The conductivity of a semiconductor increases with increase in temperature because

Options

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

Answer

**Step-by-Step Solution:**

1. **Understanding Conductivity in Semiconductors:**
- Conductivity (\( \sigma \)) in semiconductors is primarily due to the movement of charge carriers, which are electrons and holes. The conductivity can be expressed as:
\[ \sigma = n e \tau \]
where \( n \) is the charge carrier density, \( e \) is the charge of an electron, and \( \tau \) is the mean relaxation time (the average time between collisions of charge carriers).

2. **Effect of Temperature on Charge Carriers:**
- When the temperature of a semiconductor increases, thermal energy is supplied to the system. This energy can excite electrons from the valence band to the conduction band, thus increasing the number of free charge carriers (electrons).

3. **Energy Band Gap:**
- In semiconductors, there exists an energy gap (band gap) between the valence band and the conduction band. As temperature increases, more electrons gain enough energy to cross this gap and enter the conduction band.

4. **Increase in Electron Density:**
- The increase in temperature leads to an increase in the number of electrons that can reach the conduction band. This results in a higher electron density (\( n \)), which contributes positively to the conductivity.

5. **Mean Relaxation Time:**
- While the number of charge carriers increases, the mean relaxation time (\( \tau \)) tends to decrease with increasing temperature. This is because the increased thermal energy causes more frequent collisions among charge carriers.

6. **Net Effect on Conductivity:**
- Although the mean relaxation time decreases, the increase in the number density of charge carriers has a more significant effect on the overall conductivity. Therefore, the increase in conductivity with temperature is primarily due to the increase in the number of charge carriers, despite the decrease in relaxation time.

7. **Conclusion:**
- Thus, the conductivity of a semiconductor increases with an increase in temperature because the number of electrons reaching the conduction band increases significantly, outweighing the effect of the decrease in mean relaxation time.

**Final Statement:**
The conductivity of a semiconductor increases with temperature due to the increase in the number of electrons in the conduction band, despite a decrease in mean relaxation time.

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