Electronic Structure of Solids

The electronic structure of solids describes how atomic orbitals combine across a periodic lattice into continuous energy bands, and how the filling and spacing of those bands determine whether a solid is a metal, semiconductor, or insulator.

Definition

The electronic structure of a solid is the set of allowed electron energy levels — organised into bands separated by gaps — that results from the periodic potential of the crystal, and which governs the solid's electrical, optical, and magnetic behaviour.

Scope

This topic treats the electronic structure of extended solids from a chemical viewpoint: the broadening of discrete atomic levels into bands as orbitals overlap, the density of states and the Fermi level, the origin of the band gap, and the chemical bonding picture that links band structure to the building of crystals from atoms. It connects these ideas to electrical and optical properties and to the design of functional electronic materials.

Core questions

• How do overlapping atomic orbitals form energy bands in a solid?
• What determines the size of the band gap?
• Why are some solids metals, others semiconductors or insulators?
• How does electronic structure connect to chemical bonding in extended solids?

Key concepts

• Energy bands and band width
• Density of states
• Fermi level
• Band gap
• Valence and conduction bands
• Metals, semiconductors, and insulators

Key theories

Band formation from orbital overlap

As N atoms assemble into a crystal, each atomic orbital splits into N closely spaced levels that form a quasi-continuous band; the width of a band reflects the strength of orbital overlap, and band filling relative to the Fermi level governs conduction.

Band gap and the metal/insulator distinction

Whether a material conducts depends on whether the highest occupied band is partly filled (metal) or full and separated from the next empty band by a gap (semiconductor if small, insulator if large); the gap size sets optical absorption and carrier activation.

Mechanisms

Electrons in a partly filled band move under an applied field to carry current; in a material with a filled valence band, conduction requires thermal or optical excitation of carriers across the gap, so conductivity depends exponentially on the gap and temperature.

Clinical relevance

Understanding the electronic structure of solids is the basis for designing electronic and optical materials: the size and nature of the band gap determine whether a compound is useful as a transparent conductor, a semiconductor for devices, a light absorber for solar cells, or an insulating dielectric.

History

Bloch's theorem of 1928 showed that electrons in a periodic potential occupy extended states organised into bands, and Wilson in 1931 used band filling to explain the difference between metals and insulators. Later development of density functional theory by Kohn and co-workers made first-principles calculation of the electronic structure of real solids routine.

Key figures

• Felix Bloch
• Alan Herries Wilson
• Walter Kohn

Seminal works

• cox1987
• kittel2005

Frequently asked questions

Why does a solid have energy bands rather than discrete levels?

When many atoms come together, the Pauli principle forbids identical states, so each atomic orbital splits into as many slightly different levels as there are atoms. With astronomically many atoms these levels are spaced so finely that they form a continuous band of allowed energies.

What makes a material a semiconductor rather than an insulator?

Both have a filled valence band separated from an empty conduction band by a gap, but in a semiconductor the gap is small enough (roughly a few electron-volts or less) that thermal energy or light can promote a useful number of carriers across it, whereas in an insulator the gap is too large for appreciable conduction.