The study of the properties of a solid shows in general that its electrons can take only values of energy included/understood in certain intervals. One is brought to speak about energy bands , or structure of bands . According to the way in which the electrons are distributed in these bands, it is possible at least schematically to explain the differences in behaviors between an insulator, a semiconductor and a driver.

Description of a structure in energy bands

The structure of bands applies to energies of the electrons in a solid.

In an isolated atom, the electrons can have as an energy only certain discrete and well defined values. It is one of the fundamental results of quantum mechanics (see for example the resolution of the problem of the Atome of hydrogen). The energy of a perfectly free electron on the other hand is only made up of kinetic energy, it can take any positive value.

In a solid, the situation is intermediate. The energy of an electron can have values included/understood in certain intervals. One speaks then about allowed energy bands. They are separated by forbidden bands. This representation in energy bands is a representation simplified and partial of the Densité of electronic states.

The electrons of the solid are distributed in the allowed energy levels. This distribution depends on the temperature. She obeys the Statistique of Fermi-Dirac.

Within the limit where the temperature tends towards 0, two allowed energy bands play a particular part in the determination of the properties of the solid. The last band completely filled is called valence band . The energy band allowed above is called band of conduction . It can be empty or partially filled. The energy which separates the valence band from the band of conduction is called the gap .

The electrons of the valence band contribute to the local cohesion of the crystal (between close atoms). These electrons are in localized states. They cannot take part in the phenomena of conduction. Contrary, the states of the band of conduction are delocalized. These are the electrons which take part in electronic conduction. The properties electronics of the solid thus depend primarily on the distribution of the electrons in these two bands, as well as value of the gap.

The conduction and valence bands play of the roles identical to that of orbital the molecular HOMO ( Highest Occupied Molecular Orbital ) and LUMO ( Lowest Unoccupied Molecular Orbital ) in the theory of the orbital boundaries.

Metal, insulator, semiconductor

According to the filling of the bands with T 0 K

When the temperature tends towards 0, one thus distinguishes three cases according to the filling from the bands and the value from the gap.

• First case: the band of conduction is partially filled. The solid thus contains electrons likely to take part in the phenomena of conduction, it is conducting.
• Second case: the band of conduction is empty and the gap is large (about 10 eV for example). The solid then does not contain any electron able to take part in conduction. The solid is Isolant.
• Third case: the band of conduction is empty but the gap is weaker (about 1 to 2 eV). The solid is thus insulating at null temperature, but a rise in temperature makes it possible to make pass from the electrons of the valence band to the band of conduction. Conductivity increases with the temperature: it is the characteristic of a Semiconducteur.

Relationship to the level of Fermi

The occupation of the various states of energy by the electrons follows the distribution of Fermi-Dirac. There exists a characteristic energy, the Niveau of Fermi, which fixes, when the material is at a temperature of zero Kelvin degree, the energy level until where the electrons are found, i.e. the energy level of the more occupied high level. Its positioning in the diagram of the energy bands is connected to the way in which the bands are occupied.

• In the conducting S, the level of Fermi is in an allowed band which is in this case the band of conduction. The electrons can then move in the electronic system, and thus circulate of atoms in atoms.
• In the Insulator S and the Semiconductors, the level of Fermi is located in the forbidden band which separates the conduction and valence bands.

Concept of gap direct, gap indirect

The family of the materials semiconductor, insulator with forbidden band about 1eV, can be characterized by two families. Materials with gap direct, like the majority of the compounds exits of the columns III and V of the periodic table of the chemical elements and materials with gap indirect like silicon (column IV).

The concept of gap direct and indirect is related to the representation of the energy dispersion of a semiconductor: Diagram E (Energy) - K (Vector of wave). This diagram makes it possible to define the extrema spatially bands of conduction and valence. These extrema represents, in a semiconductor with balance, energy fields where the standard density of carriers p for the valence band and standard N for the band of conduction are important.

One speaks about direct semiconductor with gap, for a semiconductor of which the maximum of the valence band and the minimum of the band of conduction are at value close to the vector of wave K on the diagram E (K). One speaks about indirect semiconductor with gap, for a semiconductor of which the maximum of valence band and the minimum of the band of conduction are at values distinct from the vector of wave K on the diagram E (K).

Within the framework of the applications out of transmitter of light (interaction light/matter), one privileges materials with gap direct. Their extremums of bands being located at values of K similar, the probabilities of radiative recombinations of the carriers is more important (cf quantum yield interns) because they are in agreement with the principle of conservation of the momentum and thus of the vector of wave K.

Characteristic specific to materials with gap direct

In the field of optoelectronics, a parameter essential with the comprehension of the phenomena of generations/recombinations of carriers, is the concept of absorption coefficient. This one is two common to the direct whole of the semiconductors with gap. It first of all presents a behavior comparable at first approximation to a stair. Thus for an incidental energy lower than the energy of forbidden band, the material is " transparent" with the incidental radiation, and the absorption coefficient is very low. Starting from a value close to the energy of forbidden band, this coefficient presents a constant value to the surrounding of α ≈ 10 ⁴ cm ^-1. One speaks thus about optical threshold of absorption.

Exception of insulators of Mott

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Techniques of calculations of the energy bands

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The quasi-free gas of electrons

In the case of the quasi-free gas of electrons, one regards the periodic electrostatic potential created by the atomic nuclei as weak. One treats it like a disturbance affecting a free gas of electrons. The treatment of this problem enters within the framework of the Théorie of the disturbances.

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The theory of the strong connections