Question

An intrinsic semiconductor of band gap 1.25 eV has an electron concentration \( 10^{10} \, \text{cm}^{-3} \) at 300 K. Assume that its band gap is independent of temperature and that the electron concentration depends only exponentially on the temperature. If the electron concentration at 200 K is \( Y \times 10^N \, \text{cm}^{-3} \) (where \( 1<Y<10 \), \( N \) = integer), then the value of \( N \) is:

Hint:

For semiconductors, the electron concentration depends exponentially on the inverse of temperature. This relationship is key in understanding the temperature dependence of electronic properties.

Answer 1

Step 1: Understanding the relationship between electron concentration and temperature.
The electron concentration in a semiconductor follows an exponential dependence on temperature, described by the equation: \[ n(T) = n_0 \exp\left( -\frac{E_g}{k_B T} \right) \] where \( n_0 \) is the electron concentration at a reference temperature, \( E_g \) is the band gap, \( k_B \) is Boltzmann's constant, and \( T \) is the temperature. The electron concentration at 300 K is given as \( 10^{10} \, \text{cm}^{-3} \). This means we can calculate the value of \( n(T) \) at 200 K using the exponential model for temperature dependence.
Step 2: Deriving the exponential model.
The electron concentration at 200 K, \( n(200) \), is related to the electron concentration at 300 K, \( n(300) \), as follows: \[ n(200) = n_0 \exp\left( -\frac{E_g}{k_B \cdot 200} \right) \] \[ n(300) = n_0 \exp\left( -\frac{E_g}{k_B \cdot 300} \right) \] Taking the ratio of these two equations gives: \[ \frac{n(200)}{n(300)} = \exp\left( \frac{E_g}{k_B} \left( \frac{1}{300} - \frac{1}{200} \right) \right) \] Substitute the given band gap value of \( E_g = 1.25 \, \text{eV} \) and simplify the equation to find the ratio.
Step 3: Conclusion.
By solving the equation, we find that the value of \( N \) is 3, so the correct answer is \( N = 3 \).