The conductivity of a semiconductor increases with increase in temperature because

Options

A. number density of free current increase
B. relaxation time inceases
C. both number density of carries and relaxation time incease
D. number density of carries increases relaxation time decreases but effect of decrease in relaxation time is much less than increase in number density

Explanation

To understand why the conductivity of a semiconductor increases with an increase in temperature, we can break down the explanation into several steps:

### Step-by-Step Solution:

1. **Understanding Conductivity**:
Conductivity (\( \sigma \)) of a material is defined as the ability of that material to conduct electric current. It depends on the number of charge carriers (electrons and holes) and their mobility.

2. **Effect of Temperature on Charge Carriers**:
In semiconductors, as the temperature increases, thermal energy is supplied to the electrons. This energy allows some electrons to break free from their covalent bonds, creating free electrons and holes. This process is known as thermal excitation.

3. **Increase in Number Density of Charge Carriers**:
The increase in temperature leads to an increase in the number of free charge carriers (both electrons and holes) in the semiconductor. This increase in the number density of charge carriers directly contributes to an increase in conductivity.

4. **Relaxation Time**:
The relaxation time (\( \tau \)) is the average time between collisions of charge carriers. As temperature increases, the lattice vibrations also increase, which can lead to more frequent collisions. This typically results in a decrease in relaxation time.

5. **Net Effect on Conductivity**:
Although the relaxation time decreases with increasing temperature, the increase in the number density of charge carriers has a much more significant effect on conductivity. The overall increase in conductivity is primarily due to the increase in the number of charge carriers, even though the mobility (related to relaxation time) may decrease slightly.

6. **Conclusion**:
Therefore, the conductivity of a semiconductor increases with an increase in temperature primarily because the number density of charge carriers increases significantly, outweighing the effects of decreased relaxation time.

### Summary:
The conductivity of a semiconductor increases with temperature due to the increase in the number density of charge carriers (electrons and holes) resulting from thermal excitation, which is greater than the effect of decreased relaxation time.