Electronic band structure
From Academic Kids

In solid state physics, the electronic band structure, or simply band structure, refers to the dispersion relation (the relation between energy versus momentum) of electrons in a solid. The band structure determines a material's electronic properties, optical properties, and a variety of other properties.
Contents 
Introduction
According to quantum mechanics, the electrons of a single freestanding atom occupy atomic orbitals, which form a discrete set of energy levels. This model of discrete energy levels can also be extended to simple compound molecules, and can be most easily be observed in the emission and absorption spectra of gases.
When a large number of atoms (of order 10^{20}) are brought together to form a solid, however, the electrons no longer form a discrete set of energy levels, but instead form into a set of continuous energy bands which are separated by a set of gaps.
A simple analogy to the formation of these energy bands is to consider a single atom of a particular species. This atom has a discrete set of energy levels. When another atom of the same species is brought close to the first atom, each of the energy levels becomes degenerate and splits into two separate energy levels (which arises as a result of the Pauli exclusion principle). If this procedure is repeated 10^{20} times, each energy level will have split into 10^{20}, effectively forming a continuous band of energy states.
Although electronic band structures are usually associated with crystalline materials, quasicrystalline and amorphous materials may also exhibit band structures. However, the periodic nature and symmetrical properties of crystalline materials makes it much easier to examine the band structures of these materials theoeretically. In addition, the welldefined symmetry axes of crystalline materials makes it possible to determine the dispersion relationship between the momentum (a 3dimension vector quantity) and energy of a material. As a result, virtually all of the existing theoertical work on the electronic band structure of solids has focused on crystalline materials.
Special Bands
Although there may be an infinite number of bands in the band structure of a given material, there are two bands that are of particular significance in determining the electronic and optical properties of a material. These are the conduction band and the valence band. Note that these terms may refer to the same band in materials that are conductors.
Band Structure of Crystals
Brillouin Zone
Because electron momentum is the reciprocal of space, the dispersion relation between the energy and momentum of electrons can best be described in reciprocal space. It turns out that for crystalline structures, the dispersion relation of the electrons is periodic, and that the Brillouin Zone is the smallest repeating space within this periodic structure. For an infinitely large crystal, if the dispersion relation for an electron is defined throughout the Brillouin Zone, then it is defined throughout the entire reciprocal space.
Applications of the Band Structure
One of the most useful aspects of the band structure is the feature known as the band gap. In semiconductor and insulator materials, this is the gap between the valence band and the conduction band. The band gap and defect states created in the band gap by doping can be used to create devices such as solar cells, laser diodes, transistors, and a range of other electronic devices.
Theory of Band Structures in Crystals
Every crystal is a periodic structure can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (<math>\mathbf{b_1}<math>, <math>\mathbf{b_2}<math>, <math>\mathbf{b_3}<math>). Now, any periodic potential <math>V(\mathbf{r})<math> can be written in terms of this reciprocal lattice (according to Fourier theory) as:
<math>V(\mathbf{r}) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf{K}\cdot\mathbf{r}}}<math>
where <math>\mathbf{K} = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3 <math> for any set of integers <math>(m_1, m_2, m_3)<math>.
NearlyFree Electron Approximation
The nearlyfree electron approximation in solid state physics is similar in some respects to the Hydrogenlike atom of quantum mechanics in that interactions between electrons are completely ignored. This allows us to use Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. This can be described mathematically by:
<math>\Psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r}) <math>
where the function <math>u(\mathbf{r})<math> is periodic over the crystal lattice.
Mott Insulators
Although the nearlyfree electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a Mott insulator, and requires new theories, such as the Hubbard model, to explain the discrepency.
Other
Calculating band structures from first principles is a topic in theoretical solid state physics. A number of methods have been invented, including the nearlyfree electron approximation, the tight binding approximation, the k.p method, Green's function methods, the Hubbard model, and many more.
References
Kotai no denshiron (The theory of electrons in solids), ISBN 4621041355