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The electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have, called band gaps or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of solid-state devices such as transistors and solar cells.^[1]^[2]

A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.

Introduction

In an isolated atom, electrons occupy discrete atomic orbitals with specific energies. When many identical atoms are brought together in a crystal, these orbitals overlap and split into a very large number of closely spaced levels. Since a macroscopic solid contains an enormous number of atoms, the levels become so densely packed that they can be treated as continuous energy bands.^[3]

This band formation is especially important for the outermost, or valence, electrons, which participate in chemical bonding and electrical conduction. Inner orbitals overlap much less strongly and therefore form very narrow bands. Between bands there may be ranges of energy that contain no allowed electron states; these are the band gaps.^[4]^[5]^[6]

Why bands and band gaps occur

Two complementary pictures are often used to explain the origin of bands and band gaps.^[2]

In the nearly free electron model, electrons are treated as moving almost freely through the crystal, with the periodic lattice acting as a weak perturbation. In this case the electron states resemble plane waves, and the periodic lattice gives rise to the characteristic dispersion relations and gaps at special points in reciprocal space.^[2]

In the opposite limit, the tight binding picture begins with electrons strongly bound to individual atoms. If neighbouring atomic orbitals overlap, electrons can tunnel between atoms, and the original atomic levels split into many different energies. For a crystal containing $N$ atoms, each atomic level gives rise to $N$ closely spaced levels, which together form a band.^[3]

The widths of these bands depend on how strongly the orbitals overlap. Narrow overlap produces narrow bands, while strong overlap produces wider bands. Band gaps arise when neighbouring bands do not extend enough in energy to overlap. Core-electron bands are especially narrow, so they are often separated by large gaps, whereas higher-energy bands broaden and may overlap more strongly.^[4]

Basic concepts

Assumptions and limits of band theory

Band structure is an approximation that works best for solids made of many identical atoms or molecules arranged in a regular way. It generally assumes:

a large or effectively infinite system, so that the allowed levels form nearly continuous bands;
a chemically homogeneous material;
electrons moving in an average static potential, without fully treating interactions with other electrons, phonons, or photons.

These assumptions can break down in important cases. Near surfaces, interfaces, and junctions, the bulk band structure is altered and effects such as surface states, dopant states, and band bending become important. In very small systems such as molecules or quantum dots, the concept of a continuous band structure no longer applies. Strongly correlated materials such as Mott insulators also fall outside the normal single-electron band picture.^[2]

Crystalline symmetry and wavevectors

For a crystal, band structure calculations take advantage of lattice periodicity. The single-electron Schrödinger equation in a periodic potential has solutions of the Bloch form $ψ_{n 𝐤} (𝐫) = e^{i 𝐤 \cdot 𝐫} u_{n 𝐤} (𝐫),$ where $k$ is the wavevector and $n$ labels the band. For each value of $k$ , there are multiple allowed energies, and these vary smoothly with $k$ to produce the dispersion relation $E n (k)$ .^[7]

The relevant wavevectors lie inside the Brillouin zone, the fundamental region of reciprocal space. Band structure plots typically show energy versus wavevector along lines connecting high-symmetry points such as Γ, X, L, or K. These diagrams make it possible to distinguish, for example, between direct band gap and indirect band gap materials.^[8]^[9]

Density of states

The density of states function $g (E)$ gives the number of electronic states per unit volume and per unit energy near energy $E$ . It is central in calculations of conductivity, optical absorption, and scattering processes. Inside a band gap, the density of states is zero: $g (E) = 0 .$

Filling of bands

At thermal equilibrium, the probability that a state of energy $E$ is occupied is given by the Fermi–Dirac distribution $f (E) = \frac{1}{1 + e^{(E - μ) / (k_{B} T)}}$ where $\mu$ is the chemical potential, usually called the Fermi level, and $k B T$ is the thermal energy. The number of electrons per unit volume is obtained by integrating the occupied density of states: $N / V = \int_{- \infty}^{\infty} g (E) f (E) d E$

In semiconductors and insulators, the Fermi level lies in a band gap. The highest occupied band is called the valence band, and the lowest unoccupied band above it is the conduction band. In metals, by contrast, the Fermi level lies within one or more allowed bands, so electrons can move easily and conduct electricity.^[2]

Theoretical descriptions

Nearly free electron approximation

The nearly free electron model treats electrons as almost free, with only a weak periodic perturbation from the lattice. It is particularly useful for simple metals and can explain why gaps open at certain boundaries in reciprocal space. In some metals, such as aluminium, the resulting bands are close to those of the empty lattice approximation.^[7]

Tight binding model

The tight binding approach assumes that electrons remain largely localized around atoms and that crystal wavefunctions can be approximated as linear combinations of atomic orbitals. This model works especially well in systems where orbital overlap is limited, such as silicon, gallium arsenide, diamond, and many insulators. A related and more refined formulation uses Wannier functions.^[7]^[10]^[11]

KKR model

The Korringa–Kohn–Rostoker method (KKR), also known as multiple scattering theory, reformulates the problem using scattering matrices and Green's functions. It is particularly useful for alloys and disordered systems, and often uses the muffin-tin approximation for the crystal potential.^[12]^[13]

Density-functional theory

Many modern band calculations are carried out using density-functional theory (DFT). DFT often reproduces the overall shape of experimentally measured bands quite well, but it commonly underestimates the size of band gaps in semiconductors and insulators by about 30–40%.^[14] Although DFT is formally a ground-state theory, its Kohn–Sham solutions are widely used as practical approximations to real band structures.^[15]^[16]

Green's function and GW methods

To include many-body effects more accurately, one can use Green's function methods. In these approaches, the poles of the Green's function correspond to the quasiparticle energies of the solid. The GW approximation is especially important because it often corrects the systematic band-gap underestimation of standard DFT and gives results in closer agreement with experiment.

Dynamical mean-field theory

Some materials, such as Mott insulators, cannot be understood in terms of ordinary single-electron bands. In these cases, strong electron-electron interactions dominate. Methods such as the Hubbard model and dynamical mean-field theory are used to describe the resulting behaviour.

Band diagrams

In practical device physics, the full band structure is often simplified into a band diagram, where energy is plotted vertically and real-space position horizontally. In such diagrams, energy levels or bands may tilt with position, indicating the presence of an electric field. Band diagrams are especially useful for understanding junctions, interfaces, and semiconductor devices.

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Band structure