Explain 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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FAQs on Band Theory of Solids
1. What is the band theory of solids?
The band theory of solids explains the electrical behaviour of materials by describing how electron energy levels in an isolated atom split into continuous energy bands when atoms are brought together to form a solid crystal. According to this theory, the vast number of closely packed energy levels form bands, primarily the valence band and the conduction band, separated by a forbidden energy gap. The nature of these bands determines whether a solid is a conductor, semiconductor, or insulator.
2. How are energy bands formed in solids, unlike the discrete energy levels in isolated atoms?
In an isolated atom, electrons occupy distinct, discrete energy levels. However, when a vast number of atoms (like 1023) come close together to form a solid crystal, the electrons of one atom interact with the electrons and nucleus of neighbouring atoms. Due to the Pauli Exclusion Principle, which states that no two electrons in the same system can have the same energy state, each discrete energy level of the isolated atoms splits into a multitude of closely spaced energy levels. This collection of millions of closely packed energy levels is what we call an energy band.
3. What is the difference between a valence band and a conduction band?
The primary difference lies in their energy levels and electron occupancy:
- The valence band is the range of energy levels that are completely or partially filled with valence electrons at absolute zero temperature. Electrons in a completely filled valence band cannot move to conduct electricity.
- The conduction band is the next permitted energy band, located just above the valence band. It is typically empty at absolute zero. Electrons must gain enough energy to jump from the valence band to the conduction band to become free and participate in electrical conduction.
4. How does band theory differentiate between conductors, semiconductors, and insulators?
The differentiation is based on the size of the forbidden energy gap (Eg) between the valence and conduction bands:
- Conductors (Metals): The valence band and conduction band overlap, meaning there is no forbidden energy gap (Eg ≈ 0 eV). Electrons can easily move to the conduction band, allowing for high electrical conductivity.
- Semiconductors: There is a small, finite forbidden energy gap (Eg ≈ 1 eV). At room temperature, some electrons gain enough thermal energy to jump this gap, enabling moderate conductivity. Examples include Silicon (Si) and Germanium (Ge).
- Insulators: The forbidden energy gap is very large (Eg > 3 eV). It is extremely difficult for electrons to jump from the valence band to the conduction band, resulting in very low electrical conductivity. An example is Diamond (Eg ≈ 6 eV).
5. What is the significance of the 'forbidden energy gap' in determining a material's electrical properties?
The forbidden energy gap is the most crucial factor in band theory for determining electrical properties. It acts as an energy barrier that valence electrons must overcome to become free for conduction. A small or non-existent gap allows for easy electron movement, leading to high conductivity (conductors). A very large gap effectively traps electrons in the valence band, preventing conduction (insulators). An intermediate gap allows for controlled conduction, which is the foundational property of semiconductors used in all electronic devices.
6. Why do semiconductors behave like insulators at absolute zero temperature (0 K)?
At absolute zero (0 K), semiconductors lack thermal energy. Consequently, the valence band is completely filled with electrons, and the conduction band is completely empty. Since no electrons have enough energy to cross the forbidden energy gap and reach the conduction band, there are no free charge carriers available for electrical conduction. This makes the semiconductor behave exactly like an insulator under these conditions.
7. Can an electron exist in the forbidden energy gap? Explain why.
No, an electron cannot exist in the forbidden energy gap. This 'gap' represents a range of energy values that are not permitted for an electron to possess within the crystal lattice structure. According to quantum mechanics, the allowed energy levels for electrons in a crystal split into bands. The regions of energy between these bands are 'forbidden'. For an electron to move from the valence band to the conduction band, it must gain enough energy to 'jump' across this gap instantly; it cannot rest or exist within it.
8. How does the band structure of a metal (conductor) fundamentally differ from that of an insulator?
The fundamental difference lies in their band overlap and electron availability. In a metal (conductor), the valence band and conduction band are not separated but instead overlap. This means there is a continuous availability of empty energy states for electrons to move into with even a tiny amount of applied energy. In an insulator, the valence and conduction bands are separated by a very large forbidden energy gap (typically > 3 eV). This large gap makes it practically impossible for valence electrons to acquire enough energy to jump to the conduction band and conduct electricity.
