When two valence electron atomic orbitals in a simple molecule such as hydrogen combine to form a chemical bond, two possible molecular orbitals result. One molecular orbital is lowered in energy relative to the sum of the energies of the individual electron orbitals, and is referred to as the 'bonding' orbital. The other molecular orbital is raised in energy relative to the sum of the energies of the individual electron orbitals and is termed the 'anti-bonding' orbital.
In a solid, the same principles apply. If N valence electron atomic orbitals, all of the same energy, are taken and combined to form bonds, N possible energy levels will result. Of these, N/2 will be lowered in energy and N/2 will be raised in energy with respect to the sum of the energies of the N valence electron atomic orbitals.
However, instead of forming N/2 bonding levels all of the exact same energy, the allowed energy levels will be smeared out into energy bands. Within these energy bands local differences between energy levels are extremely small. The energy differences between the levels within the bands are much smaller than the difference between the energy of the highest bonding level and the energy of the lowest anti-bonding level. Like molecular orbitals, and also atomic orbitals, each energy level can contain at most two electrons of opposite spin.
The allowed energy levels are so close together that they are sometimes considered as being continuous. It is very important to bear in mind that, while this is a useful and reasonable approximation in some calculations, the bands are actually composed of a finite number of very closely spaced electron energy levels.
If there is one electron from each atom associated with each of the N orbitals that are combined to form the bands, then because each resulting energy level can be doubly occupied, the 'bonding' band, or valence band will be completely filled and the 'anti-bonding' band, or conduction band will be empty. This is depicted schematically in the picture above by the grey shading of the valence band.
Electrons cannot have any values of energy that lie outside these bands. An electron can only move ('be promoted') from the valence band to the conduction band if it is given an energy at least as great as the band gap energy. This can happen if, for example, the electron were to absorb a photon of sufficiently high energy.
If, as in the above one-dimensional schematic, a band is completely filled with electrons, and the band immediately above it is empty, the material has an energy band gap. This band gap is the energy difference between the highest occupied state in the valence band and the lowest unoccupied state in the conduction band. The material is either a semiconductor if the band gap is relatively small, or an insulator if the band gap is relatively large.
Electrons in metals are also arranged in bands, but in a metal the electron distribution is different - electrons are not localised on individual atoms or individual bonds. In a simple metal with one valence electron per atom, such as sodium, the valence band is not full, and so the highest occupied electron states lie some distance from the top of the valence band. Such materials are good electrical conductors, because there are empty energy states available just above the highest occupied states, so that electrons can easily gain energy from an applied electric field and jump into these empty energy states.
The distinction between an insulator and a semiconductor is less precise. In general, a material with a band gap of less than about 3 eV is regarded as a semiconductor. A material with a band gap of greater than 3 eV will commonly be regarded as an insulator. A number of ceramics such as silicon carbide (SiC), titanium dioxide (TiO2), barium titanate (BaTiO3) and zinc oxide (ZnO) have band gaps around 3 eV and are regarded by ceramicists as semiconductors. Such ceramics are often referred to as wide-band-gap semicondutors.