Band Theory of Solids

Bohr’s theory of atomic spectra says that an isolated atom possesses discrete energy levels and the energy of an electron depends on the orbit it is revolving in. However, isolated atoms don’t exist practically, but in crystals. 

Let’s take a single Silicon (Si) atom, the energy of the first electron is - 13.6 eV. Now, taking the second Si atom, the energy in its hidden electron is also - 13.6 eV, and it remains the same at n = 1. However, when atoms combine to form a crystal, the energy of electrons doesn’t remain the same. For that, we need to create energy bands in solids. Now, let’s understand the band theory of solids.

Band Theory of Solids

In crystals, electrons come close to each other (Approx. 2 to 3 Å closer) to have the following interactions with:

• Electrons of Neighboring Atoms.

• The Nucleus of Neighboring Atoms.

In crystals, each atom has a unique position. Hence, each electron has the following unique properties:

• Position

• Interactions

• Energy level even though they belong to the same subshell of an isolated atom.

• Energies are slightly greater/smaller than their original energies in an isolated state.

A straight line of the energy level splits into 1023 energy lines or levels within a width of 1 eV. These lines are so close to each other that they appear as energy bands (of crystals). These energy lines are continuous, and the difference between each is 10-23 eV.

Suppose we have a Sodium metal, where 1 mole of Na atoms has 6.022 x 1023atoms. Now, we break it into two pieces; the electron in each piece possesses different positions and interactions.

Here, the energy of an outer electron will be unique in each energy line, and this slight difference will be because each electron has a unique position and interaction. 

Now, let’s see energy bands in solids by taking an example of Na.

Energy Bands in Solids

In this context, we’ll study the band structure of solids.

We know that the electron configuration of Na = 1s22s22p63s1. Energy bands of 1s, 2s, 2p, and 3p, are shown below:

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In the upper band, i.e. 3s having electrons is the valence band, and the energy level above it, having no electrons, is the conduction band. Here, we can discern that there’s no forbidden energy gap in conductor. Now, let’s take examples of Silicon.

Let’s take a Silicon crystal having ‘n’ mole of Silicon atoms. We know that the electronic configuration of Si = 1s22s22p63s23p2. The number of electrons in the outer energy level = 4n and maximum electrons  = 8, i.e. 2 from 3s and 6 from 3p.

Similarly, the number of outer energy levels available in Si-atom = 8 

Therefore, in a crystal, there are 8n (2n from 3s and 6n from 3p) electrons, where n = 10-23. We can see that out of 8n energy levels, 4n is filled, and 4n is vacant at zero Kelvin. 

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If we look at this graph, initially, there are 2n and 6n electrons in 2n and 6n state, respectively. The interaction between outer electrons increases gradually, the energy level expands, and the time comes when both of these overlap. 

Now, it becomes hard to determine which state (2n & 6n) an electron belongs. Eventually, the distance between the atom nullifies and crystal forms.

During the mixing of energy levels (hybridization) of 2n and 6n state, the electrons from 2n state migrate to 6n, as they prefer to stay in the lower state. 

Now, after the crystal formation, we have two 4n states (8n = 4n + 4n), where the lower 4n state has filled 4 electrons, and the upper 4n state has zero electrons. The lower one is the valence band, and the upper one is the conduction band, which may/may not have electrons; however, there is a FEG or forbidden energy gap in semiconductor, i.e. Silicon. The energy band structure will be:

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Energy Gap in Insulator

If we look at the energy band diagram of an insulator such as a Diamond, the energy gap, or FEG (Eg = 6 eV)) is larger. Though the valence band is completely filled (as per Pauli’s principle); due to a large gap between the valence band (Ev) and the conduction band (Ec), these electrons can’t transfer to the conduction band.

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Since electron movement isn’t possible here, that’s why electric conductions in these materials become impossible. 

Below you can see the energy bands in different solids:

Energy Bands in Conductors Semiconductors and Insulators

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