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

\[ n = \frac{N_d - N_a}{2} + \sqrt{\left(\frac{N_d - N_a}{2}\right)^2 + n_i^2} \]

Where:
- \( N_d \) = donor concentration
- \( N_a \) = acceptor concentration
- \( n_i \) = intrinsic carrier concentration

Given:
- \( N_d = 1 \times 10^{16} \, \text{cm}^{-3} \)
- \( N_a = 1.1 \times 10^{17} \, \text{cm}^{-3} \)
- \( n_i = 1 \times 10^{12} \, \text{cm}^{-3} \)

### Step 1: Calculate \( N_d - N_a \)

\[ N_d - N_a = 1 \times 10^{16} - 1.1 \times 10^{17} = 1 \times 10^{16} - 1.1 \times 10^{17} = -1.0 \times 10^{17} + 1.0 \times 10^{16} = -1.0 \times 10^{17} + 0.1 \times 10^{17} = -0.1 \times 10^{17} = -1.0 \times 10^{16} \]

### Step 2: Calculate \( \frac{N_d - N_a}{2} \)

\[ \frac{N_d - N_a}{2} = \frac{-1.0 \times 10^{16}}{2} = -0.5 \times 10^{16} = -5.0 \times 10^{15} \]

### Step 3: Calculate \( \left(\frac{N_d - N_a}{2}\right)^2 \)

\[ \left(-5.0 \times 10^{15}\right)^2 = 25.0 \times 10^{30} = 2.5 \times 10^{31} \]

### Step 4: Calculate \( n_i^2 \)

\[ n_i^2 = (1 \times 10^{12})^2 = 1.0 \times 10^{24} \]

### Step 5: Calculate the term under the square root

\[ \left(\frac{N_d - N_a}{2}\right)^2 + n_i^2 = 2.5 \times 10^{31} + 1.0 \times 10^{24} \approx 2.5 \times 10^{31} \quad \text{(since } 2.5 \times 10^{31} \text{ is much larger)} \]

### Step 6: Calculate the square root

\[ \sqrt{2.5 \times 10^{31}} = 5.0 \times 10^{15} \sqrt{2.5} \approx 5.0 \times 10^{15} \times 1.58 \approx 7.9 \times 10^{15} \]

### Step 7: Calculate the electron density \( n \)

\[ n = -5.0 \times 10^{15} + 7.9 \times 10^{15} = 2.9 \times 10^{15} \]

### Conclusion

The electron density in the semiconductor is approximately:

\[ n \approx 2.9 \times 10^{15} \, \text{cm}^{-3} \]