Band Theory of Solids

Band theory is a quantum model in solid state physics that describes the possible energies of electrons in a solid and provides insight into electrical conductivity. It is derived from molecular orbital theory.

The allowed energy levels in a solid are confined to a band whose width, on the order of an electron volt, is determined by the crystal and the overlap of the atomic orbitals.

Solids have a band structure, which includes allowed and forbidden energy bands. The energy bands are filled using Fermi-Dirac statistics: at 0 K, electrons occupy all energy levels lower than the Fermi level. Its value is unique to the crystal under consideration.

Band Theory of Solids

The free electron model explains many important conductor properties, but it falls short in at least two areas. First, it assumes that the solid has a constant potential energy. (Recall that a constant potential energy has no forces associated with it.) contrasts the assumption of constant potential energy (dotted line) with the periodic Coulomb potential, which decreases as -1text/r at each lattice point, where r is the distance from the ion core (solid line). Second, the free electron model assumes an impenetrable surface barrier. This assumption is incorrect because electrons can escape the surface under certain conditions, such as the photoelectric effect.

Aside from these assumptions, the free electron model fails to explain the dramatic differences in electronic properties between conductors, semiconductors, and insulators. The energy band, as the name implies, is a much larger permitted region for electrons. In a nutshell, an energy band is a collection of many energy levels with varying energies.

What is an Energy Band?

Energy bands, like energy levels, are a permitted “region” in which an electron can exist in an atom or a molecule. The energy band, as the name implies, is a much larger permitted region for electrons. In a nutshell, an energy band is a collection of many energy levels with varying energies.

Thus, in multi-atomic systems such as solids, energy bands are permitted regions for an electron formed by the superposition of many similar energy levels with slightly different energies.

Why are there Energy Bands?

There are some energy levels where electrons are permitted in monoatomic systems such as gases (which have no range order), i.e. where one atom does not affect another atom.

However, in a multi-atomic system such as solids (which have a range order), energy bands are formed where one atom affects another. A nearby atom’s nucleus can form another energy level with slightly different energy, resulting in two energy levels with slightly different energies. With the presence of a third atom nearby, a third energy level is added. 

As the number of local atoms increases in solids, a large number of energy levels with slightly different energies form, forming an energy band.

How does Energy Band explain Conductivity?

The only electrons available for conduction, according to energy level theory, are those that are not bonded and are in the outermost (valence) energy level.

However, energy band theory states that conduction occurs only when some electrons are present in the conduction band.

In solids, the conduction band is the outermost energy band in which electrons are not bound to the nucleus by any energy, allowing them to move freely. As a result, the presence of electrons in the conduction band determines conduction.

However, electrons are not readily available in the conduction band. The valence band is generally defined as the outermost energy band in which electrons exist without any thermal energy (at 0k).

Depending on the nature of the solid, an energy gap exists between the conduction band and the valence band, where electrons are prohibited. It is known as a forbidden band, and its bandwidth determines the solid’s conductivity.

What is a Conduction Band?

When electrons are excited, they can jump up into the conduction band, which is a band of electron orbitals that they can enter from the valence band. These orbitals provide the electrons with enough energy to move freely through the material when they are in these positions. The movement of electrons results in the generation of an electric current. The valence band is simply the outermost electron orbital of an atom of any specific material that electrons can actually occupy when the atom is in its ground state. The band gap is defined as the difference in energy between the highest occupied energy state of the valence band and the lowest unoccupied energy state of the conduction band. It is a measure of a material’s electrical conductivity because it indicates how well it conducts electricity. Because of the large band gap, a significant amount of energy is required to excite valence electrons into the conduction band and vice versa.

What is a Forbidden Band?

This is the energy difference between the valence band and conduction. This is also known as the forbidden gap. The forbidden gap is used to calculate the electrical conductivity of a solid. Furthermore, we can determine whether the materials are conductors, semiconductors, or insulators.

What is Valence Band?

The valence band is a band of electron orbitals from which electrons can jump into the conduction band when excited. The valence band is simply the outermost electron orbital of any specific material that electrons occupy. This is related to the concept of the valence electron.

Conclusion

Band theory is a quantum model in solid state physics that describes the possible energies of electrons in a solid and provides insight into electrical conductivity. The allowed energy levels in a solid are confined to a band whose width, on the order of an electron volt, is determined by the crystal and the overlap of the atomic orbitals. The energy band, as the name implies, is a much larger permitted region for electrons. The energy band, as the name implies, is a much larger permitted region for electrons. In solids, the conduction band is the outermost energy band in which electrons are not bound to the nucleus by any energy, allowing them to move freely. Depending on the nature of the solid, an energy gap exists between the conduction band and the valence band, where electrons are prohibited.