Question

In a semiconductor, the intrinsic carrier concentration is \( 1.5 \times 10^{10} \, \text{cm}^{-3} \) at room temperature. If the energy band gap of the semiconductor is \( 1.1 \, \text{eV} \), calculate the intrinsic carrier concentration at a temperature of \( 500 \, \text{K} \). The intrinsic carrier concentration at room temperature (\( 300 \, \text{K} \)) is known to vary with temperature according to the relation: \[ n_i(T) = n_{i0} \left( \frac{T}{T_0} \right)^{3/2} \exp \left( -\frac{E_g}{2k} \left( \frac{1}{T} - \frac{1}{T_0} \right) \right) \] Where: - \( n_{i0} = 1.5 \times 10^{10} \, \text{cm}^{-3} \), - \( T_0 = 300 \, \text{K} \), - \( E_g = 1.1 \, \text{eV} \), - \( k = 8.617 \times 10^{-5} \, \text{eV/K} \), - \( T = 500 \, \text{K} \).

• A. \( 3.0 \times 10^{12} \, \text{cm}^{-3} \)
• B. \( 6.2 \times 10^{12} \, \text{cm}^{-3} \)
• C. \( 8.5 \times 10^{13} \, \text{cm}^{-3} \)
• D. \( 1.2 \times 10^{14} \, \text{cm}^{-3} \)

Hint:

The intrinsic carrier concentration in a semiconductor depends on both temperature and the energy band gap. The relation \( n_i(T) = n_{i0} \left( \frac{T}{T_0} \right)^{3/2} \exp \left( -\frac{E_g}{2k} \left( \frac{1}{T} - \frac{1}{T_0} \right) \right) \) is useful for quick calculations when temperature changes.

Answer 1

The Correct Option is B

The intrinsic carrier concentration at a different temperature can be calculated using the given formula: \[ n_i(T) = n_{i0} \left( \frac{T}{T_0} \right)^{3/2} \exp \left( -\frac{E_g}{2k} \left( \frac{1}{T} - \frac{1}{T_0} \right) \right) \] Substitute the given values into the equation: \[ n_i(500) = 1.5 \times 10^{10} \left( \frac{500}{300} \right)^{3/2} \exp \left( -\frac{1.1}{2 \times 8.617 \times 10^{-5}} \left( \frac{1}{500} - \frac{1}{300} \right) \right) \] First, calculate the factor \( \left( \frac{500}{300} \right)^{3/2} \): \[ \left( \frac{500}{300} \right)^{3/2} = \left( \frac{5}{3} \right)^{3/2} \approx 1.936 \] Now calculate the exponential term: \[ \frac{1.1}{2 \times 8.617 \times 10^{-5}} \approx 6372.17 \] \[ \frac{1}{500} - \frac{1}{300} = \frac{3 - 5}{1500} = \frac{-2}{1500} = -0.00133 \] \[ \exp \left( -6372.17 \times (-0.00133) \right) = \exp(8.47) \approx 4759.7 \] Now calculate the overall value of \( n_i(500) \): \[ n_i(500) = 1.5 \times 10^{10} \times 1.936 \times 4759.7 \approx 6.2 \times 10^{12} \, \text{cm}^{-3} \] Thus, the intrinsic carrier concentration at \( 500 \, \text{K} \) is \( 6.2 \times 10^{12} \, \text{cm}^{-3} \).