Band Theory Of Solids

The band theory of solids is different from the others because the atoms are arranged very close to each other such that the energy levels of the outermost orbital electrons are affected. But the energy level of the innermost electrons is not affected by the neighbouring atoms.

In band theory of solid, there are many energy bands but the following are the three most important energy bands in solids:

  1. Valence band
  2. Conduction band
  3. Forbidden band

Valence band: The energy band that consists of valence electrons energy levels, is known as the valence band. The valence band is present below the conduction band and the electrons of this band are loosely bound to the nucleus of the atom.

Conduction band: The energy band that consists of free electrons energy levels, is known as the conduction band. For electrons to be free, external energy must be applied such that the valence electrons get pushed to the conduction band and become free.

Forbidden band: The energy gap between the valence band and the conduction band is known as the forbidden band which is also known as the forbidden gap. The electrical conductivity of a solid is determined from the forbidden gap and also the classification of the materials as conductors, semiconductors and insulator.

What is Band Theory of Solids?

This theory explains the quantum state that an electron takes inside metal solid. Every molecule comprises of various discrete energy levels. The way electrons behave inside a molecule is well explained through this theory.

  • In atoms, electrons are filled in respective energy orbits following Pauli’s exclusion principle.
  • In molecules, Two atomic orbitals combine together to form a molecular orbit with two distinct energy levels.
  • In solids, 1023 stacked up lines confined in a tiny space would look like a band. Thereby forming energy continuum called energy bands.
  • This theory helps to visualise the difference between conductor, semiconductor and an insulator by plotting available energies for an electron in a material.

Energy Bands In Solids

Inside an atom

band theory of solids-1

Figure(1)

Consider a Sodium atom. It comprises of 11 electrons. They fill up energy level following Pauli’s exclusion principle. Refer Figure(1).

Energy levels inside a molecule made up of two atoms

What happens when two sodium atoms very close to each other almost forming a molecule? Now each atom cannot have the configuration as it followed as an individual atom If they do they will be violating Pauli’s exclusion principle and end up with a lot of electrons of the same energy levels.

When two atoms come very close to each other, What is going to happen to this system? The answer is, their respective energy bands are going to overlap on each other and transform into what we call as Molecular orbit. That is the 1s orbit of individual Sodium atom combines to form 1s molecular orbital. As two atomic orbitals are overlapping, the molecular orbit ends up having two discrete energy levels. Where the Lower energy level is called bonding orbital and Higher energy level is called anti-bonding orbital. This will repeat for all orbits. Refer figure (2) for visualisation of energy levels and molecular orbits. To learn more about the energy levels and the classification of energy bands, click on the following links:

band theory of solids-2

Figure (2)

Energy levels inside a molecule made up of three atoms

Now try to picturise, what is going to happen if we add a third sodium atom to the mix? Well, according to the theory we learned just now. here three atomic orbitals will be overlapping forming single molecular orbital with three discrete energy levels. Each molecular orbital here will inherit three energy levels. In general, the more we add atoms, more energy levels the molecular orbit going to have. Refer figure (3)

band theory of solids-3

Figure (3)

Energy levels inside a solid made up of Avogadro number of atoms

Eventually, if we have an entire solid, which is made of sodium with something like 1023 atoms packed together, Each molecular orbital of this solid will have now 1023 discrete energy levels. For better understanding purpose, Think about drawing 1s orbital of Sodium solid block, draw lower energy level and upper energy level and in between stack it with 1023 energy levels! Refer figure(4). The gaps between them will be extremely small such that, no longer we can notice individual energy levels. As a result, It is convenient to think of it as continuous energy or energy continuum. When we think in this way, we can call them as energy band instead of molecular orbit. figure (4).

Below are the links for related concepts:

  1. Energy levels
  2. Fermi energy
  3. Magnetic quantum number

Energy levels inside a solid made up of n-number of atoms

In general, If there are n-number of atoms, then there will be n discrete energy levels in each energy band. In such a system of n number of atoms, the molecular orbitals are called as energy bands. Single 1s orbital and 2s orbital can fit 2 electrons each. thus, the total number of electrons a 1s and 2s energy band can fit is 2n. A single 2p level can fit 6 electrons so 2p energy band can fit is 6n electron so on and so forth.

band theory of solids-4

Energy levels inside material of different states of matter

If you have a single atom or if you have gas. In gas, atoms are apart/infinitely apart; we can assume them as single atoms. Here every atom has a discrete energy level, if electron wants to go from one level to another, it really has to jump as no continuous energy is available(It is similar to steps).

band theory of solids-5
As atoms come close to each other and eventually form a solid, They end up forming energy continuum and we name that continuum as bands. Within the bands, energy levels which are available are continuous. Thus, the name of this theory without any surprise is, “The band theory of solids”

Using this theory, we can understand how free electrons are generated and why certain material readily have free electron available making them a conductor and why some others don’t?

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