines whether the effective mass is large (flat band, localized character) or small (dispersive band, delocalized character).
• Group velocity: $v_g = \frac{1}{\hbar}\frac{dE}{dk}$, which describes the speed of wave packet propagation.
• Density of states: derived from the dispersion, it tells you how many states are available at a given energy.

Bandwidth of energy bands

The bandwidth $W$ is the total energy range spanned by a band. For the 1D chain example above, $W = 4|t|$ (the difference between the maximum and minimum of $2t\cos(ka)$).

• Larger hopping integrals produce wider bands, meaning electrons are more delocalized.
• Smaller hopping integrals produce narrower bands, meaning electrons are more localized.

Narrow bandwidths are physically significant: when the bandwidth becomes comparable to or smaller than the electron-electron interaction energy $U$, correlation effects dominate. This is the regime where Mott insulators appear, where a material that band theory predicts to be metallic is actually insulating due to strong correlations.

Overlap integral in the tight-binding model

The overlap integral measures how much two atomic orbitals on different sites share the same region of space:

$S = \langle \phi(\mathbf{r} - \mathbf{R}) | \phi(\mathbf{r} - \mathbf{R}') \rangle$

This is distinct from the hopping integral $t$, though both depend on the distance between sites and the symmetry of the orbitals involved. In many textbook treatments, the overlap integral is set to zero (the orthogonal tight-binding approximation) to simplify the math. When $S \neq 0$, you must solve a generalized eigenvalue problem $H\psi = ES\psi$ instead of a standard one.

Larger overlap integrals generally correlate with stronger hopping and broader bands, but keeping track of $S$ separately from $t$ matters for quantitative accuracy.

Orthogonality of Wannier functions

Wannier functions form an orthonormal set:

$\langle w_n(\mathbf{r} - \mathbf{R}) | w_{n'}(\mathbf{r} - \mathbf{R}') \rangle = \delta_{nn'}\delta_{\mathbf{R}\mathbf{R}'}$

This means Wannier functions centered on different sites (or belonging to different bands) have zero overlap. This orthonormality is what makes them so convenient as a basis: the tight-binding Hamiltonian expressed in the Wannier basis has a clean structure where the matrix elements are directly the on-site energies and hopping integrals, with no overlap matrix to worry about.

This property also simplifies the calculation of physical observables like the electronic polarization (via the Berry phase formulation) and the construction of effective low-energy models.

Accuracy of the tight-binding approximation

The tight-binding model works best when electrons are well-localized, meaning the atomic orbital picture is a reasonable starting point. It tends to be most accurate for:

• d and f electron systems, where orbitals are compact
• Covalent semiconductors, where bonding can be described by a small number of orbitals
• Low-dimensional systems like graphene, where the relevant physics involves a small set of orbitals

The accuracy can be systematically improved by:

1. Including hopping to second-nearest, third-nearest neighbors, etc.
2. Adding more orbitals per site (e.g., both s and p orbitals)
3. Fitting the parameters $\varepsilon_0$ and $t$ to match first-principles (DFT) calculations or experimental data
4. Using Wannier function methods to construct maximally localized tight-binding models from ab initio results

Comparison to the nearly-free electron model

These two models represent opposite starting points for understanding band structure:

Both models produce the same qualitative result (energy bands and gaps), but each is more natural and converges faster in its respective regime.

Applications of the tight-binding model

The tight-binding model is widely used across condensed matter physics because it provides physical intuition alongside quantitative predictions. It connects the chemistry of atomic orbitals directly to the band structure of the solid.

Graphene band structure from tight-binding

Graphene is one of the most celebrated applications of the tight-binding model. The carbon atoms sit on a honeycomb lattice with two atoms per unit cell (sublattices A and B). Each carbon contributes one $p_z$ orbital perpendicular to the plane, forming the $\pi$ bands.

With nearest-neighbor hopping $t \approx -2.7$ eV, the tight-binding Hamiltonian for graphene is a $2 \times 2$ matrix:

$H(\mathbf{k}) = \begin{pmatrix} 0 & t\,f(\mathbf{k}) \\ t\,f^*(\mathbf{k}) & 0 \end{pmatrix}$

where $f(\mathbf{k}) = \sum_{\boldsymbol{\delta}} e^{i\mathbf{k}\cdot\boldsymbol{\delta}}$ sums over the three nearest-neighbor vectors. The eigenvalues give two bands:

$E(\mathbf{k}) = \pm |t\,f(\mathbf{k})|$

These bands touch at the K and K' points of the Brillouin zone (the Dirac points). Near these points, the dispersion is linear: $E \approx \hbar v_F |\mathbf{q}|$, where $v_F \approx 10^6$ m/s is the Fermi velocity and $\mathbf{q}$ is measured from the Dirac point. This linear dispersion means electrons in graphene behave as massless Dirac fermions, which underlies graphene's extraordinary electronic mobility and its unusual quantum Hall effect.

Transition metal oxides

Transition metal oxides (e.g., $\text{NiO}$, $\text{V}_2\text{O}_3$, cuprate superconductors) exhibit rich physics driven by the interplay of electron hopping and strong Coulomb repulsion.

The tight-binding model captures the essential electronic structure by including the transition metal d orbitals and their hybridization with oxygen p orbitals. The resulting bands are often narrow (small $t$), which makes the on-site Coulomb repulsion $U$ comparable to or larger than the bandwidth. This competition between $t$ and $U$ is at the heart of:

• Mott insulating states, where half-filled bands become insulating despite band theory predicting a metal
• Orbital ordering, where specific d orbitals preferentially occupy certain sites
• Magnetic ordering, driven by superexchange interactions that can be derived from the tight-binding hopping parameters

The tight-binding model, often extended to the Hubbard model (which adds an explicit $U$ term), serves as the foundation for studying these correlated electron phenomena.

Limitations of the tight-binding model

• The LCAO basis is incomplete. A finite set of atomic orbitals cannot capture all features of the true wave functions, especially in regions between atoms where the potential differs significantly from the atomic case.
• Electron-electron interactions are not included in the standard tight-binding model. For strongly correlated systems, you need extensions like the Hubbard model or dynamical mean-field theory.
• The hopping parameters are often treated as empirical fitting parameters. Without careful fitting to experiment or first-principles data, the model can give qualitatively correct but quantitatively unreliable results.
• The model struggles with nearly-free-electron metals (like sodium or aluminum) where the electrons are highly delocalized and a plane-wave basis is far more efficient.
• For complex multi-orbital systems, the number of parameters (multiple on-site energies, many distinct hopping integrals) can grow quickly, reducing the model's simplicity advantage.