Explain the Different Energy Bands of Solids?

To study the energy bands of the solids first we have to the study the Bohr’s atomic theory. According to Bohr’s theory the energy in different atoms vary with the change in shells and sub shells. In simple words we can say that different energy shells have different amount of energy. There are well maintained energy levels of electrons in each atom. If we bring two atoms near to each other then they both will affect each other. The interatomic interactions of atoms are very strong. So, it will not affect the electrons present in the inner shells but will affect the electrons in the outer shells.

To completely study this fact lets study a silicon crystal. Suppose there are N atoms in this crystal. Let the interatomic distance between the atoms is r. The diagram shown below is simply a graph between the interatomic distance and energy planted along x-axis and y-axis respectively.

The different situations which we commonly face during this process are discussed below:

1. The interatomic interaction of the atoms decrease by increasing the interatomic distance between the atoms. When the atoms are packed in the crystal each atom will act as a free atom. If the atoms act as free atoms then it is obvious that the atoms will have their individual energy levels. In case of silicon atom the electronic configuration n of atoms is: 1s22s22p63s23p2. The outer most sub shell i.e. 3s and 3p has two electrons in each sub shell. As we know that 6 electrons are needed to completely fill the p sub shell. After the complete study the result shows that there are 2N electrons which are filling the 2N possible 3s levels completely. All of these are having similar energy. Similarly, in case of 3p level there are 6N possible levels. Out of these 6N levels, 2N are filled. The filled 2N levels also have similar energy. This case study matches with the Pauli’s Exclusion principle.

2. When d is less than that of the interatomic distance which is represented by r. Then the visible splitting of the energy levels will stop.

3. When c will be equal to the interatomic distance r, then electrons present in the outermost shells i.e. 3s2 and 3p2 will start showing interaction. Then the change in energy of the electrons corresponding to 3s and 3p levels will take place. But on the other hand the energy of the inner levels will not take place.

4. In this case, we have assumed that if r lies between b and c then as a result there will be a little bit change in the energy of the electrons corresponding to the 3s and 3p. The energy of the levels is measured in volts. These levels having very less spacing between them. This is due to the spreading of the energy of 3s and 3p levels.

5. The gap of energy present between the 3s and 3p levels disappears when we make r equal to b. But the value of b must be greater than the value of a. In mathematical form it can be described as follows:

6. In this case we have assumed that if the value of r becomes equal to a then the unfilled energy band i.e. 4N will set apart from the filled 4N energy levels by some particular energy gap. This gap is known as forbidden gap. The symbol to represent the energy gap is Eg. The unfilled band is known as conduction band. The filled band is known as valence band.

Different quantum states of silicon crystal having N number of atoms are shown below:

Energy Level Total States available Total States Occupied
1s 2N 2N
2s 2N 2N
2p 6N 6N
3s 2N 2N
3p 6N 2N

Category: Conductors Insulators and Semi Conductors

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