Introduction

In this particular article The formation of Bands in Solids, we are going to discuss some basic information about Bands in solid in detail and in the easiest way possible.

The formation of Bands in solids

Atom is the basic unit of matter. Which contains protons and neutrons in its nucleus with electrons moving around the nucleus in certain allowed orbits. The number of electrons in an atom is equal to the number of protons. So that the atom is electrically neutral. However, the difference in electrical properties of the different material is due to the difference in the number of electrons, protons, and neutrons.

In an atom, all the electrons are not situated in an orbit of the one definite energy level but these but these are distributed in different orbits of allowed energy levels. The distribution in different orbits is according to Pauli’s exclusion principle. Electrons first occupy the orbit corresponding to minimum energy and then with the increase in energy they occupy higher energy orbits obeying Pauli’s exclusion principle. The outermost occupied orbit is called valence level and the electrons occupying this orbit are called valence electrons.

The valence electrons being farthest from the nucleus and being screened by the electrons in the inner orbits are loosely bound with the nucleus in the atom. These electrons are almost free to combine with other atoms. Although the valence electrons are almost free electrons and can move freely in atomic lattice their energy is so low that they can not escape. Other electrons which are located in inner orbits remain in the influence of the attractive force of the nucleus and these continue to occupy their allotted orbits. These electrons are called bound electrons. Under normal conditions, these electrons do not participate in the electrical behavior of the material.

For example

Consider an atom of Germanium (Ge). It has 32 electrons which are distributed in different orbits as :

1s²,2s² 2p⁶, 3s² 3p⁶ 3d¹⁰, 4s² 4²

First three orbits are completely filled in accordance with Pauli’s exclusion principle and these electrons can be called bound electrons. In the outermost fourth orbit, all the four electrons are called valence electrons. Thus germanium is a tetravalent element.

The motion of an electron in the bound region

In an isolated atom due to the attractive force on the electrons due to the nucleus, their potential energy is negative so that the electrons move only in the bound region formed by the potential energy curve. According to quantum mechanics, these electrons remain in the possible energy states and not in between i.e. the atom has definitely allowed energy levels which the electron can occupy. For a single Ge atom, some of the allowed energy levels are shown.

When two atoms of a solid are brought nearer the nucleus of one atom can interact with electrons of the other atom. As a result, each energy level of both the atoms due to interaction becomes a group of two energy of one level is less than the energy of the original level. And that of the other higher than the energy of the original level. Along with it, due to superposition, the height of the common wall of the potential well of the two atoms get reduced. Due to this, the valence levels of the two atoms become common. And hence the valence electrons of the atoms can move from one atom to the other.

The formation of Bands in Solids

Formation of energy bands

In the same way in solids. whose nature is usually crystalline, the atoms are arranged linearly in a regular way. The distance between these atoms in very small. So that each atom interacts with the neighboring atoms. And the energy levels of the atoms split off from groups of energy levels. The new energy levels are on both sides of the original energy levels. The separation between the new energy level is so small that the group of energy levels becomes a continuous energy band. The splitting of energy levels as a function of interatomic distance.

The energy gap between the conduction band and valence band

Thus each atom of a system of N-atoms in the split in N-energy levels. The energy levels associated with the electron-orbits have the maximum effective radius. (I.e. maximum Quantum number are the first to suffer splitting. And as the distance between the atoms goes on decreasing the energy levels corresponding to inner orbit are also split. In solids the distance between the atoms in equilibrium is definite, denoted by r. At this interatomic distance, the outer energy level becomes bands. The energy band in which the valence electrons are present is called the valence band. The allowed band above the valence band (or higher energy). Which is mostly vacant in the unexcited state is called the conduction band. Between the conduction band and valence, there is an energy gap in which there are no allowed energy levels. Hence this energy gap is called the forbidden energy gap.

The electron situated in valence band being bound with the nuclei of the atom. And due to the non-availability of empty allowed energy levels,  cannot act as charge carriers. But some of these electrons can cross over to the conduction band by excitation. These electrons are capable of taking energy and can act as charge carriers.

For the transfer of electrons from the valence band to conduction band a minimum energy equal to the energy gap between these bands is required. The difference in electrical conduction and electrical conductivity of solids can be easily explained with the help of energy band diagrams. For simplicity, these energy bands are represented by their boundaries. In the above fig, Ec represents the lowest level of the conduction band and Ev represents the highest level of the valence band. The energy gap between Ec and Ev is the forbidden energy gap ΔEg.

 

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