The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature

Options

A. increases exponentially with increasing band gap
B. decreases exponentially with increasing band gap
C. decreases with increasing temperature.
D. is independent of the temperature and band gap.

Answer

To determine the probability of electrons being found in the conduction band of an intrinsic semiconductor at a finite temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy Bands**:
- In a semiconductor, there are two main energy bands: the valence band and the conduction band. The energy gap between these two bands is known as the bandgap (Eg).

2. **Effect of Bandgap on Electron Transition**:
- Electrons need to gain energy equal to or greater than the bandgap (Eg) to transition from the valence band to the conduction band. If the bandgap increases, more energy is required for this transition.

3. **Probability of Electron Occupation**:
- The probability of finding electrons in the conduction band can be described using the Fermi-Dirac distribution. At finite temperatures, some electrons in the valence band can gain enough thermal energy to jump into the conduction band.

4. **Impact of Increasing Bandgap**:
- If the bandgap increases, fewer electrons will have enough energy to jump to the conduction band, thus decreasing the probability of finding electrons in the conduction band. Therefore, the probability decreases exponentially with an increase in the bandgap.

5. **Impact of Temperature**:
- Conversely, as the temperature increases, more electrons gain sufficient thermal energy to overcome the bandgap, leading to an increase in the probability of electrons being found in the conduction band. Thus, the probability increases with increasing temperature.

6. **Conclusion**:
- The correct interpretation is that the probability of finding electrons in the conduction band decreases with an increase in bandgap and increases with an increase in temperature.

### Final Answer:
- The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature is dependent on both the bandgap and the temperature. It decreases with an increase in bandgap and increases with an increase in temperature.