The free electron concentration (n) in the conduction band of a semiconductor at a temperature T kelvin is described in terms of `E_(g)` and T as-

Options

A. `n=Ate^(-Eg//kT)`
B. `n=AT^(2)e^(-Eg//kT)`
C. `n=AT^(2)e^(-Eg//kT)`
D. `n=AT^(2)e^(-Eg//2kT)`

Answer

To solve the problem of finding the free electron concentration (n) in the conduction band of a semiconductor at a temperature T in terms of the energy gap (E_g) and T, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Variables**:
- Let \( n \) be the free electron concentration in the conduction band.
- Let \( T \) be the temperature in Kelvin.
- Let \( E_g \) be the energy gap of the semiconductor.

2. **Identify the Formula**:
- The free electron concentration in the conduction band of a semiconductor can be expressed using the formula:
\[ n = A T^2 e^{-\frac{E_g}{2kT}} \]
where:
- \( A \) is a constant that depends on the material.
- \( k \) is the Boltzmann constant.

3. **Analyze the Formula**:
- The term \( T^2 \) indicates that the concentration increases with the square of the temperature.
- The exponential term \( e^{-\frac{E_g}{2kT}} \) shows that as the energy gap \( E_g \) increases or as the temperature \( T \) decreases, the concentration \( n \) decreases.

4. **Conclusion**:
- Based on the formula derived, we can conclude that the relationship between the free electron concentration \( n \), the energy gap \( E_g \), and the temperature \( T \) is given by:
\[ n \propto T^2 e^{-\frac{E_g}{2kT}} \]
- Therefore, the correct option that describes this relationship is option D.

### Final Answer:
The free electron concentration \( n \) in the conduction band of a semiconductor at temperature \( T \) is given by:
\[ n = A T^2 e^{-\frac{E_g}{2kT}} \]
Thus, the correct option is **D**.