Zone theory of solids. Quantum mechanics for dummies

This article describes what is the band theory of solids. It is shown, what exactly is this representation of the structure of matter. The differences between metals from dielectrics and semiconductors are presented.

Socket and button

How many times a day do we click on a variety of buttons? No one even comes to mind can not consider this - so familiar is this action. And the person does not think that all this is possible only because of how easily an electric current flows in metals. Turn on the light, boil the kettle, run the washing machine, not to mention the actions on smartphones, means to close the circuit and allow the electrons in the conductors to work instead of people. There are many explanations for such a phenomenon as conductivity. Perhaps the most visible is the band theory of solids.

Atom and teapots

Everyone who studied at school has an idea of the structure of the atom. Let us recall that around the positively charged heavy nucleus (consisting of protons and neutrons) light small electrons rotate. The number of negative particles is exactly equal to the number of positive particles . In order not to bore the readers, we will explain in the style of "quantum mechanics for dummies". Each electron has a strictly limited orbit along which it can rotate around the nucleus in a given chemical element. In turn, each type of atoms has a unique pattern of such orbits. This is how scientists-spectroscopists distinguish boron from selenium and arsenic from sodium. However, in addition to pure substances, there are innumerable quantities of various combinations in nature. Quantum mechanics (for the dummies, as the reader should remember) asserts that in complex compounds orbits intersect, merge, transform, stretch, creating connections. Their quality depends on the type: covalent and ionic stronger, hydrogen, for example, weaker.

Crystal structure

In a solid body, things are more complicated. For a model that uses the band theory of solids, an ideal crystal is usually taken. This means that it is infinite and sinless - each atom in the space allocated to it, the total charge is zero. The nuclei fluctuate around a particular equilibrium position, but the electrons, one might say, are common. Depending on how "just" one atom gives its negative particles to neighboring ones, a rigidly defined structure of dielectrics or an electronic cloud of metals is obtained. It is worth adding that when considering the assumption that all electrons occupy the minimum energy allocated to them, it means that the body is at zero Kelvin. At a higher temperature, the amplitude of oscillations of both nuclei and electrons is stronger, which means that the latter are capable of occupying higher energy levels. The distribution of negative particles becomes more "friable". In some problems this is important, but to describe this phenomenon as such, the temperature is not so important.

Principle of Pauli and the loader

The concept of the band theory of a solid body can be found only by remembering well what the Pauli principle is. If we imagine that the electrons are sacks of sugar, then, if there are many of these sacks, the conditional loader will impose them on each other. Each "bag" takes its place in space. For electrons, this means that in this particular state there can be only one in one system. This is the Pauli principle. Note that we mean ideal conditions, that is, the temperature is zero Kelvin, and the crystal is infinite. The whole system is in the same conditions: temperature, mechanical stresses, defectiveness are the same in all parts of the whole.

Electronic crystal zones

In a crystal, there are many atoms of the same type. One mole of substance contains ten in the twenty-third power of the elements. And how many moles in a kilogram, say, salt? So you can even say that even the smallest crystal contains an unrepresentable number of atoms. Each chemical element has its own pattern of electronic orbits, but what if there are several in one body? After all, according to the principle of Pauli, they all must occupy different states. The band theory of solids offers the following solution: electron orbits acquire different energies. However, the difference between them is so small that they are compressed, overlapping one another very tightly, and form a continuous zone. Thus, each level of an electron in one atom is transformed into a zone in a bulk crystal. Elements of the band theory of solids will help explain the difference between dielectrics and conductors.

Electron inside the zone

We have already discussed what happens to the set of electrons that occupy the same orbit in an atom when a crystal is formed. But their behavior within the zone has so far remained unenlightened by us. Telling this is important because it determines the difference between metals and non-metals. As mentioned above, the band theory of solids indicates that within the zone the energy levels of different orbits of individual atoms differ so little that they form an almost continuous spectrum. Thus, it is not difficult to overcome the potential barrier between them for an electron - it moves freely over them, even thermal energy is enough for this. However, every permitted zone has limits. There is always an energy level that is higher or lower than all the others.

Valent, forbidden, conductivity

Between these zones there is an energy region in which there is not one level on which an electron could be located. On the charts, it appears as a white gap. And it's called a forbidden zone. The electron can overcome this barrier only by a jerk. So, he must get the appropriate energy for this. The zone with the highest energy, in which for the given type of atoms the existence of electrons is allowed, is called valence, and the next behind it is the conductivity.

Metal, dielectric

The band theory of the conductivity of solids asserts that the presence or absence of electrons in the conduction band indicates how easily a current flows in a given substance. Thus, metals and dielectrics are different. In the first case, the conduction band already contains electrons, since it overlaps with the valence band. This means that negative particles can move freely under the influence of an electromagnetic field, without additional expenditure of energy. Therefore, the electric current in metals arises so easily, in fact - instantly, as soon as the field appears. And for the same reason, the wires are made of steel, copper, aluminum.

Materials in which the conduction band and the valence band are separated from each other energetically are called dielectrics. Their electrons are locked in the lower allowed level. The forbidden zone separates the negative particles from the level at which they could move freely. And the energy that must be communicated to the electrons in order to overcome it, will destroy the material. Or it will change its properties beyond recognition. Therefore, the plastic wire wrapping melts and burns, but does not conduct electricity.

Semiconductors

But there is an intermediate class of materials that have a forbidden zone, but in some conditions are able to conduct an electric current. They are called semiconductors. Like dielectrics, they have an energy gap between the conduction and valence bands. However, it is smaller and we will overcome some of the efforts. The classical semiconductor is silicon (in Latin - silicium). The famous Silicon Valley is famous for its technologies based on the use of crystals of this substance to create electronic equipment.