Concentration of Holes in an n-type Semiconductor

In an n-type semiconductor, the majority charge carriers are electrons, which come from the donor atoms that have been intentionally introduced into the material. However, there are also a small number of holes present due to the thermal generation of electron-hole pairs. The concentration of these holes can be determined using the given relationship.

Relationship:
The relationship to determine the concentration of holes in an n-type semiconductor is:

\(\text{Concentration of Holes} = \frac{{n_i^2}}{{N_D}}\)

where:
\(n_i\) = concentration of electrons in an intrinsic semiconductor
\(N_D\) = concentration of donor atoms

Explanation:
The concentration of holes in an n-type semiconductor is given by the ratio of the square of the concentration of electrons in an intrinsic semiconductor to the concentration of donor atoms.

- Concentration of Electrons in an Intrinsic Semiconductor (ni):
- In an intrinsic semiconductor, the concentration of electrons (ni) is determined by the material's intrinsic properties and the temperature.
- At absolute zero temperature, all electrons are bound to their respective atoms, and there are no free electrons or holes.
- As the temperature increases, some electrons gain enough energy to break free from their parent atoms, leaving behind a hole.
- The concentration of electrons in an intrinsic semiconductor increases exponentially with temperature.
- This concentration is represented by the term \(n_i\), which is a characteristic property of the semiconductor material.

- Concentration of Donor Atoms (ND):
- In an n-type semiconductor, donor atoms are intentionally introduced into the material during the doping process.
- These donor atoms have an excess electron in their valence shell and can easily donate an electron to the conduction band, creating free electrons.
- The concentration of donor atoms is represented by the term \(N_D\).
- The higher the concentration of donor atoms, the greater the number of free electrons available in the material.

- Concentration of Holes:
- The concentration of holes in an n-type semiconductor is inversely proportional to the concentration of donor atoms.
- This is because as the concentration of donor atoms (ND) increases, more free electrons are available, and the number of holes decreases.
- The square of the concentration of electrons in an intrinsic semiconductor (ni) represents the total number of electron-hole pairs generated by thermal excitation.
- Therefore, dividing this value by the concentration of donor atoms (ND) gives the concentration of holes in the material.

Hence, the correct answer is option 'C': \(\frac{{n_i^2}}{{N_D}}\), which represents the concentration of holes in an n-type semiconductor.