In solid state physics, the theory of the tapes is a modeling of the values of energy which the electrons of a solid inside this one can take. In a general way, these electrons have the possibility of taking only values of energy included ⁄ understood in certain intervals, which are separated by prohibited "tapes" of energy. This modeling results in speaking about structure or energy bands of tapes.

According to the way in which these tapes are distributed, it is possible at least schematically to explain the differences in electric behavior between an insulator, a semiconductor and a conductor.

In an isolated atom, the energy of the electrons can have only discrete and well defined values. By contrast, in the case of a perfectly free electron, it can take any positive value. In a solid, the situation is intermediate: the energy of an electron can have any value inside certain intervals. This property results in saying that the solid has allowed energy bands, separated by forbidden bands. This representation in energy bands is a representation simplified and partial of the density of electronic states. The electrons of the solid are distributed in the authorized energy levels, this distribution depends on the temperature and obeys the statistics of Fermi-Dirac function.

When the temperature of the solid tends towards the absolute zero, two allowed energy bands play a particular part. The last tape completely filled is called valence band. The allowed energy band which follows it is called tape of conduction. It can be empty or partially filled. The energy which separates the valence band from the tape of conduction is called the gap.

The electrons of the valence band contribute to the local cohesion of the solid (between close atoms) and are in localized states. They cannot take part in the phenomena of electric conduction. Contrary, the states of the tape of conduction are delocalized. These are the electrons which take part in electronic conduction. The electronic properties of the solid thus depend primarily on the distribution of the electrons in these two tapes, as well as value of the gap: in the case of the insulators, the two tapes are separated by an important gap. For the conductors, the gap does not exist and conduction bandages it is superimposed on part of the valence band. The semiconductors have as for them a sufficiently weak gap so that electrons have a considerable probability to cross it by simple thermal excitation when the temperature increases.

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.

According to the filling of the tapes with T = 0 K

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

- First case : the tape 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 tape 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 insulating.
- Third case : the tape 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 tape of conduction. Conductivity increases with the temperature: it is the characteristic of a semiconductor.

The occupation of the various states of energy by the electrons follows the distribution of Fermi-Dirac function. There exists a characteristic energy, the level of Fermi, which fixes, when the material is at a temperature of zero Kelvin, the energy level until where the electrons are found, that is to say the energy level of the more occupied high level. The level of Fermi represents the chemical potential of the system. Its positioning in the diagram of the energy bands is connected to the way in which the tapes are occupied.

- In the conductors, the level of Fermi is in an allowed tape which is in this case the tape of conduction. The electrons can then move in the electronic system, and thus circulate of atoms in atoms.
- In insulators and the semiconductors, the level of Fermi is located in the forbidden band which separates the conduction and valence bands.

Appearance of the forbidden band within the framework of a quasi-free gas of electron.

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 processing of this problem enters within the framework of the theory of the disturbances. One thus solves the equation of Schrödinger with the periodic potential created by the cores and one finds the functions clean and clean energies of electrons in the crystal. This processing is suitable in the case of noble metals, of alkaline metals and aluminum, for example.

Within the framework of the theory of the strong connections, one tries to derive the properties from the solid from the orbital atomic ones. One leaves the electronic states of the separate atoms and one considers the way in which they are modified by the vicinity of the other atoms. The effects to take into account are in particular the widening of the tapes (a state has a discrete energy within the atomic limit, but occupies an energy band in the solid) and hybridization between the close energy bands.

The family of semiconductor materials, insulator with forbidden band about 1eV, can be divided into two groups: materials with gap direct, like the majority of the compounds resulting from 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 tapes 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 tape 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 tape 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 tape 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 tapes being located at values of K similar, the probabilities of radiative recombinations of the carriers are 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.

In practice the crystal structure results in the pooling of these outer-shell electrons which then will have a much greater liberty of action.

Two types of electrons will thus be met :

- those for which the interactions of the other cores are so strong that they are practically released from their initial atom and gradually circulate in the whole of the solid under the effect of the electromagnetic fields generated by the other particles charged with the solid.
- and those which are closer to a given core to which they remain dependant in spite of the disturbance of their energy state by the actions of the other cores.

To satisfy the principle of Pauli it is noted that there is a true reduction of the energy levels of the isolated atom and in a crystal (regular assembly of atoms) this phenomenon results in the existence of quasi "energy bands" of which the conventional representation will help us to explain many properties of the semiconductor and the practical applications while rising. Let us note that two allowed energy bands are separated by a tape known as prohibited from more or less important width, and that sometimes two allowed tapes overlap while not forming whereas a single broader allowed tape characterizing different materials, semiconductor or insulating in the first case, conducting metal in the last.

One of the important components is the comprehension of the mechanisms of filling of these energy bands. An assembly of NR atoms of Z electrons thus will have NZ boxes divided into several energy bands, but the quantum numbers can take an infinity of values what means that the number of possible boxes is itself infinite.

Let us place first of all at the absolute zero, that is to say under the minimal energy conditions. The good sense indicates to us that in fact the quantum boxes of less energy will be occupied and which there will be necessarily a box beyond which all the boxes will be empty, and in on this side which all will be occupied by 2 electrons of opposite spins. This limiting level is known as level of Fermi.

Let us note that two cases are possible: That is to say this level is inside an energy band and with the top of this level the tape is empty, it will be the case of metals. Maybe, with the absolute zero, the last tape containing of the electrons will be completely filled and of course that immediately above will be completely empty and it will be the case of the semiconductors. In this last case one shows that the level, known as of Fermi, is (with the absolute zero) in the middle of the forbidden band immediately above than the last tape filled.

Any temperature different, that is to say higher than the absolute zero, by assumption the energy state is higher what implies than a certain number of boxes lower than the level of Fermi will be free while a number are equivalent to higher levels will be occupied since the number of electrons remains obviously the same one. It is rather intuitive to think that they are mainly the electrons in the vicinity of the level of Fermi (thus peripheral) which will see their energy state growing.

Quantum mechanics enables us to appreciate the density of the possible levels occupables according to energy Of (E) and particularly in the vicinity of the lower limits (such EC. for the tape known as of conduction) and higher (such Ev for the tape known as of valence) of the energy bands

expression of the density of the levels allowed in the vicinity of the limits of tapes allowed

To know the filling of the levels we need another information and they are the physicists Fermi and Dirac functions which will help us by providing us the probability of occupation of an energy level according to the energy of this level and the temperature.

Function of Fermi-Dirac function

One can then calculate the number of electrons present in a section of energy between E and E+dE which is expressed by : DNN = 2 F (E, T) Of (E) of

what integrated between EC. and the infinite one makes it possible to determine concentration N of electrons in a tape of conduction supposed to extend until the infinite one. But, as the figure shows it below, in the tape known as of Bc conduction the indeed occupied levels are of number very quickly close to zero as soon as one moves away a little from the bottom of the tape of conduction. Let us note that surface corresponding at the levels occupied in the tape of conduction is equivalent to that corresponding at the empty levels of the valence band. Before last remark, in a pure semiconductor (intrinsic) the level of Fermi is appreciably in the middle of the forbidden band.

representation of the occupied and empty levels in a semiconductor

It is also noticed that, since the temperature increases, the number of levels occupied in the tape of conduction increases (because of the evolution of the function of distribution) what amounts saying that electrons passed from the valence band to that of conduction and thus that the conductibility of the semiconductor improved.

The case of metal is appreciably different. Thus, in a metal the number of electrons of conduction (often 1 per atom) is invariant according to the temperature (only their average energy level varies according to the law of Fermi-Dirac function, but they do not change an energy band and remain in the tape of conduction which is already partially filled with the absolute zero) and, contrary to the preceding case, conduction worsens with acroissement of the temperature. Indeed when the temperature increases the oscillation of the cores around their position of balance increases slowing down the displacement of the free electrons.

To include ⁄ understand this "braking" we will remember, first of all, that the matter consist of a regularly punctuated vastness of vacuum of positively charged cores, dependant electrons and electrons quasi free. These are the latter which are concerned with the phenomena of transport. When an electron sees its energy level increasing in other words, it moves more quickly, but as it is permanently subjected to a nonuniform electric field it will accelerate and thus see its speed growing even more, but simultaneously increases also its probability of meeting (collision) with another live load (that it is another electron or a core), shock which induces a stop or at the very least a change of management and thus a limitation (reduction) of its range between two shocks.

But the orders of magnitude are very different: thus in copper there is an electron of conduction by atom, while in silicon with 25°C there is of them only 1 for approximately 2000 atoms.

So in the middle of a material the positive loads (cores) are statistically balances some with the negative ones (dependant and quasi-free electrons) to respect neutrality, it does not go from there in the same way in the vicinity of surfaces. Some free electrons having a certain kinetic energy move towards surface and leave the crystal. It results from it a defect from negative charge under surface and, obviously, a negative charge beyond surface thus a space charge R (X) positive in the crystal and negative in the vacuum.

It corresponds to him an electric field ε^{→} directed of metal towards the vacuum and a ddp - ∫ εdx such as metal is positive compared to the vacuum

evolution of ρ, ε and V in the vicinity of surface (x = 0) and energy diagram metal-vacuum

If one chooses enough like origin of the potentials the vacuum far from surface, one sees that there exists on the surface of the crystal a barrier of potential which prevents the electrons (of negative charge) from leaving material. One can plot the energy diagram of the electrons in the system metal-vacuum. The distance between EFF the highest level filled with 0K in metal and the energy level of the Ws vacuum represents with 0K the minimum of energy required to an electron so that it leaves material.

It is called the output. This barrier of potential is specifically a phenomenon of surface, which amounts saying that the least surface impurity (or pollution) will modify Ws locally.