Tight-Binding Model: Bands from Orbitals

Introduction

The nearly-free-electron model starts with waves spread through space and adds a weak periodic potential. Tight binding starts from the other end: electrons reside mainly in atomic orbitals, but neighbouring orbitals overlap enough for an electron to move between sites. Repeating these interactions across a crystal turns a discrete atomic orbital energy into a band. This viewpoint is particularly useful for covalent networks, narrow d bands and materials where named orbitals provide a chemically intuitive starting point.

Core explanation

Imagine N equally spaced atoms in a one-dimensional chain, separated by a. Each has a local orbital j⟩ with on-site energy ε₀. A Bloch-like combination is k⟩ proportional to Σ j exp(ikja) j⟩. The phase between neighbouring orbitals is set by k, and the resulting combination transforms correctly under translation. There are N distinct spatial k states for N orbitals, before considering spin. This is the extended-solid analogue of forming bonding and antibonding combinations from two atomic orbitals.

In the simplest nearest-neighbour model, the Hamiltonian has on-site matrix element ε₀ and hopping matrix element t between j and j±1. Assuming orthogonal local orbitals and no farther-neighbour hopping, the band dispersion is E(k) = ε₀ + 2t cos(ka) . Many chemistry and physics texts define a positive hopping magnitude γ = −t and write E(k) = ε₀ − 2γ cos(ka). Both are the same convention if t = −γ. The first-zone range is approximately −π/a to π/a. At k = 0, all orbital coefficients have the same phase; at the boundary they alternate signs between neighbouring atoms. For a bonding hopping sign t < 0, the in-phase state lies lower.

The cosine ranges from −1 to +1, so the band runs from ε₀ − 2 t to ε₀ + 2 t and its width is 4 t . If neighbouring atoms move farther apart, overlap and hopping usually decrease, narrowing the band toward the isolated-orbital energy ε₀. Stronger overlap generally broadens the band and can increase electron mobility, although conductivity also depends on filling and scattering. An entirely filled wide band need not conduct; an extremely narrow partially filled band may conduct poorly if disorder or electron interactions localise carriers.

The tight-binding dispersion has zero slope at the centre and boundary of this simple band. Group velocity is v g = (1/ħ)dE/dk = −(2ta/ħ)sin(ka). At a band extremum, the local curvature determines an effective mass. The model also predicts a one-dimensional DOS that becomes large near band edges, where the dispersion flattens and many nearby k values occupy a small energy range. The MIT quantum-physics notes on a tight-binding chain derive the cosine band, and a MIT materials problem set uses the equivalent ε₀−2γ cos(ka) convention.

Real materials usually have more than one relevant orbital or atom per cell. An s and a p orbital can form separate or hybridising bands; two atoms per cell can split a simple orbital family into two branches. Directional overlap makes three-dimensional bands anisotropic. Nonorthogonal atomic orbitals require an overlap matrix, and a tight-binding calculation then solves a generalised eigenvalue problem rather than the simple formula above. Electron-electron correlation can be especially important when a narrow band is partly filled. Tight binding is a controlled chemical language for building more realistic models, not a claim that electrons are permanently fixed to individual atoms.

Step-by-step reasoning

1. Choose the atomic or local orbitals relevant near the energy range of interest. 2. Assign on-site energies and permitted hopping links by geometry and symmetry. 3. Form phase-weighted Bloch combinations of those orbitals. 4. Diagonalise the resulting k-dependent matrix to obtain E n(k). 5. Count band states and electrons, then include interactions or disorder if the simple model fails.

Visual explanation

Draw a chain of identical orbital lobes. On one row, all lobes carry the same phase; on another, adjacent lobes alternate signs. Above the chain draw E(k) as a cosine from the zone centre to its edge. Mark its maximum and minimum, and label the vertical separation 4 t . A second narrower curve illustrates weaker overlap when atoms are farther apart.

Real-world analogy

Picture a row of rooms joined by doorways. Each room has the same starting energy, while the doorway width represents how easily a person moves to the next room. Wider doors spread possible collective movement patterns over a broader range. Yet the number of occupants and obstructions still determine whether traffic flows efficiently.

Real-world example

π orbitals along a conjugated molecular chain can be viewed as local p orbitals linked by hopping. As the chain length increases, discrete molecular-orbital levels crowd into bands. Bond alternation, heteroatoms and disorder modify the simple equal-site, equal-hopping model, but the orbital-based viewpoint remains useful for understanding organic conductors and polymers.

Why?

Why does the nearest-neighbour band contain cos(ka)? A site couples to both its left and right neighbours. Their Bloch phases contribute exp(ika) and exp(−ika); adding them gives 2 cos(ka). The cosine is therefore the algebraic signature of symmetric hopping in this one-dimensional model.

Common misconception

"Tight binding means electrons never leave atoms." Its local orbitals are a basis for extended Bloch states. A nonzero hopping integral explicitly represents movement between sites and makes the energy depend on k.

Worked example

Question: A one-dimensional band is E(k) = ε₀ + 2t cos(ka) with ε₀ = −5.0 eV and t = −0.50 eV. Find E at the zone centre and boundary and the band width.

Reasoning: At k = 0, cos(0) = 1, so E(0) = −5.0 + 2(−0.50) = −6.0 eV. At k = π/a, cosπ = −1, so E(π/a) = −5.0 − 2(−0.50) = −4.0 eV. Their difference is 2.0 eV, equal to 4 t . The lower in-phase state is consistent with the chosen negative hopping sign; changing the sign convention changes labels but not the width.

Answer: E(0) = −6.0 eV, E(π/a) = −4.0 eV and width = 2.0 eV.

Quick check

1. If t halves in the nearest-neighbour chain model, what happens to band width? Answer: It halves, because width is 4 t when the other assumptions stay fixed.

Exam focus

State the sign convention for hopping before locating band maxima. Derive the 4 t width from the cosine extrema. Explain that on-site orbitals produce extended Bloch combinations and that overlap, filling and scattering play different roles in material properties.

Advanced insight

The same orbital Hamiltonian can be written in real space with site indices or in reciprocal space as a compact k-dependent matrix. This makes tight binding computationally efficient for large or low-symmetry systems. Adding second-neighbour hopping introduces a cos(2ka) term and can shift dispersion extrema; adding several orbitals per cell creates hybridisation and possible avoided crossings.

Summary

Tight binding builds bands from local orbitals coupled between sites. A nearest-neighbour one-dimensional chain has E(k) = ε₀ + 2t cos(ka) and width 4 t . Band width reflects orbital interaction, while band filling, scattering, disorder and correlations determine the actual electronic response.

Practice questions

1. How many spatial Bloch combinations arise from N equivalent orbitals? Answer: N, before considering spin. 2. What is the bandwidth of E(k) = ε₀ + 2t cos(ka)? Answer: 4 t . 3. Why does larger interatomic separation usually narrow a tight-binding band? Answer: Orbital overlap and hopping generally decrease with separation. 4. Does a wide band by itself guarantee high conductivity? Answer: No. Occupation, scattering, disorder and other bands also matter.