# Band Theory of Solids

In this lecture, we are going to learn about the band theory of solids, how the energy bands form in the solids, and the types of energy bands, and in the end we will discuss the classification of solids based on the band theory. So let’s start with the formation of Energy bands in the solids.

## What is the Band Theory of Solids?

A key idea in condensed matter physics that aids in explaining the behavior of electrons in solids is the band theory of solids. It offers a framework for comprehending a material’s electrical conductivity, optical qualities, and other features.

The band theory states that a solid’s electrons are arranged into bands based on their energy levels. An electron’s possible energy levels are grouped into bands. As atoms unite to form a solid, these bands are created by the overlapping or merging of various atomic energy levels.

The doping phenomenon, in which impurities are purposefully supplied to a material in order to change its electrical properties, is likewise explained by the band theory. It is possible to alter a material’s band structure and conductivity by adding impurities with various electronic configurations.

In conclusion, the band theory of solids offers a useful framework for comprehending how electrons behave within materials, explaining their electrical and optical characteristics, and serving as the foundation for numerous technological advancements in electronics and solid-state physics.

## Formation of Energy Bands in Solids

In the case of a single isolated atom, the electron in any orbit, as shown in the figure below, has a definite energy. As a result, they occupy discrete energy levels.

The Pauli exclusion principle allows each energy level to contain only two electrons. For example, the 2s level of a single atom contains one energy level with two electrons, and the 2p level contains three energy levels with two electrons in each level thus, with a total of six electrons as shown in the figure below.

## Band Structure in Solids

Let us consider the formation of bands in solid sodium.

The single energy level of isolated sodium (Z=11) based on the electron configuration 1s2 2s2 2p6 3s1 is shown in Figure (a) below.

When another sodium atom is brought close to it, the electrons will be subjected to the effect of an additional field. As a result, each energy level is split into two, as shown in the above figure (b). Similarly, when three atoms come close together, the original level splits into three levels, and so on.

More generally, when a solid is formed by bringing N atoms together, the Pauli exclusion principle still demands that only two electrons in the entire solid should have the same energy. Hence, in a solid the different split energy levels of electrons come together to form continuous bands of energy as shown in the above figure(c)

Consequently, the 2s band in solid sodium contains N discrete energy levels and 2N electrons, two in each energy level. Similarly, each of the 2p levels contains N energy levels and 2N electrons. hence, a broad 2p band will contain 3N energy levels and 6N electrons since the three 2p bands overlap.

Hence, in general, each energy band has a total N individual levels and can hold a maximum of 2(2l+1)N electrons. [Each energy level can hold 2(2l+1) electrons. The number 2 corresponds to the electron spin and (2l+1) corresponds to the orientation of the electron orbital angular momentum.]

The result is that electrons in any orbit of the atom within a solid can have a range of energies rather than a single value. Thus, the range of energies possessed by an election in a soli is known as the energy band, i.e. Each energy level of an isolated atom becomes a band in a solid, as shown in the figure above.

Note: In general, it is the outermost energy levels that are mostly affected, whereas innermost levels barely suffer any splitting during the formation of a band. In the case of sodium, it is the outermost energy levels, i.e., 3s and 3p energy levels that are mostly affected, whereas the 1s, 2s, and 2p levels barely suffer any splitting during the formation of a band.

Also Read: Bonding In Solids

## Energy Bands in Solids

As explained above in the case of solid sodium, the discrete energy levels of an atom become bands during the formation of a solid due to the influence of the constituent atoms. Each band consists of a large number of energy levels which corresponds to a range of energy values. The energies within the bands depend on the spacing between the atoms.

The highest occupied band is called the valance band below which all the lower bands are occupied fully. The valance band may even be partially filled. In the case of sodium, the 3s energy levels are the valance band which is partially filled.

The empty band which is immediately above the valance band is called the conduction band, In the case of sodium atoms, the empty 3p energy levels which are separated from the 2s band by the energy gap is the conduction band.

The gap between the valance band and the conduction band is called the forbidden band or energy gap.

So as discussed above the most important energy bands in the solids are:

We will discuss each band in detail as follows.

### 1. Valance Band

• The band of energies occupied by the valance electrons is called as valance band.
• The electrons in the outermost orbit of an atom are known as valance electrons. In a normal atom, the valance band possesses electrons of higher energy. This band may be completely or partially filled. Electrons can be moved from the valance band to the conduction band by the application of external energy.

### 2. Forbidden Band or Energy Gap

• The gap between the valance band and the conduction band on the energy level diagram is known as the forbidden band or energy gap.
• Electrons are never found in this gap. Electrons may jump back and forth from the bottom valance band to the top conduction band, but they never come to rest in the forbidden gap.

### 3. Conduction Band

• The band of energies occupied by conduction electrons is known as the conduction band.
• This is the uppermost band, and all electrons in the conduction band are free electrons. The conduction band is empty for insulators and partially filled for conductors.

Also Read: Hall Effect in Semiconductor

## Classification of Solids on the Basis of Band Theory

Depending upon the ability to conduct electricity, materials are classified into three types.

1. Conductors
2. Semiconductors
3. Insulators

The extent of the forbidden gap determines whether the substance is a conductor, semiconductor, or insulator.

The distinction between them on the basis of the forbidden band or energy gap is discussed in the following section.

### 1. Conductors

• Conductors are characterized by high electrical conductivity. These are the solids in which plenty of free electrons are available for electrical conduction. Examples: silver, copper, iron, aluminum, etc. In general, the resistivity of conductors lies in the range of 10-9 \Omegam at room temperature.
• In a conductor the conduction band and the valence band overlap each other, as shown in Figure (a) below so that the electrons can readily pass into the conduction band, i.e., the electrons can readily move under the influence of the applied field.
• For conductors, the energy gap is of the order of 0.01 eV.

### 2. Semiconductors

• Semiconductors are solids whose electrical properties lie in between those of conductors and insulators.
• Examples of Semiconductors are Germanium and Silicon.
• The resistivity of semiconductors lies between 10-4 \Omegam to 103 \Omegam at room temperature.
• In terms of energy bands semiconductors can be defined as those materials which have almost an empty conduction band and an almost filled valence band, with a very narrow energy gap separating the two bands \approx 1 eV.
• At low temperatures, the valence band of a semiconductor is completely filled and the conduction band is completely empty, as shown in Figure (b). Therefore a semiconductor virtually behaves like an insulator at low temperatures. However, at room temperature some electrons cross over to the conduction band, giving a little conductivity to the semiconductor.
• As the temperature is increased more valence electrons cross over to the conduction band and conductivity increases. This shows that the electrical conductivity of a semiconductor increases with the increase in temperature, i.e., a semiconductor has a negative temperature coefficient of resistance.
• The forbidden energy gap for germanium is 0.7 eV and for silicon, it is 1.1 eV.

### 3. Insulators

• In an insulator, the energy gap between the valence band and conduction band is very large and approximately equal to 5 eV or more, as shown in Figure (c). Therefore, very high energy is required to push the electrons to the conduction band. For these reasons, the electrical conductivity of the insulator is extremely small and may be regarded as nil under ordinary conditions (room temperature).
• The resistivity of insulators lies between 103 to 1017 \approxm at room temperature.
• At room temperature, the valence electrons of the insulators do not have enough energy to cross over to the conduction band. However, when the temperature is raised, some of the valence electrons gain enough energy to cross the conduction band. Hence, the resistance of the insulator decreases with the increase in temperature, i.e., an insulator has a negative temperature coefficient of resistance.
• For an insulator, such as a diamond, the forbidden gap is \approx 6 eV, and for glass it is \approx 10eV.

Also Read: PN Junction Diode

## Frequently Asked Questions (FAQs)

### What is the band theory of solids?

Band theory of solids describes the quantum state that an electron takes within a metal solid.

### What is energy band theory in physics class 12?

Energy band theory explains the interaction of electrons between the outermost shell and the innermost shell.

### What is the band theory of solids explain satisfactorily?

the band theory of solids explains satisfactorily the nature of the semiconductors, insulators, and conductors.

Hello friends, my name is Trupal Bhavsar, I am the Writer and Founder of this blog. I am Electronics Engineer(2014 pass out), Currently working as Junior Telecom Officer(B.S.N.L.) also I do Project Development, PCB designing and Teaching of Electronics Subjects.

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