A particular semiconductor in equilibrium has `1xx10^(16)cm^(-3)` donor atoms, `1.1xx10^(17)cm^(-3)` acceptor atoms. If the intrinsic carrier density `(n_(i))` of the semiconductor is `10^(12)cm^(-3)`, then the electron density in it will be

Options

A. `10^(16) cm^(-3)`
B. `10^(12) cm^(-3)`
C. `1.1xx10^(17) cm^(-3)`
D. `10^(7) cm^(-3)`

Answer

To find the electron density in the given semiconductor, we can use the formula for the electron density \( n_0 \) in a semiconductor with donor and acceptor atoms:

\[ n_0 = \frac{N_D - N_A}{2} + \sqrt{\left(\frac{N_D - N_A}{2}\right)^2 + n_i^2} \]

Where:
- \( N_D \) = concentration of donor atoms
- \( N_A \) = concentration of acceptor atoms
- \( n_i \) = intrinsic carrier density

### Step-by-Step Solution:

1. **Identify the given values:**
- \( N_D = 1 \times 10^{16} \, \text{cm}^{-3} \)
- \( N_A = 1.1 \times 10^{17} \, \text{cm}^{-3} \)
- \( n_i = 1 \times 10^{12} \, \text{cm}^{-3} \)

2. **Calculate \( N_D - N_A \):**
\[ N_D - N_A = 1 \times 10^{16} - 1.1 \times 10^{17} = -1.0 \times 10^{17} + 1.0 \times 10^{16} = -1.0 \times 10^{17} + 0.1 \times 10^{17} = -1.0 \times 10^{17} \]

3. **Calculate \( \frac{N_D - N_A}{2} \):**
\[ \frac{N_D - N_A}{2} = \frac{-1.0 \times 10^{17}}{2} = -0.5 \times 10^{17} = -5.0 \times 10^{16} \]

4. **Calculate \( \left(\frac{N_D - N_A}{2}\right)^2 \):**
\[ \left(-5.0 \times 10^{16}\right)^2 = 25.0 \times 10^{32} = 2.5 \times 10^{33} \]

5. **Calculate \( n_i^2 \):**
\[ n_i^2 = (1 \times 10^{12})^2 = 1 \times 10^{24} \]

6. **Calculate the expression under the square root:**
\[ \left(\frac{N_D - N_A}{2}\right)^2 + n_i^2 = 2.5 \times 10^{33} + 1 \times 10^{24} \approx 2.5 \times 10^{33} \quad (\text{since } 2.5 \times 10^{33} \text{ is much larger than } 1 \times 10^{24}) \]

7. **Take the square root:**
\[ \sqrt{2.5 \times 10^{33}} = 5.0 \times 10^{16} \sqrt{2.5} \approx 7.5 \times 10^{16} \]

8. **Combine the results to find \( n_0 \):**
\[ n_0 = -5.0 \times 10^{16} + 7.5 \times 10^{16} = 2.5 \times 10^{16} \]

### Final Result:
The electron density \( n_0 \) in the semiconductor is approximately \( 2.5 \times 10^{16} \, \text{cm}^{-3} \).