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 \( Y \times 10^N \, \text{cm}^{-3} \) (1<Y<10; N = integer), then the value of N is:

Hint:

When solving temperature-dependent exponential problems, always use the Boltzmann constant \( k \) and the temperature in Kelvin.

Answer 1

Correct Answer: 4

Step 1: Understand the relationship between electron concentration and temperature.
The electron concentration \( n(T) \) is given by an exponential dependence on temperature: \[ n(T) = n_0 e^{-\frac{E_g}{kT}} \] where \( n_0 \) is a constant, \( E_g \) is the band gap, \( k \) is the Boltzmann constant, and \( T \) is the temperature. The concentration at two different temperatures can be written as: \[ \frac{n(T_2)}{n(T_1)} = e^{\left(\frac{E_g}{k} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\right)} \] Given that the concentration at 300 K is \( 10^{10} \, \text{cm}^{-3} \) and at 200 K it is \( Y \times 10^N \), solve for N. Step 2: Calculation of N.
Using the above equation and the given data, the value of \( N \) can be calculated to be 4.